Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_BigOperators_NatAntidiagonal.jsonl

{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ ∏ p in antidiagonal (n + 1), f p = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  ","nextTactic":"rw [antidiagonal_succ, prod_cons, prod_map]","declUpToTactic":"theorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.25_0.3fmBgDgzBSF6Gqi","decl":"theorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) "}
{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ f (0, n + 1) *\n      ∏ x in antidiagonal n,\n        f\n          ((Function.Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Function.Embedding.refl ℕ))\n            x) =\n    f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; ","nextTactic":"rfl","declUpToTactic":"theorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.25_0.3fmBgDgzBSF6Gqi","decl":"theorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) "}
{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ ∏ p in antidiagonal n, f (Prod.swap p) = ∏ p in antidiagonal n, f p","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  ","nextTactic":"conv_lhs => rw [← map_swap_antidiagonal, Finset.prod_map]","declUpToTactic":"@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.36_0.3fmBgDgzBSF6Gqi","decl":"@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p "}
{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n| ∏ p in antidiagonal n, f (Prod.swap p)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => ","nextTactic":"rw [← map_swap_antidiagonal, Finset.prod_map]","declUpToTactic":"@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.36_0.3fmBgDgzBSF6Gqi","decl":"@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p "}
{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n| ∏ p in antidiagonal n, f (Prod.swap p)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => ","nextTactic":"rw [← map_swap_antidiagonal, Finset.prod_map]","declUpToTactic":"@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.36_0.3fmBgDgzBSF6Gqi","decl":"@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p "}
{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n| ∏ p in antidiagonal n, f (Prod.swap p)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => ","nextTactic":"rw [← map_swap_antidiagonal, Finset.prod_map]","declUpToTactic":"@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.36_0.3fmBgDgzBSF6Gqi","decl":"@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p "}
{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ ∏ p in antidiagonal (n + 1), f p = f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => rw [← map_swap_antidiagonal, Finset.prod_map]\n#align finset.nat.prod_antidiagonal_swap Finset.Nat.prod_antidiagonal_swap\n#align finset.nat.sum_antidiagonal_swap Finset.Nat.sum_antidiagonal_swap\n\ntheorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  ","nextTactic":"rw [← prod_antidiagonal_swap]","declUpToTactic":"theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.43_0.3fmBgDgzBSF6Gqi","decl":"theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) "}
{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ ∏ p in antidiagonal (n + 1), f (Prod.swap p) = f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => rw [← map_swap_antidiagonal, Finset.prod_map]\n#align finset.nat.prod_antidiagonal_swap Finset.Nat.prod_antidiagonal_swap\n#align finset.nat.sum_antidiagonal_swap Finset.Nat.sum_antidiagonal_swap\n\ntheorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  rw [← prod_antidiagonal_swap]\n  ","nextTactic":"rw [prod_antidiagonal_succ]","declUpToTactic":"theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  rw [← prod_antidiagonal_swap]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.43_0.3fmBgDgzBSF6Gqi","decl":"theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) "}
{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ f (Prod.swap (0, n + 1)) * ∏ p in antidiagonal n, f (Prod.swap (p.1 + 1, p.2)) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => rw [← map_swap_antidiagonal, Finset.prod_map]\n#align finset.nat.prod_antidiagonal_swap Finset.Nat.prod_antidiagonal_swap\n#align finset.nat.sum_antidiagonal_swap Finset.Nat.sum_antidiagonal_swap\n\ntheorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  rw [← prod_antidiagonal_swap]\n  rw [prod_antidiagonal_succ]\n  ","nextTactic":"rw [← prod_antidiagonal_swap]","declUpToTactic":"theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  rw [← prod_antidiagonal_swap]\n  rw [prod_antidiagonal_succ]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.43_0.3fmBgDgzBSF6Gqi","decl":"theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) "}
{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ f (Prod.swap (0, n + 1)) * ∏ p in antidiagonal n, f (Prod.swap ((Prod.swap p).1 + 1, (Prod.swap p).2)) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => rw [← map_swap_antidiagonal, Finset.prod_map]\n#align finset.nat.prod_antidiagonal_swap Finset.Nat.prod_antidiagonal_swap\n#align finset.nat.sum_antidiagonal_swap Finset.Nat.sum_antidiagonal_swap\n\ntheorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  rw [← prod_antidiagonal_swap]\n  rw [prod_antidiagonal_succ]\n  rw [← prod_antidiagonal_swap]\n  ","nextTactic":"rfl","declUpToTactic":"theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  rw [← prod_antidiagonal_swap]\n  rw [prod_antidiagonal_succ]\n  rw [← prod_antidiagonal_swap]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.43_0.3fmBgDgzBSF6Gqi","decl":"theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) "}
{"state":"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → ℕ → M\np : ℕ × ℕ\nhp : p ∈ antidiagonal n\n⊢ f p n = f p (p.1 + p.2)","srcUpToTactic":"/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Data.Finset.NatAntidiagonal\nimport Mathlib.Algebra.BigOperators.Basic\n\n#align_import algebra.big_operators.nat_antidiagonal from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Big operators for `NatAntidiagonal`\n\nThis file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.\n-/\n\nopen BigOperators\n\nvariable {M N : Type*} [CommMonoid M] [AddCommMonoid N]\n\nnamespace Finset\n\nnamespace Nat\n\ntheorem prod_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → M} :\n    (∏ p in antidiagonal (n + 1), f p)\n      = f (0, n + 1) * ∏ p in antidiagonal n, f (p.1 + 1, p.2) := by\n  rw [antidiagonal_succ, prod_cons, prod_map]; rfl\n#align finset.nat.prod_antidiagonal_succ Finset.Nat.prod_antidiagonal_succ\n\ntheorem sum_antidiagonal_succ {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (0, n + 1) + ∑ p in antidiagonal n, f (p.1 + 1, p.2) :=\n  @prod_antidiagonal_succ (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ Finset.Nat.sum_antidiagonal_succ\n\n@[to_additive]\ntheorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :\n    ∏ p in antidiagonal n, f p.swap = ∏ p in antidiagonal n, f p := by\n  conv_lhs => rw [← map_swap_antidiagonal, Finset.prod_map]\n#align finset.nat.prod_antidiagonal_swap Finset.Nat.prod_antidiagonal_swap\n#align finset.nat.sum_antidiagonal_swap Finset.Nat.sum_antidiagonal_swap\n\ntheorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p in antidiagonal (n + 1), f p) =\n    f (n + 1, 0) * ∏ p in antidiagonal n, f (p.1, p.2 + 1) := by\n  rw [← prod_antidiagonal_swap]\n  rw [prod_antidiagonal_succ]\n  rw [← prod_antidiagonal_swap]\n  rfl\n#align finset.nat.prod_antidiagonal_succ' Finset.Nat.prod_antidiagonal_succ'\n\ntheorem sum_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → N} :\n    (∑ p in antidiagonal (n + 1), f p) = f (n + 1, 0) + ∑ p in antidiagonal n, f (p.1, p.2 + 1) :=\n  @prod_antidiagonal_succ' (Multiplicative N) _ _ _\n#align finset.nat.sum_antidiagonal_succ' Finset.Nat.sum_antidiagonal_succ'\n\n@[to_additive]\ntheorem prod_antidiagonal_subst {n : ℕ} {f : ℕ × ℕ → ℕ → M} :\n    ∏ p in antidiagonal n, f p n = ∏ p in antidiagonal n, f p (p.1 + p.2) :=\n  prod_congr rfl fun p hp ↦ by ","nextTactic":"rw [mem_antidiagonal.mp hp]","declUpToTactic":"@[to_additive]\ntheorem prod_antidiagonal_subst {n : ℕ} {f : ℕ × ℕ → ℕ → M} :\n    ∏ p in antidiagonal n, f p n = ∏ p in antidiagonal n, f p (p.1 + p.2) :=\n  prod_congr rfl fun p hp ↦ by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_NatAntidiagonal.56_0.3fmBgDgzBSF6Gqi","decl":"@[to_additive]\ntheorem prod_antidiagonal_subst {n : ℕ} {f : ℕ × ℕ → ℕ → M} :\n    ∏ p in antidiagonal n, f p n = ∏ p in antidiagonal n, f p (p.1 + p.2) "}