diff --git "a/Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_BigOperators_Fin.jsonl" "b/Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_BigOperators_Fin.jsonl"
new file mode 100644--- /dev/null
+++ "b/Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_BigOperators_Fin.jsonl"
@@ -0,0 +1,218 @@
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nn : ℕ\nf : Fin n → β\n⊢ ∏ i : Fin n, f i = List.prod (List.map f (List.finRange n))","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by ","nextTactic":"simp [univ_def]","declUpToTactic":"@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.46_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nn : ℕ\nf : Fin n → β\n⊢ List.prod (List.ofFn f) = ∏ i : Fin n, f i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  ","nextTactic":"rw [List.ofFn_eq_map]","declUpToTactic":"@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.52_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nn : ℕ\nf : Fin n → β\n⊢ List.prod (List.map f (List.finRange n)) = ∏ i : Fin n, f i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  ","nextTactic":"rw [prod_univ_def]","declUpToTactic":"@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.52_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nn : ℕ\nf : Fin (n + 1) → β\nx : Fin (n + 1)\n⊢ ∏ i : Fin (n + 1), f i = f x * ∏ i : Fin n, f (succAbove x i)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  ","nextTactic":"rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]","declUpToTactic":"/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.66_0.QyPgLR0eRR4fDXZ","decl":"/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nn : ℕ\nf : Fin (n + 1) → β\nx : Fin (n + 1)\n⊢ f x * ∏ x_1 : Fin n, f ((succAboveEmb x) x_1) = f x * ∏ i : Fin n, f (succAbove x i)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  ","nextTactic":"rfl","declUpToTactic":"/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.66_0.QyPgLR0eRR4fDXZ","decl":"/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nn : ℕ\nf : Fin (n + 1) → β\n⊢ ∏ i : Fin (n + 1), f i = (∏ i : Fin n, f (castSucc i)) * f (last n)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  ","nextTactic":"simpa [mul_comm] using prod_univ_succAbove f (last n)","declUpToTactic":"/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.88_0.QyPgLR0eRR4fDXZ","decl":"/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nn : ℕ\nx : β\nf : Fin n → β\n⊢ ∏ i : Fin (Nat.succ n), cons x f i = x * ∏ i : Fin n, f i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  ","nextTactic":"simp_rw [prod_univ_succ, cons_zero, cons_succ]","declUpToTactic":"@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.98_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 1 → β\n⊢ ∏ i : Fin 1, f i = f 0","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by ","nextTactic":"simp","declUpToTactic":"@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.105_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 2 → β\n⊢ ∏ i : Fin 2, f i = f 0 * f 1","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  ","nextTactic":"simp [prod_univ_succ]","declUpToTactic":"@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.110_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive (attr "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 3 → β\n⊢ ∏ i : Fin 3, f i = f 0 * f 1 * f 2","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  ","nextTactic":"rw [prod_univ_castSucc]","declUpToTactic":"@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.116_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 3 → β\n⊢ (∏ i : Fin 2, f (castSucc i)) * f (last 2) = f 0 * f 1 * f 2","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  ","nextTactic":"rw [prod_univ_two]","declUpToTactic":"@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.116_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 3 → β\n⊢ f (castSucc 0) * f (castSucc 1) * f (last 2) = f 0 * f 1 * f 2","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  ","nextTactic":"rfl","declUpToTactic":"@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.116_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 4 → β\n⊢ ∏ i : Fin 4, f i = f 0 * f 1 * f 2 * f 3","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  ","nextTactic":"rw [prod_univ_castSucc]","declUpToTactic":"@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.124_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 4 → β\n⊢ (∏ i : Fin 3, f (castSucc i)) * f (last 3) = f 0 * f 1 * f 2 * f 3","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  ","nextTactic":"rw [prod_univ_three]","declUpToTactic":"@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.124_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 4 → β\n⊢ f (castSucc 0) * f (castSucc 1) * f (castSucc 2) * f (last 3) = f 0 * f 1 * f 2 * f 3","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  ","nextTactic":"rfl","declUpToTactic":"@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.124_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 5 → β\n⊢ ∏ i : Fin 5, f i = f 0 * f 1 * f 2 * f 3 * f 4","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  ","nextTactic":"rw [prod_univ_castSucc]","declUpToTactic":"@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.132_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 5 → β\n⊢ (∏ i : Fin 4, f (castSucc i)) * f (last 4) = f 0 * f 1 * f 2 * f 3 * f 4","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  ","nextTactic":"rw [prod_univ_four]","declUpToTactic":"@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.132_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 5 → β\n⊢ f (castSucc 0) * f (castSucc 1) * f (castSucc 2) * f (castSucc 3) * f (last 4) = f 0 * f 1 * f 2 * f 3 * f 4","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  ","nextTactic":"rfl","declUpToTactic":"@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.132_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 6 → β\n⊢ ∏ i : Fin 6, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  ","nextTactic":"rw [prod_univ_castSucc]","declUpToTactic":"@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.141_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 6 → β\n⊢ (∏ i : Fin 5, f (castSucc i)) * f (last 5) = f 0 * f 1 * f 2 * f 3 * f 4 * f 5","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  ","nextTactic":"rw [prod_univ_five]","declUpToTactic":"@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.141_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 6 → β\n⊢ f (castSucc 0) * f (castSucc 1) * f (castSucc 2) * f (castSucc 3) * f (castSucc 4) * f (last 5) =\n    f 0 * f 1 * f 2 * f 3 * f 4 * f 5","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  ","nextTactic":"rfl","declUpToTactic":"@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.141_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 7 → β\n⊢ ∏ i : Fin 7, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  ","nextTactic":"rw [prod_univ_castSucc]","declUpToTactic":"@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.150_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 7 → β\n⊢ (∏ i : Fin 6, f (castSucc i)) * f (last 6) = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  ","nextTactic":"rw [prod_univ_six]","declUpToTactic":"@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.150_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 7 → β\n⊢ f (castSucc 0) * f (castSucc 1) * f (castSucc 2) * f (castSucc 3) * f (castSucc 4) * f (castSucc 5) * f (last 6) =\n    f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  ","nextTactic":"rfl","declUpToTactic":"@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.150_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 8 → β\n⊢ ∏ i : Fin 8, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  ","nextTactic":"rw [prod_univ_castSucc]","declUpToTactic":"@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.159_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 8 → β\n⊢ (∏ i : Fin 7, f (castSucc i)) * f (last 7) = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  ","nextTactic":"rw [prod_univ_seven]","declUpToTactic":"@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.159_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nf : Fin 8 → β\n⊢ f (castSucc 0) * f (castSucc 1) * f (castSucc 2) * f (castSucc 3) * f (castSucc 4) * f (castSucc 5) * f (castSucc 6) *\n      f (last 7) =\n    f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  ","nextTactic":"rfl","declUpToTactic":"@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.159_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 "}
+{"state":"α : Type u_1\nβ : Type u_2\nn : ℕ\nR : Type u_3\ninst✝ : CommSemiring R\na b : R\n⊢ ∑ s : Finset (Fin n), a ^ card s * b ^ (n - card s) = (a + b) ^ n","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  ","nextTactic":"simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b","declUpToTactic":"theorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.168_0.QyPgLR0eRR4fDXZ","decl":"theorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nx : α\n⊢ ∏ _i : Fin n, x = x ^ n","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by ","nextTactic":"simp","declUpToTactic":"theorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.173_0.QyPgLR0eRR4fDXZ","decl":"theorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : AddCommMonoid α\nn : ℕ\nx : α\n⊢ ∑ _i : Fin n, x = n • x","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by ","nextTactic":"simp","declUpToTactic":"theorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.176_0.QyPgLR0eRR4fDXZ","decl":"theorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\nn : ℕ\nv : Fin (Nat.succ n) → M\n⊢ ∏ i in Ioi 0, v i = ∏ j : Fin n, v (succ j)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  ","nextTactic":"rw [Ioi_zero_eq_map]","declUpToTactic":"@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.179_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\nn : ℕ\nv : Fin (Nat.succ n) → M\n⊢ ∏ i in map (succEmbedding n).toEmbedding univ, v i = ∏ j : Fin n, v (succ j)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  ","nextTactic":"rw [Finset.prod_map]","declUpToTactic":"@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.179_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\nn : ℕ\nv : Fin (Nat.succ n) → M\n⊢ ∏ x : Fin n, v ((succEmbedding n).toEmbedding x) = ∏ j : Fin n, v (succ j)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  ","nextTactic":"rw [RelEmbedding.coe_toEmbedding]","declUpToTactic":"@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.179_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\nn : ℕ\nv : Fin (Nat.succ n) → M\n⊢ ∏ x : Fin n, v ((succEmbedding n) x) = ∏ j : Fin n, v (succ j)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  ","nextTactic":"rw [val_succEmbedding]","declUpToTactic":"@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.179_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\nn : ℕ\ni : Fin n\nv : Fin (Nat.succ n) → M\n⊢ ∏ j in Ioi (succ i), v j = ∏ j in Ioi i, v (succ j)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  ","nextTactic":"rw [Ioi_succ]","declUpToTactic":"@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.189_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\nn : ℕ\ni : Fin n\nv : Fin (Nat.succ n) → M\n⊢ ∏ j in map (succEmbedding n).toEmbedding (Ioi i), v j = ∏ j in Ioi i, v (succ j)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  ","nextTactic":"rw [Finset.prod_map]","declUpToTactic":"@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.189_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\nn : ℕ\ni : Fin n\nv : Fin (Nat.succ n) → M\n⊢ ∏ x in Ioi i, v ((succEmbedding n).toEmbedding x) = ∏ j in Ioi i, v (succ j)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  ","nextTactic":"rw [RelEmbedding.coe_toEmbedding]","declUpToTactic":"@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.189_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\nn : ℕ\ni : Fin n\nv : Fin (Nat.succ n) → M\n⊢ ∏ x in Ioi i, v ((succEmbedding n) x) = ∏ j in Ioi i, v (succ j)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  ","nextTactic":"rw [val_succEmbedding]","declUpToTactic":"@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.189_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\na b : ℕ\nf : Fin b → M\nh : a = b\n⊢ ∏ i : Fin a, f (cast h i) = ∏ i : Fin b, f i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  ","nextTactic":"subst h","declUpToTactic":"@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.199_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\na : ℕ\nf : Fin a → M\n⊢ ∏ i : Fin a, f (cast (_ : a = a) i) = ∏ i : Fin a, f i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  ","nextTactic":"congr","declUpToTactic":"@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.199_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\na b : ℕ\nf : Fin (a + b) → M\n⊢ ∏ i : Fin (a + b), f i = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 �� β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  ","nextTactic":"rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]","declUpToTactic":"@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.207_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\na b : ℕ\nf : Fin (a + b) → M\n⊢ ∏ x : Fin a ⊕ Fin b, f (Equiv.toFun finSumFinEquiv x) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · ","nextTactic":"apply Fintype.prod_sum_type","declUpToTactic":"@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.207_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\na b : ℕ\nf : Fin (a + b) → M\n⊢ ∀ (x : Fin (a + b)), f x = f (Equiv.toFun finSumFinEquiv (finSumFinEquiv.symm x))","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · ","nextTactic":"intro x","declUpToTactic":"@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.207_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\na b : ℕ\nf : Fin (a + b) → M\nx : Fin (a + b)\n⊢ f x = f (Equiv.toFun finSumFinEquiv (finSumFinEquiv.symm x))","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    ","nextTactic":"simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]","declUpToTactic":"@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.207_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\na b : ℕ\nf : Fin (a + b) → M\nhf : ∀ (j : Fin b), f (natAdd a j) = 1\n⊢ ∏ i : Fin (a + b), f i = ∏ i : Fin a, f (castLE (_ : a ≤ a + b) i)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  ","nextTactic":"rw [prod_univ_add]","declUpToTactic":"@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.217_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\na b : ℕ\nf : Fin (a + b) → M\nhf : ∀ (j : Fin b), f (natAdd a j) = 1\n⊢ (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) = ∏ i : Fin a, f (castLE (_ : a ≤ a + b) i)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  ","nextTactic":"rw [Fintype.prod_eq_one _ hf]","declUpToTactic":"@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.217_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\na b : ℕ\nf : Fin (a + b) → M\nhf : ∀ (j : Fin b), f (natAdd a j) = 1\n⊢ (∏ i : Fin a, f (castAdd b i)) * 1 = ∏ i : Fin a, f (castLE (_ : a ≤ a + b) i)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  ","nextTactic":"rw [mul_one]","declUpToTactic":"@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.217_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\na b : ℕ\nf : Fin (a + b) → M\nhf : ∀ (j : Fin b), f (natAdd a j) = 1\n⊢ ∏ i : Fin a, f (castAdd b i) = ∏ i : Fin a, f (castLE (_ : a ≤ a + b) i)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  ","nextTactic":"rfl","declUpToTactic":"@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.217_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : Monoid α\nn : ℕ\nf : Fin n → α\n⊢ partialProd f 0 = 1","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by ","nextTactic":"simp [partialProd]","declUpToTactic":"@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.240_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive (attr "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : Monoid α\nn : ℕ\nf : Fin n → α\nj : Fin n\n⊢ partialProd f (succ j) = partialProd f (castSucc j) * f j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  ","nextTactic":"simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]","declUpToTactic":"@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.245_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : Monoid α\nn : ℕ\nf : Fin (n + 1) → α\nj : Fin (n + 1)\n⊢ partialProd f (succ j) = f 0 * partialProd (tail f) j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  ","nextTactic":"simp [partialProd]","declUpToTactic":"@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.252_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : Monoid α\nn : ℕ\nf : Fin (n + 1) → α\nj : Fin (n + 1)\n⊢ f 0 * List.prod (List.take (↑j) (List.ofFn fun i => f (succ i))) =\n    f 0 * List.prod (List.take (↑j) (List.ofFn (tail f)))","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  ","nextTactic":"rfl","declUpToTactic":"@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.252_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin (n + 1) → G\nx : Fin (n + 1)\n⊢ (f 0 • partialProd fun i => (f ↑↑i)⁻¹ * f (succ i)) 0 = f 0","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by ","nextTactic":"simp","declUpToTactic":"@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.260_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin (n + 1) → G\nx✝ : Fin (n + 1)\nx : Fin n\nhx : (f 0 • partialProd fun i => (f ↑↑i)⁻¹ * f (succ i)) (castSucc x) = f (castSucc x)\n⊢ (f 0 • partialProd fun i => (f ↑↑i)⁻¹ * f (succ i)) (succ x) = f (succ x)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 ��� β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    ","nextTactic":"simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢","declUpToTactic":"@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.260_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin (n + 1) → G\nx✝ : Fin (n + 1)\nx : Fin n\nhx : f 0 * partialProd (fun i => (f (castSucc i))⁻¹ * f (succ i)) (castSucc x) = f (castSucc x)\n⊢ f 0 * partialProd (fun i => (f (castSucc i))⁻¹ * f (succ i)) (succ x) = f (succ x)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    ","nextTactic":"rw [partialProd_succ]","declUpToTactic":"@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.260_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin (n + 1) → G\nx✝ : Fin (n + 1)\nx : Fin n\nhx : f 0 * partialProd (fun i => (f (castSucc i))⁻¹ * f (succ i)) (castSucc x) = f (castSucc x)\n⊢ f 0 * (partialProd (fun i => (f (castSucc i))⁻¹ * f (succ i)) (castSucc x) * ((f (castSucc x))⁻¹ * f (succ x))) =\n    f (succ x)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    ","nextTactic":"rw [← mul_assoc]","declUpToTactic":"@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.260_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin (n + 1) → G\nx✝ : Fin (n + 1)\nx : Fin n\nhx : f 0 * partialProd (fun i => (f (castSucc i))⁻¹ * f (succ i)) (castSucc x) = f (castSucc x)\n⊢ f 0 * partialProd (fun i => (f (castSucc i))⁻¹ * f (succ i)) (castSucc x) * ((f (castSucc x))⁻¹ * f (succ x)) =\n    f (succ x)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    ","nextTactic":"rw [hx]","declUpToTactic":"@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.260_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin (n + 1) → G\nx✝ : Fin (n + 1)\nx : Fin n\nhx : f 0 * partialProd (fun i => (f (castSucc i))⁻¹ * f (succ i)) (castSucc x) = f (castSucc x)\n⊢ f (castSucc x) * ((f (castSucc x))⁻¹ * f (succ x)) = f (succ x)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    ","nextTactic":"rw [mul_inv_cancel_left]","declUpToTactic":"@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.260_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : Fin n\n⊢ (partialProd f (castSucc i))⁻¹ * partialProd f (succ i) = f i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  ","nextTactic":"cases' i with i hn","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"case mk\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn : i < n\n⊢ (partialProd f (castSucc { val := i, isLt := hn }))⁻¹ * partialProd f (succ { val := i, isLt := hn }) =\n    f { val := i, isLt := hn }","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  ","nextTactic":"induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"case mk\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn : i < n\n⊢ (partialProd f (castSucc { val := i, isLt := hn }))⁻¹ * partialProd f (succ { val := i, isLt := hn }) =\n    f { val := i, isLt := hn }","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  ","nextTactic":"induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"case mk.zero\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\nhn : Nat.zero < n\n⊢ (partialProd f (castSucc { val := Nat.zero, isLt := hn }))⁻¹ * partialProd f (succ { val := Nat.zero, isLt := hn }) =\n    f { val := Nat.zero, isLt := hn }","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  ","nextTactic":"| zero => simp [-Fin.succ_mk, partialProd_succ]","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"case mk.zero\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\nhn : Nat.zero < n\n⊢ (partialProd f (castSucc { val := Nat.zero, isLt := hn }))⁻¹ * partialProd f (succ { val := Nat.zero, isLt := hn }) =\n    f { val := Nat.zero, isLt := hn }","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => ","nextTactic":"simp [-Fin.succ_mk, partialProd_succ]","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"case mk.succ\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhi :\n  ∀ (hn : i < n),\n    (partialProd f (castSucc { val := i, isLt := hn }))⁻¹ * partialProd f (succ { val := i, isLt := hn }) =\n      f { val := i, isLt := hn }\nhn : Nat.succ i < n\n⊢ (partialProd f (castSucc { val := Nat.succ i, isLt := hn }))⁻¹ *\n      partialProd f (succ { val := Nat.succ i, isLt := hn }) =\n    f { val := Nat.succ i, isLt := hn }","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  ","nextTactic":"| succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"case mk.succ\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhi :\n  ∀ (hn : i < n),\n    (partialProd f (castSucc { val := i, isLt := hn }))⁻¹ * partialProd f (succ { val := i, isLt := hn }) =\n      f { val := i, isLt := hn }\nhn : Nat.succ i < n\n⊢ (partialProd f (castSucc { val := Nat.succ i, isLt := hn }))⁻¹ *\n      partialProd f (succ { val := Nat.succ i, isLt := hn }) =\n    f { val := Nat.succ i, isLt := hn }","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    ","nextTactic":"specialize hi (lt_trans (Nat.lt_succ_self i) hn)","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"case mk.succ\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn : Nat.succ i < n\nhi :\n  (partialProd f (castSucc { val := i, isLt := (_ : i < n) }))⁻¹ *\n      partialProd f (succ { val := i, isLt := (_ : i < n) }) =\n    f { val := i, isLt := (_ : i < n) }\n⊢ (partialProd f (castSucc { val := Nat.succ i, isLt := hn }))⁻¹ *\n      partialProd f (succ { val := Nat.succ i, isLt := hn }) =\n    f { val := Nat.succ i, isLt := hn }","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    ","nextTactic":"simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"case mk.succ\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn : Nat.succ i < n\nhi :\n  (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })⁻¹ *\n      partialProd f { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) } =\n    f { val := i, isLt := (_ : i < n) }\n⊢ (partialProd f { val := Nat.succ i, isLt := (_ : Nat.succ i < Nat.succ n) })⁻¹ *\n      partialProd f { val := Nat.succ i + 1, isLt := (_ : Nat.succ (Nat.succ i) < Nat.succ n) } =\n    f { val := Nat.succ i, isLt := hn }","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    ","nextTactic":"rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"case mk.succ\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn : Nat.succ i < n\nhi :\n  (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })⁻¹ *\n      partialProd f { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) } =\n    f { val := i, isLt := (_ : i < n) }\n⊢ (partialProd f (succ { val := i, isLt := (_ : i < n) }))⁻¹ *\n      partialProd f { val := Nat.succ i + 1, isLt := (_ : Nat.succ (Nat.succ i) < Nat.succ n) } =\n    f { val := Nat.succ i, isLt := hn }","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    ","nextTactic":"rw [← Fin.succ_mk]","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"case mk.succ\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn : Nat.succ i < n\nhi :\n  (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })⁻¹ *\n      partialProd f { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) } =\n    f { val := i, isLt := (_ : i < n) }\n⊢ (partialProd f (succ { val := i, isLt := (_ : i < n) }))⁻¹ *\n      partialProd f (succ { val := i + 1, isLt := ?mk.succ.h }) =\n    f { val := Nat.succ i, isLt := hn }\ncase mk.succ.h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn : Nat.succ i < n\nhi :\n  (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })⁻¹ *\n      partialProd f { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) } =\n    f { val := i, isLt := (_ : i < n) }\n⊢ i + 1 < n","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    ","nextTactic":"rw [Nat.succ_eq_add_one] at hn","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"case mk.succ\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn✝ : Nat.succ i < n\nhn : i + 1 < n\nhi :\n  (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })⁻¹ *\n      partialProd f { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) } =\n    f { val := i, isLt := (_ : i < n) }\n⊢ (partialProd f (succ { val := i, isLt := (_ : i < n) }))⁻¹ *\n      partialProd f (succ { val := i + 1, isLt := (_ : i + 1 < n) }) =\n    f { val := Nat.succ i, isLt := hn✝ }","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 �� β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    ","nextTactic":"simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"case mk.succ\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn✝ : Nat.succ i < n\nhn : i + 1 < n\nhi :\n  (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })⁻¹ *\n      partialProd f { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) } =\n    f { val := i, isLt := (_ : i < n) }\n⊢ (f { val := i, isLt := (_ : i < n) })⁻¹ * (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })⁻¹ *\n      (partialProd f { val := i + 1, isLt := (_ : i + 1 < Nat.succ n) } * f { val := i + 1, isLt := (_ : i + 1 < n) }) =\n    f { val := Nat.succ i, isLt := hn✝ }","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    ","nextTactic":"rw [← mul_assoc]","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"case mk.succ\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn✝ : Nat.succ i < n\nhn : i + 1 < n\nhi :\n  (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })⁻¹ *\n      partialProd f { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) } =\n    f { val := i, isLt := (_ : i < n) }\n⊢ (f { val := i, isLt := (_ : i < n) })⁻¹ * (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })⁻¹ *\n        partialProd f { val := i + 1, isLt := (_ : i + 1 < Nat.succ n) } *\n      f { val := i + 1, isLt := (_ : i + 1 < n) } =\n    f { val := Nat.succ i, isLt := hn✝ }\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn : Nat.succ i < n\n⊢ (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })⁻¹ *\n        partialProd f { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) } =\n      f { val := i, isLt := (_ : i < n) } →\n    i + 1 < n","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    ","nextTactic":"rw [mul_left_eq_self]","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"case mk.succ\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn✝ : Nat.succ i < n\nhn : i + 1 < n\nhi :\n  (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })⁻¹ *\n      partialProd f { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) } =\n    f { val := i, isLt := (_ : i < n) }\n⊢ (f { val := i, isLt := (_ : i < n) })⁻¹ * (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })⁻¹ *\n      partialProd f { val := i + 1, isLt := (_ : i + 1 < Nat.succ n) } =\n    1","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ �� M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    ","nextTactic":"rw [mul_assoc]","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"case mk.succ\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn✝ : Nat.succ i < n\nhn : i + 1 < n\nhi :\n  (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })⁻¹ *\n      partialProd f { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) } =\n    f { val := i, isLt := (_ : i < n) }\n⊢ (f { val := i, isLt := (_ : i < n) })⁻¹ *\n      ((partialProd f { val := i, isLt := (_ : i < Nat.succ n) })⁻¹ *\n        partialProd f { val := i + 1, isLt := (_ : i + 1 < Nat.succ n) }) =\n    1","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    ","nextTactic":"rw [hi]","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"case mk.succ\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn✝ : Nat.succ i < n\nhn : i + 1 < n\nhi :\n  (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })⁻¹ *\n      partialProd f { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) } =\n    f { val := i, isLt := (_ : i < n) }\n⊢ (f { val := i, isLt := (_ : i < n) })⁻¹ * f { val := i, isLt := (_ : i < n) } = 1","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    ","nextTactic":"rw [mul_left_inv]","declUpToTactic":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.272_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\n⊢ (partialProd g (succAbove (succ j) (castSucc k)))⁻¹ * partialProd g (succ (succAbove j k)) =\n    contractNth j (fun x x_1 => x * x_1) g k","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  ","nextTactic":"rcases lt_trichotomy (k : ℕ) j with (h | h | h)","declUpToTactic":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.295_0.QyPgLR0eRR4fDXZ","decl":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k "}
+{"state":"case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑k < ↑j\n⊢ (partialProd g (succAbove (succ j) (castSucc k)))⁻¹ * partialProd g (succ (succAbove j k)) =\n    contractNth j (fun x x_1 => x * x_1) g k","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · ","nextTactic":"rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]","declUpToTactic":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.295_0.QyPgLR0eRR4fDXZ","decl":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k "}
+{"state":"case inl.h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑k < ↑j\n⊢ castSucc k < j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · ","nextTactic":"assumption","declUpToTactic":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.295_0.QyPgLR0eRR4fDXZ","decl":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k "}
+{"state":"case inl.h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑k < ↑j\n⊢ castSucc (castSucc k) < succ j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · ","nextTactic":"rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]","declUpToTactic":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.295_0.QyPgLR0eRR4fDXZ","decl":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k "}
+{"state":"case inl.h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑k < ↑j\n⊢ ↑(castSucc k) ≤ ↑j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      ","nextTactic":"exact le_of_lt h","declUpToTactic":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.295_0.QyPgLR0eRR4fDXZ","decl":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k "}
+{"state":"case inr.inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑k = ↑j\n⊢ (partialProd g (succAbove (succ j) (castSucc k)))⁻¹ * partialProd g (succ (succAbove j k)) =\n    contractNth j (fun x x_1 => x * x_1) g k","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · ","nextTactic":"rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]","declUpToTactic":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.295_0.QyPgLR0eRR4fDXZ","decl":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k "}
+{"state":"case inr.inl.h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑k = ↑j\n⊢ j ≤ castSucc k","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · ","nextTactic":"simp [le_iff_val_le_val, ← h]","declUpToTactic":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.295_0.QyPgLR0eRR4fDXZ","decl":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k "}
+{"state":"case inr.inl.h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑k = ↑j\n⊢ castSucc (castSucc k) < succ j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · ","nextTactic":"rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]","declUpToTactic":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.295_0.QyPgLR0eRR4fDXZ","decl":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k "}
+{"state":"case inr.inl.h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑k = ↑j\n⊢ ↑(castSucc k) ≤ ↑j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      ","nextTactic":"exact le_of_eq h","declUpToTactic":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.295_0.QyPgLR0eRR4fDXZ","decl":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k "}
+{"state":"case inr.inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑j < ↑k\n⊢ (partialProd g (succAbove (succ j) (castSucc k)))⁻¹ * partialProd g (succ (succAbove j k)) =\n    contractNth j (fun x x_1 => x * x_1) g k","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · ","nextTactic":"rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]","declUpToTactic":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.295_0.QyPgLR0eRR4fDXZ","decl":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k "}
+{"state":"case inr.inr.h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑j < ↑k\n⊢ j ≤ castSucc k","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · ","nextTactic":"exact le_iff_val_le_val.2 (le_of_lt h)","declUpToTactic":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.295_0.QyPgLR0eRR4fDXZ","decl":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k "}
+{"state":"case inr.inr.h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑j < ↑k\n⊢ succ j ≤ castSucc (castSucc k)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · ","nextTactic":"rw [le_iff_val_le_val, val_succ]","declUpToTactic":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.295_0.QyPgLR0eRR4fDXZ","decl":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ���₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k "}
+{"state":"case inr.inr.h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑j < ↑k\n⊢ ↑j + 1 ≤ ↑(castSucc (castSucc k))","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      ","nextTactic":"exact Nat.succ_le_of_lt h","declUpToTactic":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.295_0.QyPgLR0eRR4fDXZ","decl":"/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k "}
+{"state":"α : Type u_1\nβ : Type u_2\nm n : ℕ\n⊢ Fintype.card (Fin n → Fin m) = Fintype.card (Fin (m ^ n))","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by ","nextTactic":"simp_rw [Fintype.card_fun, Fintype.card_fin]","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm n : ℕ\nf : Fin n → Fin m\n⊢ ∑ i : Fin n, ↑(f i) * m ^ ↑i < m ^ n","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      ","nextTactic":"induction' n with n ih","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case zero\nα : Type u_1\nβ : Type u_2\nm : ℕ\nf : Fin Nat.zero → Fin m\n⊢ ∑ i : Fin Nat.zero, ↑(f i) * m ^ ↑i < m ^ Nat.zero","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · ","nextTactic":"simp","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ\nα : Type u_1\nβ : Type u_2\nm n : ℕ\nih : ∀ (f : Fin n → Fin m), ∑ i : Fin n, ↑(f i) * m ^ ↑i < m ^ n\nf : Fin (Nat.succ n) → Fin m\n⊢ ∑ i : Fin (Nat.succ n), ↑(f i) * m ^ ↑i < m ^ Nat.succ n","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      ","nextTactic":"cases m","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ.zero\nα : Type u_1\nβ : Type u_2\nn : ℕ\nih : ∀ (f : Fin n → Fin Nat.zero), ∑ i : Fin n, ↑(f i) * Nat.zero ^ ↑i < Nat.zero ^ n\nf : Fin (Nat.succ n) → Fin Nat.zero\n⊢ ∑ i : Fin (Nat.succ n), ↑(f i) * Nat.zero ^ ↑i < Nat.zero ^ Nat.succ n","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · ","nextTactic":"dsimp only [Nat.zero_eq] at f","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ.zero\nα : Type u_1\nβ : Type u_2\nn : ℕ\nih : ∀ (f : Fin n → Fin Nat.zero), ∑ i : Fin n, ↑(f i) * Nat.zero ^ ↑i < Nat.zero ^ n\nf : Fin (Nat.succ n) → Fin 0\n⊢ ∑ i : Fin (Nat.succ n), ↑(f i) * Nat.zero ^ ↑i < Nat.zero ^ Nat.succ n","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        ","nextTactic":"exact isEmptyElim (f <| Fin.last _)","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ.succ\nα : Type u_1\nβ : Type u_2\nn n✝ : ℕ\nih : ∀ (f : Fin n → Fin (Nat.succ n✝)), ∑ i : Fin n, ↑(f i) * Nat.succ n✝ ^ ↑i < Nat.succ n✝ ^ n\nf : Fin (Nat.succ n) → Fin (Nat.succ n✝)\n⊢ ∑ i : Fin (Nat.succ n), ↑(f i) * Nat.succ n✝ ^ ↑i < Nat.succ n✝ ^ Nat.succ n","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      ","nextTactic":"simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ.succ\nα : Type u_1\nβ : Type u_2\nn n✝ : ℕ\nih : ∀ (f : Fin n → Fin (Nat.succ n✝)), ∑ i : Fin n, ↑(f i) * Nat.succ n✝ ^ ↑i < Nat.succ n✝ ^ n\nf : Fin (Nat.succ n) → Fin (Nat.succ n✝)\n⊢ ∑ x : Fin n, ↑(f (Fin.castSucc x)) * Nat.succ n✝ ^ ↑x + ↑(f (Fin.last n)) * Nat.succ n✝ ^ n < Nat.succ n✝ ^ Nat.succ n","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ��� i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      ","nextTactic":"refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ.succ\nα : Type u_1\nβ : Type u_2\nn n✝ : ℕ\nih : ∀ (f : Fin n → Fin (Nat.succ n✝)), ∑ i : Fin n, ↑(f i) * Nat.succ n✝ ^ ↑i < Nat.succ n✝ ^ n\nf : Fin (Nat.succ n) → Fin (Nat.succ n✝)\n⊢ Nat.succ n✝ ^ n + n✝ * Nat.succ n✝ ^ n = Nat.succ n✝ ^ Nat.succ n","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      ","nextTactic":"rw [← one_add_mul (_ : ℕ)]","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ.succ\nα : Type u_1\nβ : Type u_2\nn n✝ : ℕ\nih : ∀ (f : Fin n → Fin (Nat.succ n✝)), ∑ i : Fin n, ↑(f i) * Nat.succ n✝ ^ ↑i < Nat.succ n✝ ^ n\nf : Fin (Nat.succ n) → Fin (Nat.succ n✝)\n⊢ (1 + n✝) * Nat.succ n✝ ^ n = Nat.succ n✝ ^ Nat.succ n","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      ","nextTactic":"rw [add_comm]","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ.succ\nα : Type u_1\nβ : Type u_2\nn n✝ : ℕ\nih : ∀ (f : Fin n → Fin (Nat.succ n✝)), ∑ i : Fin n, ↑(f i) * Nat.succ n✝ ^ ↑i < Nat.succ n✝ ^ n\nf : Fin (Nat.succ n) → Fin (Nat.succ n✝)\n⊢ (n✝ + 1) * Nat.succ n✝ ^ n = Nat.succ n✝ ^ Nat.succ n","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      ","nextTactic":"rw [pow_succ]","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ.succ\nα : Type u_1\nβ : Type u_2\nn n✝ : ℕ\nih : ∀ (f : Fin n → Fin (Nat.succ n✝)), ∑ i : Fin n, ↑(f i) * Nat.succ n✝ ^ ↑i < Nat.succ n✝ ^ n\nf : Fin (Nat.succ n) → Fin (Nat.succ n✝)\n⊢ (n✝ + 1) * Nat.succ n✝ ^ n = Nat.succ n✝ * Nat.succ n✝ ^ n","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      ","nextTactic":"rfl","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm n : ℕ\na : Fin (m ^ n)\nb : Fin n\n⊢ ↑a / m ^ ↑b % m < m","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      ","nextTactic":"cases' n with n","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case zero\nα : Type u_1\nβ : Type u_2\nm : ℕ\na : Fin (m ^ Nat.zero)\nb : Fin Nat.zero\n⊢ ↑a / m ^ ↑b % m < m","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · ","nextTactic":"exact b.elim0","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ\nα : Type u_1\nβ : Type u_2\nm n : ℕ\na : Fin (m ^ Nat.succ n)\nb : Fin (Nat.succ n)\n⊢ ↑a / m ^ ↑b % m < m","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      ","nextTactic":"cases' m with m","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ.zero\nα : Type u_1\nβ : Type u_2\nn : ℕ\nb : Fin (Nat.succ n)\na : Fin (Nat.zero ^ Nat.succ n)\n⊢ ↑a / Nat.zero ^ ↑b % Nat.zero < Nat.zero","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · ","nextTactic":"dsimp only [Nat.zero_eq] at a","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ.zero\nα : Type u_1\nβ : Type u_2\nn : ℕ\nb : Fin (Nat.succ n)\na : Fin (0 ^ Nat.succ n)\n⊢ ↑a / Nat.zero ^ ↑b % Nat.zero < Nat.zero","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        ","nextTactic":"rw [zero_pow n.succ_pos] at a","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ.zero\nα : Type u_1\nβ : Type u_2\nn : ℕ\nb : Fin (Nat.succ n)\na✝ : Fin (0 ^ Nat.succ n)\na : Fin 0\n⊢ ↑a✝ / Nat.zero ^ ↑b % Nat.zero < Nat.zero","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        ","nextTactic":"exact a.elim0","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ.succ\nα : Type u_1\nβ : Type u_2\nn : ℕ\nb : Fin (Nat.succ n)\nm : ℕ\na : Fin (Nat.succ m ^ Nat.succ n)\n⊢ ↑a / Nat.succ m ^ ↑b % Nat.succ m < Nat.succ m","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · ","nextTactic":"exact Nat.mod_lt _ m.succ_pos","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm n : ℕ\na : Fin (m ^ n)\n⊢ (fun f => { val := ∑ i : Fin n, ↑(f i) * m ^ ↑i, isLt := (_ : ∑ i : Fin n, ↑(f i) * m ^ ↑i < m ^ n) })\n      ((fun a b => { val := ↑a / m ^ ↑b % m, isLt := (_ : ↑a / m ^ ↑b % m < m) }) a) =\n    a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      ","nextTactic":"dsimp","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm n : ℕ\na : Fin (m ^ n)\n⊢ { val := ∑ i : Fin n, ↑a / m ^ ↑i % m * m ^ ↑i,\n      isLt := (_ : ∑ i : Fin n, ↑{ val := ↑a / m ^ ↑i % m, isLt := (_ : ↑a / m ^ ↑i % m < m) } * m ^ ↑i < m ^ n) } =\n    a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : �� → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      ","nextTactic":"induction' n with n ih","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case zero\nα : Type u_1\nβ : Type u_2\nm : ℕ\na : Fin (m ^ Nat.zero)\n⊢ { val := ∑ i : Fin Nat.zero, ↑a / m ^ ↑i % m * m ^ ↑i,\n      isLt :=\n        (_ :\n          ∑ i : Fin Nat.zero, ↑{ val := ↑a / m ^ ↑i % m, isLt := (_ : ↑a / m ^ ↑i % m < m) } * m ^ ↑i <\n            m ^ Nat.zero) } =\n    a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) ��� α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · ","nextTactic":"haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case zero\nα : Type u_1\nβ : Type u_2\nm : ℕ\na : Fin (m ^ Nat.zero)\nthis : Subsingleton (Fin (m ^ 0))\n⊢ { val := ∑ i : Fin Nat.zero, ↑a / m ^ ↑i % m * m ^ ↑i,\n      isLt :=\n        (_ :\n          ∑ i : Fin Nat.zero, ↑{ val := ↑a / m ^ ↑i % m, isLt := (_ : ↑a / m ^ ↑i % m < m) } * m ^ ↑i <\n            m ^ Nat.zero) } =\n    a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        ","nextTactic":"exact Subsingleton.elim _ _","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ\nα : Type u_1\nβ : Type u_2\nm n : ℕ\nih :\n  ∀ (a : Fin (m ^ n)),\n    { val := ∑ i : Fin n, ↑a / m ^ ↑i % m * m ^ ↑i,\n        isLt := (_ : ∑ i : Fin n, ↑{ val := ↑a / m ^ ↑i % m, isLt := (_ : ↑a / m ^ ↑i % m < m) } * m ^ ↑i < m ^ n) } =\n      a\na : Fin (m ^ Nat.succ n)\n⊢ { val := ∑ i : Fin (Nat.succ n), ↑a / m ^ ↑i % m * m ^ ↑i,\n      isLt :=\n        (_ :\n          ∑ i : Fin (Nat.succ n), ↑{ val := ↑a / m ^ ↑i % m, isLt := (_ : ↑a / m ^ ↑i % m < m) } * m ^ ↑i <\n            m ^ Nat.succ n) } =\n    a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      ","nextTactic":"simp_rw [Fin.forall_iff, Fin.ext_iff] at ih","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ\nα : Type u_1\nβ : Type u_2\nm n : ℕ\na : Fin (m ^ Nat.succ n)\nih : ∀ i < m ^ n, ∑ x : Fin n, i / m ^ ↑x % m * m ^ ↑x = i\n⊢ { val := ∑ i : Fin (Nat.succ n), ↑a / m ^ ↑i % m * m ^ ↑i,\n      isLt :=\n        (_ :\n          ∑ i : Fin (Nat.succ n), ↑{ val := ↑a / m ^ ↑i % m, isLt := (_ : ↑a / m ^ ↑i % m < m) } * m ^ ↑i <\n            m ^ Nat.succ n) } =\n    a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ","nextTactic":"ext","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ.h\nα : Type u_1\nβ : Type u_2\nm n : ℕ\na : Fin (m ^ Nat.succ n)\nih : ∀ i < m ^ n, ∑ x : Fin n, i / m ^ ↑x % m * m ^ ↑x = i\n⊢ ↑{ val := ∑ i : Fin (Nat.succ n), ↑a / m ^ ↑i % m * m ^ ↑i,\n        isLt :=\n          (_ :\n            ∑ i : Fin (Nat.succ n), ↑{ val := ↑a / m ^ ↑i % m, isLt := (_ : ↑a / m ^ ↑i % m < m) } * m ^ ↑i <\n              m ^ Nat.succ n) } =\n    ↑a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      ","nextTactic":"simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ.h\nα : Type u_1\nβ : Type u_2\nm n : ℕ\na : Fin (m ^ Nat.succ n)\nih : ∀ i < m ^ n, ∑ x : Fin n, i / m ^ ↑x % m * m ^ ↑x = i\n⊢ ↑a % m + m * ∑ x : Fin n, ↑a / m / m ^ ↑x % m * m ^ ↑x = ↑a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      ","nextTactic":"rw [ih _ (Nat.div_lt_of_lt_mul ?_)]","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"case succ.h\nα : Type u_1\nβ : Type u_2\nm n : ℕ\na : Fin (m ^ Nat.succ n)\nih : ∀ i < m ^ n, ∑ x : Fin n, i / m ^ ↑x % m * m ^ ↑x = i\n⊢ ↑a % m + m * (↑a / m) = ↑a\nα : Type u_1\nβ : Type u_2\nm n : ℕ\na : Fin (m ^ Nat.succ n)\nih : ∀ i < m ^ n, ∑ x : Fin n, i / m ^ ↑x % m * m ^ ↑x = i\n⊢ ↑a < m * m ^ n","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      ","nextTactic":"rw [Nat.mod_add_div]","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm n : ℕ\na : Fin (m ^ Nat.succ n)\nih : ∀ i < m ^ n, ∑ x : Fin n, i / m ^ ↑x % m * m ^ ↑x = i\n⊢ ↑a < m * m ^ n","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ��₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      ","nextTactic":"exact a.is_lt.trans_eq (pow_succ _ _)","declUpToTactic":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.332_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\n⊢ ↑(finFunctionFinEquiv (Pi.single i j)) = ↑j * m ^ ↑i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  ","nextTactic":"rw [finFunctionFinEquiv_apply]","declUpToTactic":"theorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.378_0.QyPgLR0eRR4fDXZ","decl":"theorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\n⊢ ∑ i_1 : Fin n, ↑(Pi.single i j i_1) * m ^ ↑i_1 = ↑j * m ^ ↑i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  ","nextTactic":"rw [Fintype.sum_eq_single i]","declUpToTactic":"theorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.378_0.QyPgLR0eRR4fDXZ","decl":"theorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\n⊢ ↑(Pi.single i j i) * m ^ ↑i = ↑j * m ^ ↑i\nα : Type u_1\nβ : Type u_2\nm n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\n⊢ ∀ (x : Fin n), x ≠ i → ↑(Pi.single i j x) * m ^ ↑x = 0","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  ","nextTactic":"rw [Pi.single_eq_same]","declUpToTactic":"theorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.378_0.QyPgLR0eRR4fDXZ","decl":"theorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\n⊢ ∀ (x : Fin n), x ≠ i → ↑(Pi.single i j x) * m ^ ↑x = 0","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  ","nextTactic":"rintro x hx","declUpToTactic":"theorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.378_0.QyPgLR0eRR4fDXZ","decl":"theorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\nx : Fin n\nhx : x ≠ i\n⊢ ↑(Pi.single i j x) * m ^ ↑x = 0","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  ","nextTactic":"rw [Pi.single_eq_of_ne hx]","declUpToTactic":"theorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.378_0.QyPgLR0eRR4fDXZ","decl":"theorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\nx : Fin n\nhx : x ≠ i\n⊢ ↑0 * m ^ ↑x = 0","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  ","nextTactic":"rw [Fin.val_zero']","declUpToTactic":"theorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.378_0.QyPgLR0eRR4fDXZ","decl":"theorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\nx : Fin n\nhx : x ≠ i\n⊢ 0 * m ^ ↑x = 0","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  ","nextTactic":"rw [zero_mul]","declUpToTactic":"theorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.378_0.QyPgLR0eRR4fDXZ","decl":"theorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\n⊢ Fintype.card ((i : Fin m) → Fin (n i)) = Fintype.card (Fin (∏ i : Fin m, n i))","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by ","nextTactic":"simp_rw [Fintype.card_pi, Fintype.card_fin]","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\nf : (i : Fin m) → Fin (n i)\n⊢ ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      ","nextTactic":"induction' m with m ih","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case zero\nα : Type u_1\nβ : Type u_2\nn : Fin Nat.zero → ℕ\nf : (i : Fin Nat.zero) → Fin (n i)\n⊢ ∑ i : Fin Nat.zero, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ Nat.zero) j) < ∏ i : Fin Nat.zero, n i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · ","nextTactic":"simp","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case succ\nα : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn : Fin (Nat.succ m) → ℕ\nf : (i : Fin (Nat.succ m)) → Fin (n i)\n⊢ ∑ i : Fin (Nat.succ m), ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ Nat.succ m) j) < ∏ i : Fin (Nat.succ m), n i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      ","nextTactic":"rw [Fin.prod_univ_castSucc]","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case succ\nα : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn : Fin (Nat.succ m) → ℕ\nf : (i : Fin (Nat.succ m)) → Fin (n i)\n⊢ ∑ i : Fin (Nat.succ m), ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ Nat.succ m) j) <\n    (∏ i : Fin m, n (Fin.castSucc i)) * n (Fin.last m)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      ","nextTactic":"rw [Fin.sum_univ_castSucc]","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case succ\nα : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn : Fin (Nat.succ m) → ℕ\nf : (i : Fin (Nat.succ m)) → Fin (n i)\n⊢ ∑ i : Fin m,\n        ↑(f (Fin.castSucc i)) * ∏ j : Fin ↑(Fin.castSucc i), n (Fin.castLE (_ : ↑(Fin.castSucc i) ≤ Nat.succ m) j) +\n      ↑(f (Fin.last m)) * ∏ j : Fin ↑(Fin.last m), n (Fin.castLE (_ : ↑(Fin.last m) ≤ Nat.succ m) j) <\n    (∏ i : Fin m, n (Fin.castSucc i)) * n (Fin.last m)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      ","nextTactic":"suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn : Fin (Nat.succ m) → ℕ\nf : (i : Fin (Nat.succ m)) → Fin (n i)\nthis :\n  ∀ (n : Fin m → ℕ) (nn : ℕ) (f : (i : Fin m) → Fin (n i)) (fn : Fin nn),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) + ↑fn * ∏ j : Fin m, n j < (∏ i : Fin m, n i) * nn\n⊢ ∑ i : Fin m,\n        ↑(f (Fin.castSucc i)) * ∏ j : Fin ↑(Fin.castSucc i), n (Fin.castLE (_ : ↑(Fin.castSucc i) ≤ Nat.succ m) j) +\n      ↑(f (Fin.last m)) * ∏ j : Fin ↑(Fin.last m), n (Fin.castLE (_ : ↑(Fin.last m) ≤ Nat.succ m) j) <\n    (∏ i : Fin m, n (Fin.castSucc i)) * n (Fin.last m)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        ","nextTactic":"replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn : Fin (Nat.succ m) → ℕ\nf : (i : Fin (Nat.succ m)) → Fin (n i)\nthis :\n  ∑ i : Fin m, ↑(Fin.init f i) * ∏ j : Fin ↑i, Fin.init n (Fin.castLE (_ : ↑i ≤ m) j) +\n      ↑(f (Fin.last m)) * ∏ j : Fin m, Fin.init n j <\n    (∏ i : Fin m, Fin.init n i) * n (Fin.last m)\n⊢ ∑ i : Fin m,\n        ↑(f (Fin.castSucc i)) * ∏ j : Fin ↑(Fin.castSucc i), n (Fin.castLE (_ : ↑(Fin.castSucc i) ≤ Nat.succ m) j) +\n      ↑(f (Fin.last m)) * ∏ j : Fin ↑(Fin.last m), n (Fin.castLE (_ : ↑(Fin.last m) ≤ Nat.succ m) j) <\n    (∏ i : Fin m, n (Fin.castSucc i)) * n (Fin.last m)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        ","nextTactic":"rw [← Fin.snoc_init_self f]","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn : Fin (Nat.succ m) → ℕ\nf : (i : Fin (Nat.succ m)) → Fin (n i)\nthis :\n  ∑ i : Fin m, ↑(Fin.init f i) * ∏ j : Fin ↑i, Fin.init n (Fin.castLE (_ : ↑i ≤ m) j) +\n      ↑(f (Fin.last m)) * ∏ j : Fin m, Fin.init n j <\n    (∏ i : Fin m, Fin.init n i) * n (Fin.last m)\n⊢ ∑ i : Fin m,\n        ↑(Fin.snoc (Fin.init f) (f (Fin.last m)) (Fin.castSucc i)) *\n          ∏ j : Fin ↑(Fin.castSucc i), n (Fin.castLE (_ : ↑(Fin.castSucc i) ≤ Nat.succ m) j) +\n      ↑(Fin.snoc (Fin.init f) (f (Fin.last m)) (Fin.last m)) *\n        ∏ j : Fin ↑(Fin.last m), n (Fin.castLE (_ : ↑(Fin.last m) ≤ Nat.succ m) j) <\n    (∏ i : Fin m, n (Fin.castSucc i)) * n (Fin.last m)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        ","nextTactic":"simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn : Fin (Nat.succ m) → ℕ\nf : (i : Fin (Nat.succ m)) → Fin (n i)\nthis :\n  ∑ i : Fin m, ↑(Fin.init f i) * ∏ j : Fin ↑i, Fin.init n (Fin.castLE (_ : ↑i ≤ m) j) +\n      ↑(f (Fin.last m)) * ∏ j : Fin m, Fin.init n j <\n    (∏ i : Fin m, Fin.init n i) * n (Fin.last m)\n⊢ ∑ x : Fin m,\n        ↑(Fin.snoc (Fin.init f) (f (Fin.last m)) (Fin.castSucc x)) *\n          ∏ x_1 : Fin ↑(Fin.castSucc x),\n            Fin.snoc (Fin.init n) (n (Fin.last m)) (Fin.castLE (_ : ↑(Fin.castSucc x) ≤ Nat.succ m) x_1) +\n      ↑(Fin.snoc (Fin.init f) (f (Fin.last m)) (Fin.last m)) *\n        ∏ x : Fin ↑(Fin.last m),\n          Fin.snoc (Fin.init n) (n (Fin.last m)) (Fin.castLE (_ : ↑(Fin.last m) ≤ Nat.succ m) x) <\n    (∏ x : Fin m, Fin.snoc (Fin.init n) (n (Fin.last m)) (Fin.castSucc x)) *\n      Fin.snoc (Fin.init n) (n (Fin.last m)) (Fin.last m)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) ��� β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        ","nextTactic":"simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn : Fin (Nat.succ m) → ℕ\nf : (i : Fin (Nat.succ m)) → Fin (n i)\nthis :\n  ∑ i : Fin m, ↑(Fin.init f i) * ∏ j : Fin ↑i, Fin.init n (Fin.castLE (_ : ↑i ≤ m) j) +\n      ↑(f (Fin.last m)) * ∏ j : Fin m, Fin.init n j <\n    (∏ i : Fin m, Fin.init n i) * n (Fin.last m)\n⊢ ∑ x : Fin m,\n        ↑(Fin.init f x) * ∏ x_1 : Fin ↑(Fin.castSucc x), n (Fin.castLE (_ : ↑(Fin.castSucc x) ≤ Nat.succ m) x_1) +\n      ↑(f (Fin.last m)) * ∏ x : Fin ↑(Fin.last m), n (Fin.castLE (_ : ↑(Fin.last m) ≤ Nat.succ m) x) <\n    (∏ x : Fin m, Fin.init n x) * n (Fin.last m)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        ","nextTactic":"exact this","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case succ\nα : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn : Fin (Nat.succ m) → ℕ\nf : (i : Fin (Nat.succ m)) → Fin (n i)\n⊢ ∀ (n : Fin m → ℕ) (nn : ℕ) (f : (i : Fin m) → Fin (n i)) (fn : Fin nn),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) + ↑fn * ∏ j : Fin m, n j < (∏ i : Fin m, n i) * nn","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      ","nextTactic":"intro n nn f fn","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case succ\nα : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn✝ : Fin (Nat.succ m) → ℕ\nf✝ : (i : Fin (Nat.succ m)) → Fin (n✝ i)\nn : Fin m → ℕ\nnn : ℕ\nf : (i : Fin m) → Fin (n i)\nfn : Fin nn\n⊢ ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) + ↑fn * ∏ j : Fin m, n j < (∏ i : Fin m, n i) * nn","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      ","nextTactic":"cases nn","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case succ.zero\nα : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn✝ : Fin (Nat.succ m) → ℕ\nf✝ : (i : Fin (Nat.succ m)) → Fin (n✝ i)\nn : Fin m → ℕ\nf : (i : Fin m) → Fin (n i)\nfn : Fin Nat.zero\n⊢ ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) + ↑fn * ∏ j : Fin m, n j <\n    (∏ i : Fin m, n i) * Nat.zero","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · ","nextTactic":"dsimp only [Nat.zero_eq] at fn","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case succ.zero\nα : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn✝ : Fin (Nat.succ m) → ℕ\nf✝ : (i : Fin (Nat.succ m)) → Fin (n✝ i)\nn : Fin m → ℕ\nf : (i : Fin m) → Fin (n i)\nfn : Fin 0\n⊢ ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) + ↑fn * ∏ j : Fin m, n j <\n    (∏ i : Fin m, n i) * Nat.zero","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        ","nextTactic":"exact isEmptyElim fn","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case succ.succ\nα : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn✝¹ : Fin (Nat.succ m) → ℕ\nf✝ : (i : Fin (Nat.succ m)) → Fin (n✝¹ i)\nn : Fin m → ℕ\nf : (i : Fin m) → Fin (n i)\nn✝ : ℕ\nfn : Fin (Nat.succ n✝)\n⊢ ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) + ↑fn * ∏ j : Fin m, n j <\n    (∏ i : Fin m, n i) * Nat.succ n✝","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻�� * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      ","nextTactic":"refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case succ.succ\nα : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn✝¹ : Fin (Nat.succ m) → ℕ\nf✝ : (i : Fin (Nat.succ m)) → Fin (n✝¹ i)\nn : Fin m → ℕ\nf : (i : Fin m) → Fin (n i)\nn✝ : ℕ\nfn : Fin (Nat.succ n✝)\n⊢ ∏ i : Fin m, n i + n✝ * ∏ j : Fin m, n j = (∏ i : Fin m, n i) * Nat.succ n✝","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      ","nextTactic":"rw [← one_add_mul (_ : ℕ)]","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case succ.succ\nα : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn✝¹ : Fin (Nat.succ m) → ℕ\nf✝ : (i : Fin (Nat.succ m)) → Fin (n✝¹ i)\nn : Fin m → ℕ\nf : (i : Fin m) → Fin (n i)\nn✝ : ℕ\nfn : Fin (Nat.succ n✝)\n⊢ (1 + n✝) * ∏ i : Fin m, n i = (∏ i : Fin m, n i) * Nat.succ n✝","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      ","nextTactic":"rw [mul_comm]","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case succ.succ\nα : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn✝¹ : Fin (Nat.succ m) → ℕ\nf✝ : (i : Fin (Nat.succ m)) → Fin (n✝¹ i)\nn : Fin m → ℕ\nf : (i : Fin m) → Fin (n i)\nn✝ : ℕ\nfn : Fin (Nat.succ n✝)\n⊢ (∏ i : Fin m, n i) * (1 + n✝) = (∏ i : Fin m, n i) * Nat.succ n✝","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      ","nextTactic":"rw [add_comm]","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case succ.succ\nα : Type u_1\nβ : Type u_2\nm : ℕ\nih :\n  ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)),\n    ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i\nn✝¹ : Fin (Nat.succ m) → ℕ\nf✝ : (i : Fin (Nat.succ m)) → Fin (n✝¹ i)\nn : Fin m → ℕ\nf : (i : Fin m) → Fin (n i)\nn✝ : ℕ\nfn : Fin (Nat.succ n✝)\n⊢ (∏ i : Fin m, n i) * (n✝ + 1) = (∏ i : Fin m, n i) * Nat.succ n✝","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      ","nextTactic":"rfl","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\na : Fin (∏ i : Fin m, n i)\nb : Fin m\n⊢ (↑a / ∏ j : Fin ↑b, n (Fin.castLE (_ : ↑b ≤ m) j)) % n b < n b","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      ","nextTactic":"cases m","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case zero\nα : Type u_1\nβ : Type u_2\nn : Fin Nat.zero → ℕ\na : Fin (∏ i : Fin Nat.zero, n i)\nb : Fin Nat.zero\n⊢ (↑a / ∏ j : Fin ↑b, n (Fin.castLE (_ : ↑b ≤ Nat.zero) j)) % n b < n b","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · ","nextTactic":"exact b.elim0","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case succ\nα : Type u_1\nβ : Type u_2\nn✝ : ℕ\nn : Fin (Nat.succ n✝) → ℕ\na : Fin (∏ i : Fin (Nat.succ n✝), n i)\nb : Fin (Nat.succ n✝)\n⊢ (↑a / ∏ j : Fin ↑b, n (Fin.castLE (_ : ↑b ≤ Nat.succ n✝) j)) % n b < n b","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      ","nextTactic":"cases' h : n b with nb","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case succ.zero\nα : Type u_1\nβ : Type u_2\nn✝ : ℕ\nn : Fin (Nat.succ n✝) → ℕ\na : Fin (∏ i : Fin (Nat.succ n✝), n i)\nb : Fin (Nat.succ n✝)\nh : n b = Nat.zero\n⊢ (↑a / ∏ j : Fin ↑b, n (Fin.castLE (_ : ↑b ≤ Nat.succ n✝) j)) % Nat.zero < Nat.zero","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · ","nextTactic":"rw [prod_eq_zero (Finset.mem_univ _) h] at a","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case succ.zero\nα : Type u_1\nβ : Type u_2\nn✝ : ℕ\nn : Fin (Nat.succ n✝) → ℕ\na✝ : Fin (∏ i : Fin (Nat.succ n✝), n i)\na : Fin 0\nb : Fin (Nat.succ n✝)\nh : n b = Nat.zero\n⊢ (↑a✝ / ∏ j : Fin ↑b, n (Fin.castLE (_ : ↑b ≤ Nat.succ n✝) j)) % Nat.zero < Nat.zero","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        ","nextTactic":"exact isEmptyElim a","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case succ.succ\nα : Type u_1\nβ : Type u_2\nn✝ : ℕ\nn : Fin (Nat.succ n✝) → ℕ\na : Fin (∏ i : Fin (Nat.succ n✝), n i)\nb : Fin (Nat.succ n✝)\nnb : ℕ\nh : n b = Nat.succ nb\n⊢ (↑a / ∏ j : Fin ↑b, n (Fin.castLE (_ : ↑b ≤ Nat.succ n✝) j)) % Nat.succ nb < Nat.succ nb","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      ","nextTactic":"exact Nat.mod_lt _ nb.succ_pos","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\n⊢ Function.RightInverse\n    (fun a b =>\n      { val := (↑a / ∏ j : Fin ↑b, n (Fin.castLE (_ : ↑b ≤ m) j)) % n b,\n        isLt := (_ : (↑a / ∏ j : Fin ↑b, n (Fin.castLE (_ : ↑b ≤ m) j)) % n b < n b) })\n    fun f =>\n    { val := ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j),\n      isLt := (_ : ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i) }","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      ","nextTactic":"intro a","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\na : Fin (∏ i : Fin m, n i)\n⊢ (fun f =>\n        { val := ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j),\n          isLt := (_ : ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i) })\n      ((fun a b =>\n          { val := (↑a / ∏ j : Fin ↑b, n (Fin.castLE (_ : ↑b ≤ m) j)) % n b,\n            isLt := (_ : (↑a / ∏ j : Fin ↑b, n (Fin.castLE (_ : ↑b ≤ m) j)) % n b < n b) })\n        a) =\n    a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g��g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; ","nextTactic":"revert a","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\n⊢ ∀ (a : Fin (∏ i : Fin m, n i)),\n    (fun f =>\n          { val := ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j),\n            isLt := (_ : ∑ i : Fin m, ↑(f i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) < ∏ i : Fin m, n i) })\n        ((fun a b =>\n            { val := (↑a / ∏ j : Fin ↑b, n (Fin.castLE (_ : ↑b ≤ m) j)) % n b,\n              isLt := (_ : (↑a / ∏ j : Fin ↑b, n (Fin.castLE (_ : ↑b ≤ m) j)) % n b < n b) })\n          a) =\n      a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; ","nextTactic":"dsimp only [Fin.val_mk]","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\n⊢ ∀ (a : Fin (∏ i : Fin m, n i)),\n    {\n        val :=\n          ∑ i : Fin m,\n            (↑a / ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j)) % n i * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j),\n        isLt :=\n          (_ :\n            ∑ i : Fin m,\n                ↑{ val := (↑a / ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j)) % n i,\n                      isLt := (_ : (↑a / ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j)) % n i < n i) } *\n                  ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) <\n              ∏ i : Fin m, n i) } =\n      a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      ","nextTactic":"refine' Fin.consInduction _ _ n","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case refine'_1\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\n⊢ ∀ (a : Fin (∏ i : Fin 0, Fin.elim0 i)),\n    {\n        val :=\n          ∑ i : Fin 0,\n            (↑a / ∏ j : Fin ↑i, Fin.elim0 (Fin.castLE (_ : ↑i ≤ 0) j)) % Fin.elim0 i *\n              ∏ j : Fin ↑i, Fin.elim0 (Fin.castLE (_ : ↑i ≤ 0) j),\n        isLt :=\n          (_ :\n            ∑ i : Fin 0,\n                ↑{ val := (↑a / ∏ j : Fin ↑i, Fin.elim0 (Fin.castLE (_ : ↑i ≤ 0) j)) % Fin.elim0 i,\n                      isLt :=\n                        (_ : (↑a / ∏ j : Fin ↑i, Fin.elim0 (Fin.castLE (_ : ↑i ≤ 0) j)) % Fin.elim0 i < Fin.elim0 i) } *\n                  ∏ j : Fin ↑i, Fin.elim0 (Fin.castLE (_ : ↑i ≤ 0) j) <\n              ∏ i : Fin 0, Fin.elim0 i) } =\n      a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · ","nextTactic":"intro a","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case refine'_1\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\na : Fin (∏ i : Fin 0, Fin.elim0 i)\n⊢ {\n      val :=\n        ∑ i : Fin 0,\n          (↑a / ∏ j : Fin ↑i, Fin.elim0 (Fin.castLE (_ : ↑i ≤ 0) j)) % Fin.elim0 i *\n            ∏ j : Fin ↑i, Fin.elim0 (Fin.castLE (_ : ↑i ≤ 0) j),\n      isLt :=\n        (_ :\n          ∑ i : Fin 0,\n              ↑{ val := (↑a / ∏ j : Fin ↑i, Fin.elim0 (Fin.castLE (_ : ↑i ≤ 0) j)) % Fin.elim0 i,\n                    isLt :=\n                      (_ : (↑a / ∏ j : Fin ↑i, Fin.elim0 (Fin.castLE (_ : ↑i ≤ 0) j)) % Fin.elim0 i < Fin.elim0 i) } *\n                ∏ j : Fin ↑i, Fin.elim0 (Fin.castLE (_ : ↑i ≤ 0) j) <\n            ∏ i : Fin 0, Fin.elim0 i) } =\n    a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        ","nextTactic":"haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case refine'_1\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\na : Fin (∏ i : Fin 0, Fin.elim0 i)\nthis : Subsingleton (Fin (∏ i : Fin 0, Fin.elim0 i))\n⊢ {\n      val :=\n        ∑ i : Fin 0,\n          (↑a / ∏ j : Fin ↑i, Fin.elim0 (Fin.castLE (_ : ↑i ≤ 0) j)) % Fin.elim0 i *\n            ∏ j : Fin ↑i, Fin.elim0 (Fin.castLE (_ : ↑i ≤ 0) j),\n      isLt :=\n        (_ :\n          ∑ i : Fin 0,\n              ↑{ val := (↑a / ∏ j : Fin ↑i, Fin.elim0 (Fin.castLE (_ : ↑i ≤ 0) j)) % Fin.elim0 i,\n                    isLt :=\n                      (_ : (↑a / ∏ j : Fin ↑i, Fin.elim0 (Fin.castLE (_ : ↑i ≤ 0) j)) % Fin.elim0 i < Fin.elim0 i) } *\n                ∏ j : Fin ↑i, Fin.elim0 (Fin.castLE (_ : ↑i ≤ 0) j) <\n            ∏ i : Fin 0, Fin.elim0 i) } =\n    a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        ","nextTactic":"exact Subsingleton.elim _ _","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case refine'_2\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\n⊢ ∀ {n : ℕ} (x₀ : ℕ) (x : Fin n → ℕ),\n    (∀ (a : Fin (∏ i : Fin n, x i)),\n        {\n            val :=\n              ∑ i : Fin n,\n                (↑a / ∏ j : Fin ↑i, x (Fin.castLE (_ : ↑i ≤ n) j)) % x i * ∏ j : Fin ↑i, x (Fin.castLE (_ : ↑i ≤ n) j),\n            isLt :=\n              (_ :\n                ∑ i : Fin n,\n                    ↑{ val := (↑a / ∏ j : Fin ↑i, x (Fin.castLE (_ : ↑i ≤ n) j)) % x i,\n                          isLt := (_ : (↑a / ∏ j : Fin ↑i, x (Fin.castLE (_ : ↑i ≤ n) j)) % x i < x i) } *\n                      ∏ j : Fin ↑i, x (Fin.castLE (_ : ↑i ≤ n) j) <\n                  ∏ i : Fin n, x i) } =\n          a) →\n      ∀ (a : Fin (∏ i : Fin (n + 1), Fin.cons x₀ x i)),\n        {\n            val :=\n              ∑ i : Fin (n + 1),\n                (↑a / ∏ j : Fin ↑i, Fin.cons x₀ x (Fin.castLE (_ : ↑i ≤ n + 1) j)) % Fin.cons x₀ x i *\n                  ∏ j : Fin ↑i, Fin.cons x₀ x (Fin.castLE (_ : ↑i ≤ n + 1) j),\n            isLt :=\n              (_ :\n                ∑ i : Fin (n + 1),\n                    ↑{ val := (↑a / ∏ j : Fin ↑i, Fin.cons x₀ x (Fin.castLE (_ : ↑i ≤ n + 1) j)) % Fin.cons x₀ x i,\n                          isLt :=\n                            (_ :\n                              (↑a / ∏ j : Fin ↑i, Fin.cons x₀ x (Fin.castLE (_ : ↑i ≤ n + 1) j)) % Fin.cons x₀ x i <\n                                Fin.cons x₀ x i) } *\n                      ∏ j : Fin ↑i, Fin.cons x₀ x (Fin.castLE (_ : ↑i ≤ n + 1) j) <\n                  ∏ i : Fin (n + 1), Fin.cons x₀ x i) } =\n          a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · ","nextTactic":"intro n x xs ih a","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case refine'_2\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn✝ : Fin m → ℕ\nn x : ℕ\nxs : Fin n → ℕ\nih :\n  ∀ (a : Fin (∏ i : Fin n, xs i)),\n    {\n        val :=\n          ∑ i : Fin n,\n            (↑a / ∏ j : Fin ↑i, xs (Fin.castLE (_ : ↑i ≤ n) j)) % xs i * ∏ j : Fin ↑i, xs (Fin.castLE (_ : ↑i ≤ n) j),\n        isLt :=\n          (_ :\n            ∑ i : Fin n,\n                ↑{ val := (↑a / ∏ j : Fin ↑i, xs (Fin.castLE (_ : ↑i ≤ n) j)) % xs i,\n                      isLt := (_ : (↑a / ∏ j : Fin ↑i, xs (Fin.castLE (_ : ↑i ≤ n) j)) % xs i < xs i) } *\n                  ∏ j : Fin ↑i, xs (Fin.castLE (_ : ↑i ≤ n) j) <\n              ∏ i : Fin n, xs i) } =\n      a\na : Fin (∏ i : Fin (n + 1), Fin.cons x xs i)\n⊢ {\n      val :=\n        ∑ i : Fin (n + 1),\n          (↑a / ∏ j : Fin ↑i, Fin.cons x xs (Fin.castLE (_ : ↑i ≤ n + 1) j)) % Fin.cons x xs i *\n            ∏ j : Fin ↑i, Fin.cons x xs (Fin.castLE (_ : ↑i ≤ n + 1) j),\n      isLt :=\n        (_ :\n          ∑ i : Fin (n + 1),\n              ↑{ val := (↑a / ∏ j : Fin ↑i, Fin.cons x xs (Fin.castLE (_ : ↑i ≤ n + 1) j)) % Fin.cons x xs i,\n                    isLt :=\n                      (_ :\n                        (↑a / ∏ j : Fin ↑i, Fin.cons x xs (Fin.castLE (_ : ↑i ≤ n + 1) j)) % Fin.cons x xs i <\n                          Fin.cons x xs i) } *\n                ∏ j : Fin ↑i, Fin.cons x xs (Fin.castLE (_ : ↑i ≤ n + 1) j) <\n            ∏ i : Fin (n + 1), Fin.cons x xs i) } =\n    a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        ","nextTactic":"simp_rw [Fin.forall_iff, Fin.ext_iff] at ih","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case refine'_2\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn✝ : Fin m → ℕ\nn x : ℕ\nxs : Fin n → ℕ\na : Fin (∏ i : Fin (n + 1), Fin.cons x xs i)\nih :\n  ∀ i < ∏ i : Fin n, xs i,\n    ∑ x : Fin n,\n        (i / ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j)) % xs x * ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j) =\n      i\n⊢ {\n      val :=\n        ∑ i : Fin (n + 1),\n          (↑a / ∏ j : Fin ↑i, Fin.cons x xs (Fin.castLE (_ : ↑i ≤ n + 1) j)) % Fin.cons x xs i *\n            ∏ j : Fin ↑i, Fin.cons x xs (Fin.castLE (_ : ↑i ≤ n + 1) j),\n      isLt :=\n        (_ :\n          ∑ i : Fin (n + 1),\n              ↑{ val := (↑a / ∏ j : Fin ↑i, Fin.cons x xs (Fin.castLE (_ : ↑i ≤ n + 1) j)) % Fin.cons x xs i,\n                    isLt :=\n                      (_ :\n                        (↑a / ∏ j : Fin ↑i, Fin.cons x xs (Fin.castLE (_ : ↑i ≤ n + 1) j)) % Fin.cons x xs i <\n                          Fin.cons x xs i) } *\n                ∏ j : Fin ↑i, Fin.cons x xs (Fin.castLE (_ : ↑i ≤ n + 1) j) <\n            ∏ i : Fin (n + 1), Fin.cons x xs i) } =\n    a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ","nextTactic":"ext","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case refine'_2.h\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn✝ : Fin m → ℕ\nn x : ℕ\nxs : Fin n → ℕ\na : Fin (∏ i : Fin (n + 1), Fin.cons x xs i)\nih :\n  ∀ i < ∏ i : Fin n, xs i,\n    ∑ x : Fin n,\n        (i / ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j)) % xs x * ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j) =\n      i\n⊢ ↑{\n        val :=\n          ∑ i : Fin (n + 1),\n            (↑a / ∏ j : Fin ↑i, Fin.cons x xs (Fin.castLE (_ : ↑i ≤ n + 1) j)) % Fin.cons x xs i *\n              ∏ j : Fin ↑i, Fin.cons x xs (Fin.castLE (_ : ↑i ≤ n + 1) j),\n        isLt :=\n          (_ :\n            ∑ i : Fin (n + 1),\n                ↑{ val := (↑a / ∏ j : Fin ↑i, Fin.cons x xs (Fin.castLE (_ : ↑i ≤ n + 1) j)) % Fin.cons x xs i,\n                      isLt :=\n                        (_ :\n                          (↑a / ∏ j : Fin ↑i, Fin.cons x xs (Fin.castLE (_ : ↑i ≤ n + 1) j)) % Fin.cons x xs i <\n                            Fin.cons x xs i) } *\n                  ∏ j : Fin ↑i, Fin.cons x xs (Fin.castLE (_ : ↑i ≤ n + 1) j) <\n              ∏ i : Fin (n + 1), Fin.cons x xs i) } =\n    ↑a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        ","nextTactic":"simp_rw [Fin.sum_univ_succ, Fin.cons_succ]","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case refine'_2.h\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn✝ : Fin m → ℕ\nn x : ℕ\nxs : Fin n → ℕ\na : Fin (∏ i : Fin (n + 1), Fin.cons x xs i)\nih :\n  ∀ i < ∏ i : Fin n, xs i,\n    ∑ x : Fin n,\n        (i / ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j)) % xs x * ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j) =\n      i\n⊢ (↑a / ∏ j : Fin ↑0, Fin.cons x xs (Fin.castLE (_ : ↑0 ≤ n + 1) j)) % Fin.cons x xs 0 *\n        ∏ j : Fin ↑0, Fin.cons x xs (Fin.castLE (_ : ↑0 ≤ n + 1) j) +\n      ∑ x_1 : Fin n,\n        (↑a / ∏ j : Fin ↑(Fin.succ x_1), Fin.cons x xs (Fin.castLE (_ : ↑(Fin.succ x_1) ≤ n + 1) j)) % xs x_1 *\n          ∏ j : Fin ↑(Fin.succ x_1), Fin.cons x xs (Fin.castLE (_ : ↑(Fin.succ x_1) ≤ n + 1) j) =\n    ↑a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        ","nextTactic":"have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case refine'_2.h\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn✝ : Fin m → ℕ\nn x : ℕ\nxs : Fin n → ℕ\na : Fin (∏ i : Fin (n + 1), Fin.cons x xs i)\nih :\n  ∀ i < ∏ i : Fin n, xs i,\n    ∑ x : Fin n,\n        (i / ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j)) % xs x * ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j) =\n      i\nthis :\n  ∀ (i : Fin n),\n    ∏ x_1 : Fin ↑(Fin.succ i), Fin.cons x xs (Fin.castLE (_ : ↑(Fin.succ i) ≤ n + 1) x_1) =\n      ∏ x_1 : Fin (↑i + 1), Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑i ≤ Nat.succ n) x_1)\n⊢ (↑a / ∏ j : Fin ↑0, Fin.cons x xs (Fin.castLE (_ : ↑0 ≤ n + 1) j)) % Fin.cons x xs 0 *\n        ∏ j : Fin ↑0, Fin.cons x xs (Fin.castLE (_ : ↑0 ≤ n + 1) j) +\n      ∑ x_1 : Fin n,\n        (↑a / ∏ j : Fin ↑(Fin.succ x_1), Fin.cons x xs (Fin.castLE (_ : ↑(Fin.succ x_1) ≤ n + 1) j)) % xs x_1 *\n          ∏ j : Fin ↑(Fin.succ x_1), Fin.cons x xs (Fin.castLE (_ : ↑(Fin.succ x_1) ≤ n + 1) j) =\n    ↑a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        ","nextTactic":"simp_rw [this]","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case refine'_2.h\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn✝ : Fin m → ℕ\nn x : ℕ\nxs : Fin n → ℕ\na : Fin (∏ i : Fin (n + 1), Fin.cons x xs i)\nih :\n  ∀ i < ∏ i : Fin n, xs i,\n    ∑ x : Fin n,\n        (i / ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j)) % xs x * ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j) =\n      i\nthis :\n  ∀ (i : Fin n),\n    ∏ x_1 : Fin ↑(Fin.succ i), Fin.cons x xs (Fin.castLE (_ : ↑(Fin.succ i) ≤ n + 1) x_1) =\n      ∏ x_1 : Fin (↑i + 1), Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑i ≤ Nat.succ n) x_1)\n⊢ (↑a / ∏ j : Fin ↑0, Fin.cons x xs (Fin.castLE (_ : ↑0 ≤ n + 1) j)) % Fin.cons x xs 0 *\n        ∏ j : Fin ↑0, Fin.cons x xs (Fin.castLE (_ : ↑0 ≤ n + 1) j) +\n      ∑ x_1 : Fin n,\n        (↑a / ∏ x_2 : Fin (↑x_1 + 1), Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑x_1 ≤ Nat.succ n) x_2)) % xs x_1 *\n          ∏ x_2 : Fin (↑x_1 + 1), Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑x_1 ≤ Nat.succ n) x_2) =\n    ↑a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        ","nextTactic":"clear this","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case refine'_2.h\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn✝ : Fin m → ℕ\nn x : ℕ\nxs : Fin n → ℕ\na : Fin (∏ i : Fin (n + 1), Fin.cons x xs i)\nih :\n  ∀ i < ∏ i : Fin n, xs i,\n    ∑ x : Fin n,\n        (i / ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j)) % xs x * ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j) =\n      i\n⊢ (↑a / ∏ j : Fin ↑0, Fin.cons x xs (Fin.castLE (_ : ↑0 ≤ n + 1) j)) % Fin.cons x xs 0 *\n        ∏ j : Fin ↑0, Fin.cons x xs (Fin.castLE (_ : ↑0 ≤ n + 1) j) +\n      ∑ x_1 : Fin n,\n        (↑a / ∏ x_2 : Fin (↑x_1 + 1), Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑x_1 ≤ Nat.succ n) x_2)) % xs x_1 *\n          ∏ x_2 : Fin (↑x_1 + 1), Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑x_1 ≤ Nat.succ n) x_2) =\n    ↑a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        ","nextTactic":"dsimp only [Fin.val_zero]","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case refine'_2.h\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn✝ : Fin m → ℕ\nn x : ℕ\nxs : Fin n → ℕ\na : Fin (∏ i : Fin (n + 1), Fin.cons x xs i)\nih :\n  ∀ i < ∏ i : Fin n, xs i,\n    ∑ x : Fin n,\n        (i / ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j)) % xs x * ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j) =\n      i\n⊢ (↑a / ∏ j : Fin 0, Fin.cons x xs (Fin.castLE (_ : 0 ≤ n + 1) j)) % Fin.cons x xs 0 *\n        ∏ j : Fin 0, Fin.cons x xs (Fin.castLE (_ : 0 ≤ n + 1) j) +\n      ∑ x_1 : Fin n,\n        (↑a / ∏ x_2 : Fin (↑x_1 + 1), Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑x_1 ≤ Nat.succ n) x_2)) % xs x_1 *\n          ∏ x_2 : Fin (↑x_1 + 1), Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑x_1 ≤ Nat.succ n) x_2) =\n    ↑a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        ","nextTactic":"simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case refine'_2.h\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn✝ : Fin m → ℕ\nn x : ℕ\nxs : Fin n → ℕ\na : Fin (∏ i : Fin (n + 1), Fin.cons x xs i)\nih :\n  ∀ i < ∏ i : Fin n, xs i,\n    ∑ x : Fin n,\n        (i / ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j)) % xs x * ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j) =\n      i\n⊢ ↑a % x +\n      ∑ x_1 : Fin n,\n        ↑a /\n              (Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑x_1 ≤ Nat.succ n) 0) *\n                ∏ i : Fin ↑x_1, Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑x_1 ≤ Nat.succ n) (Fin.succ i))) %\n            xs x_1 *\n          (Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑x_1 ≤ Nat.succ n) 0) *\n            ∏ i : Fin ↑x_1, Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑x_1 ≤ Nat.succ n) (Fin.succ i))) =\n    ↑a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        ","nextTactic":"change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case refine'_2.h\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn✝ : Fin m → ℕ\nn x : ℕ\nxs : Fin n → ℕ\na : Fin (∏ i : Fin (n + 1), Fin.cons x xs i)\nih :\n  ∀ i < ∏ i : Fin n, xs i,\n    ∑ x : Fin n,\n        (i / ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j)) % xs x * ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j) =\n      i\n⊢ ↑a % x +\n      ∑ y : Fin n,\n        ↑a / (x * ∏ i : Fin ↑y, Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑y ≤ Nat.succ n) (Fin.succ i))) % xs y *\n          (x * ∏ i : Fin ↑y, Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑y ≤ Nat.succ n) (Fin.succ i))) =\n    ↑a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        ","nextTactic":"simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case refine'_2.h\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn✝ : Fin m → ℕ\nn x : ℕ\nxs : Fin n → ℕ\na : Fin (∏ i : Fin (n + 1), Fin.cons x xs i)\nih :\n  ∀ i < ∏ i : Fin n, xs i,\n    ∑ x : Fin n,\n        (i / ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j)) % xs x * ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j) =\n      i\n⊢ ↑a % x +\n      x *\n        ∑ x_1 : Fin n,\n          (↑a / x / ∏ i : Fin ↑x_1, Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑x_1 ≤ Nat.succ n) (Fin.succ i))) % xs x_1 *\n            ∏ i : Fin ↑x_1, Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑x_1 ≤ Nat.succ n) (Fin.succ i)) =\n    ↑a","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        ","nextTactic":"convert Nat.mod_add_div _ _","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case h.e'_2.h.e'_6.h.e'_6\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn✝ : Fin m → ℕ\nn x : ℕ\nxs : Fin n → ℕ\na : Fin (∏ i : Fin (n + 1), Fin.cons x xs i)\nih :\n  ∀ i < ∏ i : Fin n, xs i,\n    ∑ x : Fin n,\n        (i / ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j)) % xs x * ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j) =\n      i\n⊢ ∑ x_1 : Fin n,\n      (↑a / x / ∏ i : Fin ↑x_1, Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑x_1 ≤ Nat.succ n) (Fin.succ i))) % xs x_1 *\n        ∏ i : Fin ↑x_1, Fin.cons x xs (Fin.castLE (_ : Nat.succ ↑x_1 ≤ Nat.succ n) (Fin.succ i)) =\n    ↑a / x","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        ","nextTactic":"refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"case h.e'_2.h.e'_6.h.e'_6\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn✝ : Fin m → ℕ\nn x : ℕ\nxs : Fin n → ℕ\na : Fin (∏ i : Fin (n + 1), Fin.cons x xs i)\nih :\n  ∀ i < ∏ i : Fin n, xs i,\n    ∑ x : Fin n,\n        (i / ∏ j : Fin ↑x, xs (Fin.castLE (_ : ���x ≤ n) j)) % xs x * ∏ j : Fin ↑x, xs (Fin.castLE (_ : ↑x ≤ n) j) =\n      i\n⊢ ∏ i : Fin (n + 1), Fin.cons x xs i = x * ∏ i : Fin n, xs i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        ","nextTactic":"exact Fin.prod_univ_succ _","declUpToTactic":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.389_0.QyPgLR0eRR4fDXZ","decl":"/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\ninst✝ : ∀ (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\n⊢ ↑(finPiFinEquiv (Pi.single i j)) = ↑j * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  ","nextTactic":"rw [finPiFinEquiv_apply]","declUpToTactic":"theorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.467_0.QyPgLR0eRR4fDXZ","decl":"theorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\ninst✝ : ∀ (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\n⊢ ∑ i_1 : Fin m, ↑(Pi.single i j i_1) * ∏ j : Fin ↑i_1, n (Fin.castLE (_ : ↑i_1 ≤ m) j) =\n    ↑j * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  ","nextTactic":"rw [Fintype.sum_eq_single i]","declUpToTactic":"theorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.467_0.QyPgLR0eRR4fDXZ","decl":"theorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\ninst✝ : ∀ (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\n⊢ ↑(Pi.single i j i) * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j) = ↑j * ∏ j : Fin ↑i, n (Fin.castLE (_ : ↑i ≤ m) j)\nα : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\ninst✝ : ∀ (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\n⊢ ∀ (x : Fin m), x ≠ i → ↑(Pi.single i j x) * ∏ j : Fin ↑x, n (Fin.castLE (_ : ↑x ≤ m) j) = 0","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  ","nextTactic":"rw [Pi.single_eq_same]","declUpToTactic":"theorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.467_0.QyPgLR0eRR4fDXZ","decl":"theorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\ninst✝ : ∀ (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\n⊢ ∀ (x : Fin m), x ≠ i → ↑(Pi.single i j x) * ∏ j : Fin ↑x, n (Fin.castLE (_ : ↑x ≤ m) j) = 0","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n �� Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  ","nextTactic":"rintro x hx","declUpToTactic":"theorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.467_0.QyPgLR0eRR4fDXZ","decl":"theorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\ninst✝ : ∀ (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\nx : Fin m\nhx : x ≠ i\n⊢ ↑(Pi.single i j x) * ∏ j : Fin ↑x, n (Fin.castLE (_ : ↑x ≤ m) j) = 0","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  ","nextTactic":"rw [Pi.single_eq_of_ne hx]","declUpToTactic":"theorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.467_0.QyPgLR0eRR4fDXZ","decl":"theorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\ninst✝ : ∀ (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\nx : Fin m\nhx : x ≠ i\n⊢ ↑0 * ∏ j : Fin ↑x, n (Fin.castLE (_ : ↑x ≤ m) j) = 0","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  ","nextTactic":"rw [Fin.val_zero']","declUpToTactic":"theorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.467_0.QyPgLR0eRR4fDXZ","decl":"theorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) "}
+{"state":"α : Type u_1\nβ : Type u_2\nm : ℕ\nn : Fin m → ℕ\ninst✝ : ∀ (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\nx : Fin m\nhx : x ≠ i\n⊢ 0 * ∏ j : Fin ↑x, n (Fin.castLE (_ : ↑x ≤ m) j) = 0","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  ","nextTactic":"rw [zero_mul]","declUpToTactic":"theorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.467_0.QyPgLR0eRR4fDXZ","decl":"theorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\n⊢ prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  ","nextTactic":"induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      simp [← A, B, IH]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\n⊢ prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  ","nextTactic":"induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      simp [← A, B, IH]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"case zero\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\n⊢ prod (take Nat.zero (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < Nat.zero) univ, f j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  ","nextTactic":"| zero =>\n    simp","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"case zero\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\n⊢ prod (take Nat.zero (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < Nat.zero) univ, f j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    ","nextTactic":"simp","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"case succ\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\n⊢ prod (take (Nat.succ i) (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  ","nextTactic":"| succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      simp [← A, B, IH]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"case succ\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\n⊢ prod (take (Nat.succ i) (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m �� ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    ","nextTactic":"by_cases h : i < n","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"case pos\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : i < n\n⊢ prod (take (Nat.succ i) (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · ","nextTactic":"have : i < length (ofFn f) := by rwa [length_ofFn f]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : i < n\n⊢ i < length (ofFn f)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by ","nextTactic":"rwa [length_ofFn f]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"case pos\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\n⊢ prod (take (Nat.succ i) (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      ","nextTactic":"rw [prod_take_succ _ _ this]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"case pos\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\n⊢ prod (take i (ofFn f)) * get (ofFn f) { val := i, isLt := this } =\n    ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      ","nextTactic":"have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\n⊢ Finset.filter (fun j => ↑j < i + 1) univ = Finset.filter (fun j => ↑j < i) univ ∪ {{ val := i, isLt := h }}","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ","nextTactic":"ext ⟨_, _⟩","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"case a.mk\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\nval✝ : ℕ\nisLt✝ : val✝ < n\n⊢ { val := val✝, isLt := isLt✝ } ∈ Finset.filter (fun j => ↑j < i + 1) univ ↔\n    { val := val✝, isLt := isLt✝ } ∈ Finset.filter (fun j => ↑j < i) univ ∪ {{ val := i, isLt := h }}","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        ","nextTactic":"simp [Nat.lt_succ_iff_lt_or_eq]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"case pos\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\nA : Finset.filter (fun j => ↑j < i + 1) univ = Finset.filter (fun j => ↑j < i) univ ∪ {{ val := i, isLt := h }}\n⊢ prod (take i (ofFn f)) * get (ofFn f) { val := i, isLt := this } =\n    ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      ","nextTactic":"have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\nA : Finset.filter (fun j => ↑j < i + 1) univ = Finset.filter (fun j => ↑j < i) univ ∪ {{ val := i, isLt := h }}\n⊢ _root_.Disjoint (Finset.filter (fun j => ↑j < i) univ) {{ val := i, isLt := h }}","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by ","nextTactic":"simp","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"case pos\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\nA : Finset.filter (fun j => ↑j < i + 1) univ = Finset.filter (fun j => ↑j < i) univ ∪ {{ val := i, isLt := h }}\nB : _root_.Disjoint (Finset.filter (fun j => ↑j < i) univ) {{ val := i, isLt := h }}\n⊢ prod (take i (ofFn f)) * get (ofFn f) { val := i, isLt := this } =\n    ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    �� assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      ","nextTactic":"rw [A]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"case pos\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\nA : Finset.filter (fun j => ↑j < i + 1) univ = Finset.filter (fun j => ↑j < i) univ ∪ {{ val := i, isLt := h }}\nB : _root_.Disjoint (Finset.filter (fun j => ↑j < i) univ) {{ val := i, isLt := h }}\n⊢ prod (take i (ofFn f)) * get (ofFn f) { val := i, isLt := this } =\n    ∏ j in Finset.filter (fun j => ↑j < i) univ ∪ {{ val := i, isLt := h }}, f j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      ","nextTactic":"rw [Finset.prod_union B]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"case pos\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\nA : Finset.filter (fun j => ↑j < i + 1) univ = Finset.filter (fun j => ↑j < i) univ ∪ {{ val := i, isLt := h }}\nB : _root_.Disjoint (Finset.filter (fun j => ↑j < i) univ) {{ val := i, isLt := h }}\n⊢ prod (take i (ofFn f)) * get (ofFn f) { val := i, isLt := this } =\n    (∏ x in Finset.filter (fun j => ↑j < i) univ, f x) * ∏ x in {{ val := i, isLt := h }}, f x","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      ","nextTactic":"rw [IH]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"case pos\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\nA : Finset.filter (fun j => ↑j < i + 1) univ = Finset.filter (fun j => ↑j < i) univ ∪ {{ val := i, isLt := h }}\nB : _root_.Disjoint (Finset.filter (fun j => ↑j < i) univ) {{ val := i, isLt := h }}\n⊢ (∏ j in Finset.filter (fun j => ↑j < i) univ, f j) * get (ofFn f) { val := i, isLt := this } =\n    (∏ x in Finset.filter (fun j => ↑j < i) univ, f x) * ∏ x in {{ val := i, isLt := h }}, f x","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      ","nextTactic":"simp","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"case neg\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : ¬i < n\n⊢ prod (take (Nat.succ i) (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · ","nextTactic":"have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : ¬i < n\n⊢ take i (ofFn f) = take (Nat.succ i) (ofFn f)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        ","nextTactic":"rw [← length_ofFn f] at h","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : ¬i < length (ofFn f)\n⊢ take i (ofFn f) = take (Nat.succ i) (ofFn f)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        ","nextTactic":"have : length (ofFn f) ≤ i := not_lt.mp h","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : ¬i < length (ofFn f)\nthis : length (ofFn f) ≤ i\n⊢ take i (ofFn f) = take (Nat.succ i) (ofFn f)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        ","nextTactic":"rw [take_all_of_le this]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : ¬i < length (ofFn f)\nthis : length (ofFn f) ≤ i\n⊢ ofFn f = take (Nat.succ i) (ofFn f)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m ��� ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        ","nextTactic":"rw [take_all_of_le (le_trans this (Nat.le_succ _))]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"case neg\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : ¬i < n\nA : take i (ofFn f) = take (Nat.succ i) (ofFn f)\n⊢ prod (take (Nat.succ i) (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ��¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      ","nextTactic":"have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : ¬i < n\nA : take i (ofFn f) = take (Nat.succ i) (ofFn f)\n⊢ ∀ (j : Fin n), (↑j < Nat.succ i) = (↑j < i)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        ","nextTactic":"intro j","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : ¬i < n\nA : take i (ofFn f) = take (Nat.succ i) (ofFn f)\nj : Fin n\n⊢ (↑j < Nat.succ i) = (↑j < i)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        ","nextTactic":"have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : ¬i < n\nA : take i (ofFn f) = take (Nat.succ i) (ofFn f)\nj : Fin n\nthis : ↑j < i\n⊢ (↑j < Nat.succ i) = (↑j < i)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        ","nextTactic":"simp [this, lt_trans this (Nat.lt_succ_self _)]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"case neg\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\ni : ℕ\nIH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j\nh : ¬i < n\nA : take i (ofFn f) = take (Nat.succ i) (ofFn f)\nB : ∀ (j : Fin n), (↑j < Nat.succ i) = (↑j < i)\n⊢ prod (take (Nat.succ i) (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      ","nextTactic":"simp [← A, B, IH]","declUpToTactic":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.486_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j "}
+{"state":"α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\n⊢ prod (ofFn f) = ∏ i : Fin n, f i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      simp [← A, B, IH]\n#align list.prod_take_of_fn List.prod_take_ofFn\n#align list.sum_take_of_fn List.sum_take_ofFn\n\n@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by\n  ","nextTactic":"convert prod_take_ofFn f n","declUpToTactic":"@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.519_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i "}
+{"state":"case h.e'_2.h.e'_4\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\n⊢ ofFn f = take n (ofFn f)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid ��] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      simp [← A, B, IH]\n#align list.prod_take_of_fn List.prod_take_ofFn\n#align list.sum_take_of_fn List.sum_take_ofFn\n\n@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by\n  convert prod_take_ofFn f n\n  · ","nextTactic":"rw [take_all_of_le (le_of_eq (length_ofFn f))]","declUpToTactic":"@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by\n  convert prod_take_ofFn f n\n  · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.519_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i "}
+{"state":"case h.e'_3.h\nα : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\nf : Fin n → α\n⊢ univ = Finset.filter (fun j => ↑j < n) univ","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      simp [← A, B, IH]\n#align list.prod_take_of_fn List.prod_take_ofFn\n#align list.sum_take_of_fn List.sum_take_ofFn\n\n@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by\n  convert prod_take_ofFn f n\n  · rw [take_all_of_le (le_of_eq (length_ofFn f))]\n  · ","nextTactic":"simp","declUpToTactic":"@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by\n  convert prod_take_ofFn f n\n  · rw [take_all_of_le (le_of_eq (length_ofFn f))]\n  · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.519_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i "}
+{"state":"α : Type u_1\nβ : Type u_2\nG : Type u_3\ninst✝ : CommGroup G\n⊢ alternatingProd [] = ∏ i : Fin (length []), get [] i ^ (-1) ^ ↑i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      simp [← A, B, IH]\n#align list.prod_take_of_fn List.prod_take_ofFn\n#align list.sum_take_of_fn List.sum_take_ofFn\n\n@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by\n  convert prod_take_ofFn f n\n  · rw [take_all_of_le (le_of_eq (length_ofFn f))]\n  · simp\n#align list.prod_of_fn List.prod_ofFn\n#align list.sum_of_fn List.sum_ofFn\n\nend CommMonoid\n\n-- Porting note: Statement had deprecated `L.nthLe i i.is_lt` instead of `L.get i`.\n@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    ","nextTactic":"rw [alternatingProd]","declUpToTactic":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.530_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; simp\n  | g::h::L =>\n    calc g * h⁻¹ * L.alternatingProd\n      = g * h⁻¹ * ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ) "}
+{"state":"α : Type u_1\nβ : Type u_2\nG : Type u_3\ninst✝ : CommGroup G\n⊢ 1 = ∏ i : Fin (length []), get [] i ^ (-1) ^ ↑i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (�� i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → ��) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      simp [← A, B, IH]\n#align list.prod_take_of_fn List.prod_take_ofFn\n#align list.sum_take_of_fn List.sum_take_ofFn\n\n@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by\n  convert prod_take_ofFn f n\n  · rw [take_all_of_le (le_of_eq (length_ofFn f))]\n  · simp\n#align list.prod_of_fn List.prod_ofFn\n#align list.sum_of_fn List.sum_ofFn\n\nend CommMonoid\n\n-- Porting note: Statement had deprecated `L.nthLe i i.is_lt` instead of `L.get i`.\n@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    ","nextTactic":"rw [Finset.prod_eq_one]","declUpToTactic":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.530_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; simp\n  | g::h::L =>\n    calc g * h⁻¹ * L.alternatingProd\n      = g * h⁻¹ * ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ) "}
+{"state":"α : Type u_1\nβ : Type u_2\nG : Type u_3\ninst✝ : CommGroup G\n⊢ ∀ x ∈ univ, get [] x ^ (-1) ^ ↑x = 1","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      simp [← A, B, IH]\n#align list.prod_take_of_fn List.prod_take_ofFn\n#align list.sum_take_of_fn List.sum_take_ofFn\n\n@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by\n  convert prod_take_ofFn f n\n  · rw [take_all_of_le (le_of_eq (length_ofFn f))]\n  · simp\n#align list.prod_of_fn List.prod_ofFn\n#align list.sum_of_fn List.sum_ofFn\n\nend CommMonoid\n\n-- Porting note: Statement had deprecated `L.nthLe i i.is_lt` instead of `L.get i`.\n@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    ","nextTactic":"rintro ⟨i, ⟨⟩⟩","declUpToTactic":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.530_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; simp\n  | g::h::L =>\n    calc g * h⁻¹ * L.alternatingProd\n      = g * h⁻¹ * ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ) "}
+{"state":"α : Type u_1\nβ : Type u_2\nG : Type u_3\ninst✝ : CommGroup G\ng : G\n⊢ alternatingProd [g] = ∏ i : Fin (length [g]), get [g] i ^ (-1) ^ ↑i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      simp [← A, B, IH]\n#align list.prod_take_of_fn List.prod_take_ofFn\n#align list.sum_take_of_fn List.sum_take_ofFn\n\n@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by\n  convert prod_take_ofFn f n\n  · rw [take_all_of_le (le_of_eq (length_ofFn f))]\n  · simp\n#align list.prod_of_fn List.prod_ofFn\n#align list.sum_of_fn List.sum_ofFn\n\nend CommMonoid\n\n-- Porting note: Statement had deprecated `L.nthLe i i.is_lt` instead of `L.get i`.\n@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    ","nextTactic":"show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)","declUpToTactic":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.530_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; simp\n  | g::h::L =>\n    calc g * h⁻¹ * L.alternatingProd\n      = g * h⁻¹ * ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ) "}
+{"state":"α : Type u_1\nβ : Type u_2\nG : Type u_3\ninst✝ : CommGroup G\ng : G\n⊢ g = ∏ i : Fin 1, get [g] i ^ (-1) ^ ↑i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      simp [← A, B, IH]\n#align list.prod_take_of_fn List.prod_take_ofFn\n#align list.sum_take_of_fn List.sum_take_ofFn\n\n@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by\n  convert prod_take_ofFn f n\n  · rw [take_all_of_le (le_of_eq (length_ofFn f))]\n  · simp\n#align list.prod_of_fn List.prod_ofFn\n#align list.sum_of_fn List.sum_ofFn\n\nend CommMonoid\n\n-- Porting note: Statement had deprecated `L.nthLe i i.is_lt` instead of `L.get i`.\n@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    ","nextTactic":"rw [Fin.prod_univ_succ]","declUpToTactic":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.530_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; simp\n  | g::h::L =>\n    calc g * h⁻¹ * L.alternatingProd\n      = g * h⁻¹ * ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ) "}
+{"state":"α : Type u_1\nβ : Type u_2\nG : Type u_3\ninst✝ : CommGroup G\ng : G\n⊢ g = get [g] 0 ^ (-1) ^ ↑0 * ∏ i : Fin 0, get [g] (Fin.succ i) ^ (-1) ^ ↑(Fin.succ i)","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 �� β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + g���₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      simp [← A, B, IH]\n#align list.prod_take_of_fn List.prod_take_ofFn\n#align list.sum_take_of_fn List.sum_take_ofFn\n\n@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by\n  convert prod_take_ofFn f n\n  · rw [take_all_of_le (le_of_eq (length_ofFn f))]\n  · simp\n#align list.prod_of_fn List.prod_ofFn\n#align list.sum_of_fn List.sum_ofFn\n\nend CommMonoid\n\n-- Porting note: Statement had deprecated `L.nthLe i i.is_lt` instead of `L.get i`.\n@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; ","nextTactic":"simp","declUpToTactic":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.530_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; simp\n  | g::h::L =>\n    calc g * h⁻¹ * L.alternatingProd\n      = g * h⁻¹ * ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ) "}
+{"state":"α : Type u_1\nβ : Type u_2\nG : Type u_3\ninst✝ : CommGroup G\ng h : G\nL : List G\n⊢ g * h⁻¹ * ∏ i : Fin (length L), get L i ^ (-1) ^ ↑i = ∏ i : Fin (length L + 2), get (g :: h :: L) i ^ (-1) ^ ↑i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      simp [← A, B, IH]\n#align list.prod_take_of_fn List.prod_take_ofFn\n#align list.sum_take_of_fn List.sum_take_ofFn\n\n@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by\n  convert prod_take_ofFn f n\n  · rw [take_all_of_le (le_of_eq (length_ofFn f))]\n  · simp\n#align list.prod_of_fn List.prod_ofFn\n#align list.sum_of_fn List.sum_ofFn\n\nend CommMonoid\n\n-- Porting note: Statement had deprecated `L.nthLe i i.is_lt` instead of `L.get i`.\n@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; simp\n  | g::h::L =>\n    calc g * h⁻¹ * L.alternatingProd\n      = g * h⁻¹ * ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ) :=\n        congr_arg _ (alternatingProd_eq_finset_prod _)\n    _ = ∏ i : Fin (L.length + 2), List.get (g::h::L) i ^ (-1 : ℤ) ^ (i : ℕ) := by\n        ","nextTactic":"{ rw [Fin.prod_univ_succ, Fin.prod_univ_succ, mul_assoc]\n          simp [Nat.succ_eq_add_one, pow_add]}","declUpToTactic":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; simp\n  | g::h::L =>\n    calc g * h⁻¹ * L.alternatingProd\n      = g * h⁻¹ * ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ) :=\n        congr_arg _ (alternatingProd_eq_finset_prod _)\n    _ = ∏ i : Fin (L.length + 2), List.get (g::h::L) i ^ (-1 : ℤ) ^ (i : ℕ) := by\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.530_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; simp\n  | g::h::L =>\n    calc g * h⁻¹ * L.alternatingProd\n      = g * h⁻¹ * ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ) "}
+{"state":"α : Type u_1\nβ : Type u_2\nG : Type u_3\ninst✝ : CommGroup G\ng h : G\nL : List G\n⊢ g * h⁻¹ * ∏ i : Fin (length L), get L i ^ (-1) ^ ↑i = ∏ i : Fin (length L + 2), get (g :: h :: L) i ^ (-1) ^ ↑i","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      simp [← A, B, IH]\n#align list.prod_take_of_fn List.prod_take_ofFn\n#align list.sum_take_of_fn List.sum_take_ofFn\n\n@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by\n  convert prod_take_ofFn f n\n  · rw [take_all_of_le (le_of_eq (length_ofFn f))]\n  · simp\n#align list.prod_of_fn List.prod_ofFn\n#align list.sum_of_fn List.sum_ofFn\n\nend CommMonoid\n\n-- Porting note: Statement had deprecated `L.nthLe i i.is_lt` instead of `L.get i`.\n@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; simp\n  | g::h::L =>\n    calc g * h⁻¹ * L.alternatingProd\n      = g * h⁻¹ * ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ) :=\n        congr_arg _ (alternatingProd_eq_finset_prod _)\n    _ = ∏ i : Fin (L.length + 2), List.get (g::h::L) i ^ (-1 : ℤ) ^ (i : ℕ) := by\n        { ","nextTactic":"rw [Fin.prod_univ_succ, Fin.prod_univ_succ, mul_assoc]","declUpToTactic":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; simp\n  | g::h::L =>\n    calc g * h⁻¹ * L.alternatingProd\n      = g * h⁻¹ * ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ) :=\n        congr_arg _ (alternatingProd_eq_finset_prod _)\n    _ = ∏ i : Fin (L.length + 2), List.get (g::h::L) i ^ (-1 : ℤ) ^ (i : ℕ) := by\n        { ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.530_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; simp\n  | g::h::L =>\n    calc g * h⁻¹ * L.alternatingProd\n      = g * h⁻¹ * ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ) "}
+{"state":"α : Type u_1\nβ : Type u_2\nG : Type u_3\ninst✝ : CommGroup G\ng h : G\nL : List G\n⊢ g * (h⁻¹ * ∏ i : Fin (length L), get L i ^ (-1) ^ ↑i) =\n    get (g :: h :: L) 0 ^ (-1) ^ ↑0 *\n      (get (g :: h :: L) (Fin.succ 0) ^ (-1) ^ ↑(Fin.succ 0) *\n        ∏ i : Fin (length L), get (g :: h :: L) (Fin.succ (Fin.succ i)) ^ (-1) ^ ↑(Fin.succ (Fin.succ i)))","srcUpToTactic":"/-\nCopyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Anne Baanen\n-/\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Fintype.Fin\nimport Mathlib.Data.List.FinRange\nimport Mathlib.Logic.Equiv.Fin\n\n#align_import algebra.big_operators.fin from \"leanprover-community/mathlib\"@\"cc5dd6244981976cc9da7afc4eee5682b037a013\"\n\n/-!\n# Big operators and `Fin`\n\nSome results about products and sums over the type `Fin`.\n\nThe most important results are the induction formulas `Fin.prod_univ_castSucc`\nand `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a\nconstant function. These results have variants for sums instead of products.\n\n## Main declarations\n\n* `finFunctionFinEquiv`: An explicit equivalence between `Fin n → Fin m` and `Fin (m ^ n)`.\n-/\n\nopen BigOperators\n\nopen Finset\n\nvariable {α : Type*} {β : Type*}\n\nnamespace Finset\n\n@[to_additive]\ntheorem prod_range [CommMonoid β] {n : ℕ} (f : ℕ → β) :\n    ∏ i in Finset.range n, f i = ∏ i : Fin n, f i :=\n  (Fin.prod_univ_eq_prod_range _ _).symm\n#align finset.prod_range Finset.prod_range\n#align finset.sum_range Finset.sum_range\n\nend Finset\n\nnamespace Fin\n\n@[to_additive]\ntheorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :\n    ∏ i, f i = ((List.finRange n).map f).prod := by simp [univ_def]\n#align fin.prod_univ_def Fin.prod_univ_def\n#align fin.sum_univ_def Fin.sum_univ_def\n\n@[to_additive]\ntheorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by\n  rw [List.ofFn_eq_map]\n  rw [prod_univ_def]\n#align fin.prod_of_fn Fin.prod_ofFn\n#align fin.sum_of_fn Fin.sum_ofFn\n\n/-- A product of a function `f : Fin 0 → β` is `1` because `Fin 0` is empty -/\n@[to_additive \"A sum of a function `f : Fin 0 → β` is `0` because `Fin 0` is empty\"]\ntheorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=\n  rfl\n#align fin.prod_univ_zero Fin.prod_univ_zero\n#align fin.sum_univ_zero Fin.sum_univ_zero\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f x`, for some `x : Fin (n + 1)` plus the remaining product\"]\ntheorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :\n    ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by\n  rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding,\n    RelEmbedding.coe_toEmbedding]\n  rfl\n#align fin.prod_univ_succ_above Fin.prod_univ_succAbove\n#align fin.sum_univ_succ_above Fin.sum_univ_succAbove\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f 0` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f 0` plus the remaining product\"]\ntheorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=\n  prod_univ_succAbove f 0\n#align fin.prod_univ_succ Fin.prod_univ_succ\n#align fin.sum_univ_succ Fin.sum_univ_succ\n\n/-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`\nis the product of `f (Fin.last n)` plus the remaining product -/\n@[to_additive \"A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of\n`f (Fin.last n)` plus the remaining sum\"]\ntheorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :\n    ∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by\n  simpa [mul_comm] using prod_univ_succAbove f (last n)\n#align fin.prod_univ_cast_succ Fin.prod_univ_castSucc\n#align fin.sum_univ_cast_succ Fin.sum_univ_castSucc\n\n@[to_additive]\ntheorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :\n    (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by\n  simp_rw [prod_univ_succ, cons_zero, cons_succ]\n#align fin.prod_cons Fin.prod_cons\n#align fin.sum_cons Fin.sum_cons\n\n@[to_additive sum_univ_one]\ntheorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp\n#align fin.prod_univ_one Fin.prod_univ_one\n#align fin.sum_univ_one Fin.sum_univ_one\n\n@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by\n  simp [prod_univ_succ]\n#align fin.prod_univ_two Fin.prod_univ_two\n#align fin.sum_univ_two Fin.sum_univ_two\n\n@[to_additive]\ntheorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_two]\n  rfl\n#align fin.prod_univ_three Fin.prod_univ_three\n#align fin.sum_univ_three Fin.sum_univ_three\n\n@[to_additive]\ntheorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_three]\n  rfl\n#align fin.prod_univ_four Fin.prod_univ_four\n#align fin.sum_univ_four Fin.sum_univ_four\n\n@[to_additive]\ntheorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_four]\n  rfl\n#align fin.prod_univ_five Fin.prod_univ_five\n#align fin.sum_univ_five Fin.sum_univ_five\n\n@[to_additive]\ntheorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_five]\n  rfl\n#align fin.prod_univ_six Fin.prod_univ_six\n#align fin.sum_univ_six Fin.sum_univ_six\n\n@[to_additive]\ntheorem prod_univ_seven [CommMonoid β] (f : Fin 7 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_six]\n  rfl\n#align fin.prod_univ_seven Fin.prod_univ_seven\n#align fin.sum_univ_seven Fin.sum_univ_seven\n\n@[to_additive]\ntheorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :\n    ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by\n  rw [prod_univ_castSucc]\n  rw [prod_univ_seven]\n  rfl\n#align fin.prod_univ_eight Fin.prod_univ_eight\n#align fin.sum_univ_eight Fin.sum_univ_eight\n\ntheorem sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [CommSemiring R] (a b : R) :\n    (∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n := by\n  simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n#align fin.sum_pow_mul_eq_add_pow Fin.sum_pow_mul_eq_add_pow\n\ntheorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by simp\n#align fin.prod_const Fin.prod_const\n\ntheorem sum_const [AddCommMonoid α] (n : ℕ) (x : α) : ∑ _i : Fin n, x = n • x := by simp\n#align fin.sum_const Fin.sum_const\n\n@[to_additive]\ntheorem prod_Ioi_zero {M : Type*} [CommMonoid M] {n : ℕ} {v : Fin n.succ → M} :\n    ∏ i in Ioi 0, v i = ∏ j : Fin n, v j.succ := by\n  rw [Ioi_zero_eq_map]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_zero Fin.prod_Ioi_zero\n#align fin.sum_Ioi_zero Fin.sum_Ioi_zero\n\n@[to_additive]\ntheorem prod_Ioi_succ {M : Type*} [CommMonoid M] {n : ℕ} (i : Fin n) (v : Fin n.succ → M) :\n    ∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ := by\n  rw [Ioi_succ]\n  rw [Finset.prod_map]\n  rw [RelEmbedding.coe_toEmbedding]\n  rw [val_succEmbedding]\n#align fin.prod_Ioi_succ Fin.prod_Ioi_succ\n#align fin.sum_Ioi_succ Fin.sum_Ioi_succ\n\n@[to_additive]\ntheorem prod_congr' {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin b → M) (h : a = b) :\n    (∏ i : Fin a, f (cast h i)) = ∏ i : Fin b, f i := by\n  subst h\n  congr\n#align fin.prod_congr' Fin.prod_congr'\n#align fin.sum_congr' Fin.sum_congr'\n\n@[to_additive]\ntheorem prod_univ_add {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M) :\n    (∏ i : Fin (a + b), f i) = (∏ i : Fin a, f (castAdd b i)) * ∏ i : Fin b, f (natAdd a i) := by\n  rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n  · apply Fintype.prod_sum_type\n  · intro x\n    simp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n#align fin.prod_univ_add Fin.prod_univ_add\n#align fin.sum_univ_add Fin.sum_univ_add\n\n@[to_additive]\ntheorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)\n    (hf : ∀ j : Fin b, f (natAdd a j) = 1) :\n    (∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by\n  rw [prod_univ_add]\n  rw [Fintype.prod_eq_one _ hf]\n  rw [mul_one]\n  rfl\n#align fin.prod_trunc Fin.prod_trunc\n#align fin.sum_trunc Fin.sum_trunc\n\nsection PartialProd\n\nvariable [Monoid α] {n : ℕ}\n\n/-- For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialProd f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. -/\n@[to_additive \"For `f = (a₁, ..., aₙ)` in `αⁿ`, `partialSum f` is\\n\n`(0, a₁, a₁ + a₂, ..., a₁ + ... + aₙ)` in `αⁿ⁺¹`.\"]\ndef partialProd (f : Fin n → α) (i : Fin (n + 1)) : α :=\n  ((List.ofFn f).take i).prod\n#align fin.partial_prod Fin.partialProd\n#align fin.partial_sum Fin.partialSum\n\n@[to_additive (attr := simp)]\ntheorem partialProd_zero (f : Fin n → α) : partialProd f 0 = 1 := by simp [partialProd]\n#align fin.partial_prod_zero Fin.partialProd_zero\n#align fin.partial_sum_zero Fin.partialSum_zero\n\n@[to_additive]\ntheorem partialProd_succ (f : Fin n → α) (j : Fin n) :\n    partialProd f j.succ = partialProd f (Fin.castSucc j) * f j := by\n  simp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, ← Option.coe_def]\n#align fin.partial_prod_succ Fin.partialProd_succ\n#align fin.partial_sum_succ Fin.partialSum_succ\n\n@[to_additive]\ntheorem partialProd_succ' (f : Fin (n + 1) → α) (j : Fin (n + 1)) :\n    partialProd f j.succ = f 0 * partialProd (Fin.tail f) j := by\n  simp [partialProd]\n  rfl\n#align fin.partial_prod_succ' Fin.partialProd_succ'\n#align fin.partial_sum_succ' Fin.partialSum_succ'\n\n@[to_additive]\ntheorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :\n    (f 0 • partialProd fun i : Fin n => (f i)⁻¹ * f i.succ) = f :=\n  funext fun x => Fin.inductionOn x (by simp) fun x hx => by\n    simp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx ⊢\n    rw [partialProd_succ]\n    rw [← mul_assoc]\n    rw [hx]\n    rw [mul_inv_cancel_left]\n#align fin.partial_prod_left_inv Fin.partialProd_left_inv\n#align fin.partial_sum_left_neg Fin.partialSum_left_neg\n\n@[to_additive]\ntheorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :\n    (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by\n  cases' i with i hn\n  induction i with\n  | zero => simp [-Fin.succ_mk, partialProd_succ]\n  | succ i hi =>\n    specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢\n    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn)]\n    rw [← Fin.succ_mk]\n    rw [Nat.succ_eq_add_one] at hn\n    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n    -- assoc_rw [hi, inv_mul_cancel_left]\n    rw [← mul_assoc]\n    rw [mul_left_eq_self]\n    rw [mul_assoc]\n    rw [hi]\n    rw [mul_left_inv]\n#align fin.partial_prod_right_inv Fin.partialProd_right_inv\n#align fin.partial_sum_right_neg Fin.partialSum_right_neg\n\n/-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\nThen if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.\nIf `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.\nIf `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`\nUseful for defining group cohomology. -/\n@[to_additive\n      \"Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.\n      Then if `k < j`, this says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ) = gₖ`.\n      If `k = j`, it says `-(g₀ + g₁ + ... + gₖ₋₁) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ + gₖ₊₁`.\n      If `k > j`, it says `-(g₀ + g₁ + ... + gₖ) + (g₀ + g₁ + ... + gₖ₊₁) = gₖ₊₁.`\n      Useful for defining group cohomology.\"]\ntheorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)\n    (j : Fin (n + 1)) (k : Fin n) :\n    (partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =\n      j.contractNth (· * ·) g k := by\n  rcases lt_trichotomy (k : ℕ) j with (h | h | h)\n  · rwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n    · assumption\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_lt h\n  · rwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, ← mul_assoc,\n      partialProd_right_inv, contractNth_apply_of_eq]\n    · simp [le_iff_val_le_val, ← h]\n    · rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n      exact le_of_eq h\n  · rwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ,\n      castSucc_fin_succ, partialProd_succ, inv_mul_cancel_left, contractNth_apply_of_gt]\n    · exact le_iff_val_le_val.2 (le_of_lt h)\n    · rw [le_iff_val_le_val, val_succ]\n      exact Nat.succ_le_of_lt h\n#align fin.inv_partial_prod_mul_eq_contract_nth Fin.inv_partialProd_mul_eq_contractNth\n#align fin.neg_partial_sum_add_eq_contract_nth Fin.neg_partialSum_add_eq_contractNth\n\nend PartialProd\n\nend Fin\n\n/-- Equivalence between `Fin n → Fin m` and `Fin (m ^ n)`. -/\n@[simps!]\ndef finFunctionFinEquiv {m n : ℕ} : (Fin n → Fin m) ≃ Fin (m ^ n) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_fun, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * m ^ (i : ℕ), by\n      induction' n with n ih\n      · simp\n      cases m\n      · dsimp only [Nat.zero_eq] at f -- porting note: added, wrong zero\n        exact isEmptyElim (f <| Fin.last _)\n      simp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [add_comm]\n      rw [pow_succ]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨a / m ^ (b : ℕ) % m, by\n      cases' n with n\n      · exact b.elim0\n      cases' m with m\n      · dsimp only [Nat.zero_eq] at a -- porting note: added, wrong zero\n        rw [zero_pow n.succ_pos] at a\n        exact a.elim0\n      · exact Nat.mod_lt _ m.succ_pos⟩)\n    fun a => by\n      dsimp\n      induction' n with n ih\n      · haveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n      ext\n      simp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one,\n        mul_one, pow_succ, ← Nat.div_div_eq_div_mul, mul_left_comm _ m, ← mul_sum]\n      rw [ih _ (Nat.div_lt_of_lt_mul ?_)]\n      rw [Nat.mod_add_div]\n      -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n      -- instance for `Fin`.\n      exact a.is_lt.trans_eq (pow_succ _ _)\n#align fin_function_fin_equiv finFunctionFinEquiv\n\ntheorem finFunctionFinEquiv_apply {m n : ℕ} (f : Fin n → Fin m) :\n    (finFunctionFinEquiv f : ℕ) = ∑ i : Fin n, ↑(f i) * m ^ (i : ℕ) :=\n  rfl\n#align fin_function_fin_equiv_apply finFunctionFinEquiv_apply\n\ntheorem finFunctionFinEquiv_single {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) :\n    (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) := by\n  rw [finFunctionFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_function_fin_equiv_single finFunctionFinEquiv_single\n\n/-- Equivalence between `∀ i : Fin m, Fin (n i)` and `Fin (∏ i : Fin m, n i)`. -/\ndef finPiFinEquiv {m : ℕ} {n : Fin m → ℕ} : (∀ i : Fin m, Fin (n i)) ≃ Fin (∏ i : Fin m, n i) :=\n  Equiv.ofRightInverseOfCardLE (le_of_eq <| by simp_rw [Fintype.card_pi, Fintype.card_fin])\n    (fun f => ⟨∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j), by\n      induction' m with m ih\n      · simp\n      rw [Fin.prod_univ_castSucc]\n      rw [Fin.sum_univ_castSucc]\n      suffices\n        ∀ (n : Fin m → ℕ) (nn : ℕ) (f : ∀ i : Fin m, Fin (n i)) (fn : Fin nn),\n          ((∑ i : Fin m, ↑(f i) * ∏ j : Fin i, n (Fin.castLE i.prop.le j)) + ↑fn * ∏ j, n j) <\n            (∏ i : Fin m, n i) * nn by\n        replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n        rw [← Fin.snoc_init_self f]\n        simp (config := { singlePass := true }) only [← Fin.snoc_init_self n]\n        simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n        exact this\n      intro n nn f fn\n      cases nn\n      · dsimp only [Nat.zero_eq] at fn -- porting note: added, wrong zero\n        exact isEmptyElim fn\n      refine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n      rw [← one_add_mul (_ : ℕ)]\n      rw [mul_comm]\n      rw [add_comm]\n      -- porting note: added, wrong `succ`\n      rfl⟩)\n    (fun a b => ⟨(a / ∏ j : Fin b, n (Fin.castLE b.is_lt.le j)) % n b, by\n      cases m\n      · exact b.elim0\n      cases' h : n b with nb\n      · rw [prod_eq_zero (Finset.mem_univ _) h] at a\n        exact isEmptyElim a\n      exact Nat.mod_lt _ nb.succ_pos⟩)\n    (by\n      intro a; revert a; dsimp only [Fin.val_mk]\n      refine' Fin.consInduction _ _ n\n      · intro a\n        haveI : Subsingleton (Fin (∏ i : Fin 0, i.elim0)) :=\n          (Fin.castIso <| prod_empty).toEquiv.subsingleton\n        exact Subsingleton.elim _ _\n      · intro n x xs ih a\n        simp_rw [Fin.forall_iff, Fin.ext_iff] at ih\n        ext\n        simp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n        have := fun i : Fin n =>\n          Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))\n            (fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))\n            fun j => rfl\n        simp_rw [this]\n        clear this\n        dsimp only [Fin.val_zero]\n        simp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n        change (_ + ∑ y : _, _ / (x * _) % _ * (x * _)) = _\n        simp_rw [← Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : ℕ), ← mul_sum]\n        convert Nat.mod_add_div _ _\n        -- porting note: new\n        refine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n        exact Fin.prod_univ_succ _\n        -- porting note: was:\n        /-\n        refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n        swap\n        · convert Fin.prod_univ_succ (Fin.cons x xs : ∀ _, ℕ)\n          simp_rw [Fin.cons_succ]\n        congr with i\n        congr with j\n        · cases j\n          rfl\n        · cases j\n          rfl-/)\n#align fin_pi_fin_equiv finPiFinEquiv\n\ntheorem finPiFinEquiv_apply {m : ℕ} {n : Fin m → ℕ} (f : ∀ i : Fin m, Fin (n i)) :\n    (finPiFinEquiv f : ℕ) = ∑ i, f i * ∏ j, n (Fin.castLE i.is_lt.le j) := rfl\n#align fin_pi_fin_equiv_apply finPiFinEquiv_apply\n\ntheorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)\n    (j : Fin (n i)) :\n    (finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =\n      j * ∏ j, n (Fin.castLE i.is_lt.le j) := by\n  rw [finPiFinEquiv_apply]\n  rw [Fintype.sum_eq_single i]\n  rw [Pi.single_eq_same]\n  rintro x hx\n  rw [Pi.single_eq_of_ne hx]\n  rw [Fin.val_zero']\n  rw [zero_mul]\n#align fin_pi_fin_equiv_single finPiFinEquiv_single\n\nnamespace List\n\nsection CommMonoid\n\nvariable [CommMonoid α]\n\n@[to_additive]\ntheorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :\n    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by\n  induction i with\n  | zero =>\n    simp\n  | succ i IH =>\n    by_cases h : i < n\n    · have : i < length (ofFn f) := by rwa [length_ofFn f]\n      rw [prod_take_succ _ _ this]\n      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by\n        ext ⟨_, _⟩\n        simp [Nat.lt_succ_iff_lt_or_eq]\n      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)\n          (singleton (⟨i, h⟩ : Fin n)) := by simp\n      rw [A]\n      rw [Finset.prod_union B]\n      rw [IH]\n      simp\n    · have A : (ofFn f).take i = (ofFn f).take i.succ := by\n        rw [← length_ofFn f] at h\n        have : length (ofFn f) ≤ i := not_lt.mp h\n        rw [take_all_of_le this]\n        rw [take_all_of_le (le_trans this (Nat.le_succ _))]\n      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by\n        intro j\n        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n        simp [this, lt_trans this (Nat.lt_succ_self _)]\n      simp [← A, B, IH]\n#align list.prod_take_of_fn List.prod_take_ofFn\n#align list.sum_take_of_fn List.sum_take_ofFn\n\n@[to_additive]\ntheorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by\n  convert prod_take_ofFn f n\n  · rw [take_all_of_le (le_of_eq (length_ofFn f))]\n  · simp\n#align list.prod_of_fn List.prod_ofFn\n#align list.sum_of_fn List.sum_ofFn\n\nend CommMonoid\n\n-- Porting note: Statement had deprecated `L.nthLe i i.is_lt` instead of `L.get i`.\n@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; simp\n  | g::h::L =>\n    calc g * h⁻¹ * L.alternatingProd\n      = g * h⁻¹ * ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ) :=\n        congr_arg _ (alternatingProd_eq_finset_prod _)\n    _ = ∏ i : Fin (L.length + 2), List.get (g::h::L) i ^ (-1 : ℤ) ^ (i : ℕ) := by\n        { rw [Fin.prod_univ_succ, Fin.prod_univ_succ, mul_assoc]\n          ","nextTactic":"simp [Nat.succ_eq_add_one, pow_add]","declUpToTactic":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; simp\n  | g::h::L =>\n    calc g * h⁻¹ * L.alternatingProd\n      = g * h⁻¹ * ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ) :=\n        congr_arg _ (alternatingProd_eq_finset_prod _)\n    _ = ∏ i : Fin (L.length + 2), List.get (g::h::L) i ^ (-1 : ℤ) ^ (i : ℕ) := by\n        { rw [Fin.prod_univ_succ, Fin.prod_univ_succ, mul_assoc]\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Fin.530_0.QyPgLR0eRR4fDXZ","decl":"@[to_additive]\ntheorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :\n    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)\n  | [] => by\n    rw [alternatingProd]\n    rw [Finset.prod_eq_one]\n    rintro ⟨i, ⟨⟩⟩\n  | g::[] => by\n    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)\n    rw [Fin.prod_univ_succ]; simp\n  | g::h::L =>\n    calc g * h⁻¹ * L.alternatingProd\n      = g * h⁻¹ * ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ) "}