diff --git "a/Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_BigOperators_Intervals.jsonl" "b/Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_BigOperators_Intervals.jsonl" new file mode 100644--- /dev/null +++ "b/Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_BigOperators_Intervals.jsonl" @@ -0,0 +1,114 @@ +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝ : α\ng f✝ : α → β\ninst✝³ : CommMonoid β\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\nf : α → β\na b c : α\n⊢ ∏ x in Ico a b, f (x + c) = ∏ x in Ico (a + c) (b + c), f x","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n ","nextTactic":"rw [← map_add_right_Ico]","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.33_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝ : α\ng f✝ : α → β\ninst✝³ : CommMonoid β\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\nf : α → β\na b c : α\n⊢ ∏ x in Ico a b, f (x + c) = ∏ x in map (addRightEmbedding c) (Ico a b), f x","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n ","nextTactic":"rw [prod_map]","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.33_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝ : α\ng f✝ : α → β\ninst✝³ : CommMonoid β\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\nf : α → β\na b c : α\n⊢ ∏ x in Ico a b, f (x + c) = ∏ x in Ico a b, f ((addRightEmbedding c) x)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n ","nextTactic":"rfl","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.33_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝ : α\ng f✝ : α → β\ninst✝³ : CommMonoid β\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\nf : α → β\na b c : α\n⊢ ∏ x in Ico a b, f (c + x) = ∏ x in Ico (a + c) (b + c), f x","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n ","nextTactic":"convert prod_Ico_add' f a b c using 2","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.42_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x "} +{"state":"case h.e'_2.a\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝¹ : α\ng f✝ : α → β\ninst✝³ : CommMonoid β\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\nf : α → β\na b c x✝ : α\na✝ : x✝ ∈ Ico a b\n⊢ f (c + x✝) = f (x✝ + c)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n ","nextTactic":"rw [add_comm]","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.42_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝ : α\ng f✝ : α → β\ninst✝ : CommMonoid β\na b : ℕ\nhab : a ≤ b\nf : ℕ → β\n⊢ ∏ k in Ico a (b + 1), f k = (∏ k in Ico a b, f k) * f b","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n ","nextTactic":"rw [Nat.Ico_succ_right_eq_insert_Ico hab]","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.50_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝ : α\ng f✝ : α → β\ninst✝ : CommMonoid β\na b : ℕ\nhab : a ≤ b\nf : ℕ → β\n⊢ ∏ k in insert b (Ico a b), f k = (∏ k in Ico a b, f k) * f b","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n ","nextTactic":"rw [prod_insert right_not_mem_Ico]","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.50_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝ : α\ng f✝ : α → β\ninst✝ : CommMonoid β\na b : ℕ\nhab : a ≤ b\nf : ℕ → β\n⊢ f b * ∏ x in Ico a b, f x = (∏ k in Ico a b, f k) * f b","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n ","nextTactic":"rw [mul_comm]","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.50_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝ : α\ng f✝ : α → β\ninst✝ : CommMonoid β\na b : ℕ\nhab : a < b\nf : ℕ → β\n⊢ ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n ","nextTactic":"have ha : a ∉ Ico (a + 1) b := by simp","declUpToTactic":"@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.59_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝ : α\ng f✝ : α → β\ninst✝ : CommMonoid β\na b : ℕ\nhab : a < b\nf : ℕ → β\n⊢ a ∉ Ico (a + 1) b","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by ","nextTactic":"simp","declUpToTactic":"@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.59_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝ : α\ng f✝ : α → β\ninst✝ : CommMonoid β\na b : ℕ\nhab : a < b\nf : ℕ → β\nha : a ∉ Ico (a + 1) b\n⊢ ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n ","nextTactic":"rw [← prod_insert ha]","declUpToTactic":"@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.59_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝ : α\ng f✝ : α → β\ninst✝ : CommMonoid β\na b : ℕ\nhab : a < b\nf : ℕ → β\nha : a ∉ Ico (a + 1) b\n⊢ ∏ k in Ico a b, f k = ∏ x in insert a (Ico (a + 1) b), f x","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n ","nextTactic":"rw [Nat.Ico_insert_succ_left hab]","declUpToTactic":"@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.59_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n k : ℕ\nhmn : m ≤ n\nhnk : n ≤ k\n⊢ (∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i = ∏ i in Ioc m k, f i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n ","nextTactic":"rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]","declUpToTactic":"@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.75_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n k : ℕ\nhmn : m ≤ n\nhnk : n ≤ k\n⊢ (∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i = ∏ i in Ioc m n ∪ Ioc n k, f i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n ","nextTactic":"rw [prod_union]","declUpToTactic":"@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.75_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n k : ℕ\nhmn : m ≤ n\nhnk : n ≤ k\n⊢ Disjoint (Ioc m n) (Ioc n k)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ��� i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n ","nextTactic":"apply disjoint_left.2 fun x hx h'x => _","declUpToTactic":"@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.75_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n k : ℕ\nhmn : m ≤ n\nhnk : n ≤ k\n⊢ ∀ x ∈ Ioc m n, x ∈ Ioc n k → False","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n ","nextTactic":"intros x hx h'x","declUpToTactic":"@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.75_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n k : ℕ\nhmn : m ≤ n\nhnk : n ≤ k\nx : ℕ\nhx : x ∈ Ioc m n\nh'x : x ∈ Ioc n k\n⊢ False","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n ","nextTactic":"exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)","declUpToTactic":"@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.75_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝ : α\ng f✝ : α → β\ninst✝ : CommMonoid β\na b : ℕ\nhab : a ≤ b\nf : ℕ → β\n⊢ ∏ k in Ioc a (b + 1), f k = (∏ k in Ioc a b, f k) * f (b + 1)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n ","nextTactic":"rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]","declUpToTactic":"@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.86_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝ : α\ng f✝ : α → β\ninst✝ : CommMonoid β\na b : ℕ\nhab : a ≤ b\nf : ℕ → β\n⊢ (∏ i in Ioc a b, f i) * ∏ i in Ioc b (succ b), f i = (∏ k in Ioc a b, f k) * f (b + 1)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n ","nextTactic":"rw [Nat.Ioc_succ_singleton]","declUpToTactic":"@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.86_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝ : α\ng f✝ : α → β\ninst✝ : CommMonoid β\na b : ℕ\nhab : a ≤ b\nf : ℕ → β\n⊢ (∏ i in Ioc a b, f i) * ∏ i in {b + 1}, f i = (∏ k in Ioc a b, f k) * f (b + 1)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n ","nextTactic":"rw [prod_singleton]","declUpToTactic":"@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.86_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝¹ : CommMonoid β\nδ : Type u_1\ninst✝ : CommGroup δ\nf : ℕ → δ\nm n : ℕ\nh : m ≤ n\n⊢ (∏ k in Ico m n, f k) * ∏ k in range m, f k = ∏ k in range n, f k","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (","nextTactic":"rw [mul_comm]","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.102_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝¹ : CommMonoid β\nδ : Type u_1\ninst✝ : CommGroup δ\nf : ℕ → δ\nm n : ℕ\nh : m ≤ n\n⊢ (∏ k in range m, f k) * ∏ k in Ico m n, f k = ∏ k in range n, f k","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; ","nextTactic":"exact prod_range_mul_prod_Ico f h","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.102_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝¹ : CommMonoid β\nδ : Type u_1\ninst✝ : CommGroup δ\nf : ℕ → δ\nm n : ℕ\nh : m ≤ n\n⊢ ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n ","nextTactic":"simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.109_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k "} +{"state":"α✝ : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α✝\na : α✝\ng f✝ : α✝ → β\ninst✝¹ : CommMonoid β\nα : Type u_1\ninst✝ : CommGroup α\nf : ℕ → α\nn m : ℕ\nhnm : n ≤ m\n⊢ (∏ k in range m, f k) / ∏ k in range n, f k = ∏ k in filter (fun k => n ≤ k) (range m), f k","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n ","nextTactic":"rw [← prod_Ico_eq_div f hnm]","declUpToTactic":"@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.116_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k "} +{"state":"α✝ : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α✝\na : α✝\ng f✝ : α✝ → β\ninst✝¹ : CommMonoid β\nα : Type u_1\ninst✝ : CommGroup α\nf : ℕ → α\nn m : ℕ\nhnm : n ≤ m\n⊢ ∏ k in Ico n m, f k = ∏ k in filter (fun k => n ≤ k) (range m), f k","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n ","nextTactic":"congr","declUpToTactic":"@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.116_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k "} +{"state":"case e_s\nα✝ : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α✝\na : α✝\ng f✝ : α✝ → β\ninst✝¹ : CommMonoid β\nα : Type u_1\ninst✝ : CommGroup α\nf : ℕ → α\nn m : ℕ\nhnm : n ≤ m\n⊢ Ico n m = filter (fun k => n ≤ k) (range m)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n ","nextTactic":"apply Finset.ext","declUpToTactic":"@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.116_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k "} +{"state":"case e_s.a\nα✝ : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α✝\na : α✝\ng f✝ : α✝ → β\ninst✝¹ : CommMonoid β\nα : Type u_1\ninst✝ : CommGroup α\nf : ℕ → α\nn m : ℕ\nhnm : n ≤ m\n⊢ ∀ (a : ℕ), a ∈ Ico n m ↔ a ∈ filter (fun k => n ≤ k) (range m)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n ","nextTactic":"simp only [mem_Ico, mem_filter, mem_range, *]","declUpToTactic":"@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.116_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k "} +{"state":"case e_s.a\nα✝ : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α✝\na : α✝\ng f✝ : α✝ → β\ninst✝¹ : CommMonoid β\nα : Type u_1\ninst✝ : CommGroup α\nf : ℕ → α\nn m : ℕ\nhnm : n ≤ m\n⊢ ∀ (a : ℕ), n ≤ a ∧ a < m ↔ a < m ∧ n ≤ a","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n ","nextTactic":"tauto","declUpToTactic":"@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.116_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝ : α\ng f✝ : α → β\ninst✝¹ : CommMonoid β\nM : Type u_1\ninst✝ : AddCommMonoid M\na b : ℕ\nf : ℕ → ℕ → M\n⊢ ∑ i in Ico a b, ∑ j in Ico i b, f i j = ∑ j in Ico a b, ∑ i in Ico a (j + 1), f i j","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j Ico i b, f x.fst x.snd = ∑ j in Ico a b, ∑ i in Ico a (j + 1), f i j","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j Ico i b, f x.fst x.snd =\n ∑ x in Finset.sigma (Ico a b) fun j => Ico a (j + 1), f x.snd x.fst","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl))","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j Ico i b),\n (fun x x_1 => { fst := x.snd, snd := x.fst }) ((fun x x_1 => { fst := x.snd, snd := x.fst }) a_1 ha)\n (_ : (fun x x_1 => { fst := x.snd, snd := x.fst }) a_1 ha ∈ ?m.27223) =\n a_1","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (","nextTactic":"rintro ⟨⟩ _","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j Ico i b\n⊢ (fun x x_1 => { fst := x.snd, snd := x.fst })\n ((fun x x_1 => { fst := x.snd, snd := x.fst }) { fst := fst✝, snd := snd✝ } ha✝)\n (_ : (fun x x_1 => { fst := x.snd, snd := x.fst }) { fst := fst✝, snd := snd✝ } ha✝ ∈ ?m.27223) =\n { fst := fst✝, snd := snd✝ }","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; ","nextTactic":"rfl","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j Ico a (j + 1)),\n (fun x x_1 => { fst := x.snd, snd := x.fst }) ((fun x x_1 => { fst := x.snd, snd := x.fst }) a_1 ha)\n (_ : (fun x x_1 => { fst := x.snd, snd := x.fst }) a_1 ha ∈ ?m.27222) =\n a_1","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (","nextTactic":"rintro ⟨⟩ _","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j Ico a (j + 1)\n⊢ (fun x x_1 => { fst := x.snd, snd := x.fst })\n ((fun x x_1 => { fst := x.snd, snd := x.fst }) { fst := fst✝, snd := snd✝ } ha✝)\n (_ : (fun x x_1 => { fst := x.snd, snd := x.fst }) { fst := fst✝, snd := snd✝ } ha✝ ∈ ?m.27222) =\n { fst := fst✝, snd := snd✝ }","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; ","nextTactic":"rfl","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j Ico i b),\n (fun x x_1 => { fst := x.snd, snd := x.fst }) a_1 ha ∈ Finset.sigma (Ico a b) fun j => Ico a (j + 1)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n ","nextTactic":"simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma]","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j Ico a (j + 1)),\n (fun x x_1 => { fst := x.snd, snd := x.fst }) a_1 ha ∈ Finset.sigma (Ico a b) fun i => Ico i b","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n ","nextTactic":"simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma]","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n ","nextTactic":"rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n ","nextTactic":"rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n ","nextTactic":"refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n ","nextTactic":"refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n ","nextTactic":"linarith","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n ","nextTactic":"linarith","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n ","nextTactic":"linarith","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n ","nextTactic":"linarith","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n ","nextTactic":"linarith","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n ","nextTactic":"linarith","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n ","nextTactic":"linarith","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n ","nextTactic":"linarith","declUpToTactic":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.128_0.jTt8VzqpIRePtoS","decl":"/-- The two ways of summing over `(i,j)` in the range `a<=i<=j _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩��� <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n ","nextTactic":"by_cases h : m ≤ n","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.144_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) "} +{"state":"case pos\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n : ℕ\nh : m ≤ n\n⊢ ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · ","nextTactic":"rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.144_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) "} +{"state":"case neg\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n : ℕ\nh : ¬m ≤ n\n⊢ ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · ","nextTactic":"replace h : n ≤ m := le_of_not_ge h","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.144_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) "} +{"state":"case neg\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n : ℕ\nh : n ≤ m\n⊢ ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n ","nextTactic":"rw [Ico_eq_empty_of_le h]","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.144_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) "} +{"state":"case neg\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n : ℕ\nh : n ≤ m\n⊢ ∏ k in ∅, f k = ∏ k in range (n - m), f (m + k)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n ","nextTactic":"rw [tsub_eq_zero_iff_le.mpr h]","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.144_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) "} +{"state":"case neg\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n : ℕ\nh : n ≤ m\n⊢ ∏ k in ∅, f k = ∏ k in range 0, f (m + k)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n ","nextTactic":"rw [range_zero]","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.144_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) "} +{"state":"case neg\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n : ℕ\nh : n ≤ m\n⊢ ∏ k in ∅, f k = ∏ k in ∅, f (m + k)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n ","nextTactic":"rw [prod_empty]","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.144_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) "} +{"state":"case neg\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n : ℕ\nh : n ≤ m\n⊢ 1 = ∏ k in ∅, f (m + k)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n ","nextTactic":"rw [prod_empty]","declUpToTactic":"@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.144_0.jTt8VzqpIRePtoS","decl":"@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nk m n : ℕ\nh : m ≤ n + 1\n⊢ ∏ j in Ico k m, f (n - j) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n ","nextTactic":"have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)","declUpToTactic":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.158_0.jTt8VzqpIRePtoS","decl":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nk m n : ℕ\nh : m ≤ n + 1\n⊢ ∀ i < m, i ≤ n","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n ","nextTactic":"intro i hi","declUpToTactic":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.158_0.jTt8VzqpIRePtoS","decl":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nk m n : ℕ\nh : m ≤ n + 1\ni : ℕ\nhi : i < m\n⊢ i ≤ n","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m �� n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n ","nextTactic":"exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)","declUpToTactic":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.158_0.jTt8VzqpIRePtoS","decl":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nk m n : ℕ\nh : m ≤ n + 1\nthis : ∀ i < m, i ≤ n\n⊢ ∏ j in Ico k m, f (n - j) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n ","nextTactic":"cases' lt_or_le k m with hkm hkm","declUpToTactic":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.158_0.jTt8VzqpIRePtoS","decl":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j "} +{"state":"case inl\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nk m n : ℕ\nh : m ≤ n + 1\nthis : ∀ i < m, i ≤ n\nhkm : k < m\n⊢ ∏ j in Ico k m, f (n - j) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · ","nextTactic":"rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]","declUpToTactic":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.158_0.jTt8VzqpIRePtoS","decl":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j "} +{"state":"case inl\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nk m n : ℕ\nh : m ≤ n + 1\nthis : ∀ i < m, i ≤ n\nhkm : k < m\n⊢ ∏ j in Ico k m, f (n - j) = ∏ j in image (fun x => n - x) (Ico k m), f j","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n ","nextTactic":"refine' (prod_image _).symm","declUpToTactic":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.158_0.jTt8VzqpIRePtoS","decl":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j "} +{"state":"case inl\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nk m n : ℕ\nh : m ≤ n + 1\nthis : ∀ i < m, i ≤ n\nhkm : k < m\n⊢ ∀ x ∈ Ico k m, ∀ y ∈ Ico k m, n - x = n - y → x = y","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n ","nextTactic":"simp only [mem_Ico]","declUpToTactic":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.158_0.jTt8VzqpIRePtoS","decl":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j "} +{"state":"case inl\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nk m n : ℕ\nh : m ≤ n + 1\nthis : ∀ i < m, i ≤ n\nhkm : k < m\n⊢ ∀ (x : ℕ), k ≤ x ∧ x < m → ∀ (y : ℕ), k ≤ y ∧ y < m → n - x = n - y → x = y","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n ","nextTactic":"rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij","declUpToTactic":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.158_0.jTt8VzqpIRePtoS","decl":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j "} +{"state":"case inl.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nk m n : ℕ\nh : m ≤ n + 1\nthis : ∀ i < m, i ≤ n\nhkm : k < m\ni : ℕ\nleft✝¹ : k ≤ i\nim : i < m\nj : ℕ\nleft✝ : k ≤ j\njm : j < m\nHij : n - i = n - j\n⊢ i = j","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n ","nextTactic":"rw [← tsub_tsub_cancel_of_le (this _ im)]","declUpToTactic":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.158_0.jTt8VzqpIRePtoS","decl":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j "} +{"state":"case inl.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nk m n : ℕ\nh : m ≤ n + 1\nthis : ∀ i < m, i ≤ n\nhkm : k < m\ni : ℕ\nleft✝¹ : k ≤ i\nim : i < m\nj : ℕ\nleft✝ : k ≤ j\njm : j < m\nHij : n - i = n - j\n⊢ n - (n - i) = j","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n ","nextTactic":"rw [Hij]","declUpToTactic":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.158_0.jTt8VzqpIRePtoS","decl":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j "} +{"state":"case inl.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nk m n : ℕ\nh : m ≤ n + 1\nthis : ∀ i < m, i ≤ n\nhkm : k < m\ni : ℕ\nleft✝¹ : k ≤ i\nim : i < m\nj : ℕ\nleft✝ : k ≤ j\njm : j < m\nHij : n - i = n - j\n⊢ n - (n - j) = j","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n ","nextTactic":"rw [tsub_tsub_cancel_of_le (this _ jm)]","declUpToTactic":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.158_0.jTt8VzqpIRePtoS","decl":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j "} +{"state":"case inr\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nk m n : ℕ\nh : m ≤ n + 1\nthis : ∀ i < m, i ≤ n\nhkm : m ≤ k\n⊢ ∏ j in Ico k m, f (n - j) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · ","nextTactic":"have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm","declUpToTactic":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.158_0.jTt8VzqpIRePtoS","decl":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nk m n : ℕ\nh : m ≤ n + 1\nthis : ∀ i < m, i ≤ n\nhkm : m ≤ k\n⊢ n + 1 - k ≤ n + 1 - m","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n ","nextTactic":"rw [tsub_le_tsub_iff_left h]","declUpToTactic":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.158_0.jTt8VzqpIRePtoS","decl":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nk m n : ℕ\nh : m ≤ n + 1\nthis : ∀ i < m, i ≤ n\nhkm : m ≤ k\n⊢ m ≤ k","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n ","nextTactic":"exact hkm","declUpToTactic":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.158_0.jTt8VzqpIRePtoS","decl":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j "} +{"state":"case inr\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nk m n : ℕ\nh : m ≤ n + 1\nthis✝ : ∀ i < m, i ≤ n\nhkm : m ≤ k\nthis : n + 1 - k ≤ n + 1 - m\n⊢ ∏ j in Ico k m, f (n - j) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n ","nextTactic":"simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]","declUpToTactic":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.158_0.jTt8VzqpIRePtoS","decl":"theorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nn : ℕ\n⊢ ∏ j in range n, f (n - 1 - j) = ∏ j in range n, f j","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n ","nextTactic":"cases n","declUpToTactic":"theorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.183_0.jTt8VzqpIRePtoS","decl":"theorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j "} +{"state":"case zero\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\n⊢ ∏ j in range zero, f (zero - 1 - j) = ∏ j in range zero, f j","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · ","nextTactic":"simp","declUpToTactic":"theorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.183_0.jTt8VzqpIRePtoS","decl":"theorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j "} +{"state":"case succ\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nn✝ : ℕ\n⊢ ∏ j in range (succ n✝), f (succ n✝ - 1 - j) = ∏ j in range (succ n✝), f j","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · ","nextTactic":"simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]","declUpToTactic":"theorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.183_0.jTt8VzqpIRePtoS","decl":"theorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j "} +{"state":"case succ\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nn✝ : ℕ\n⊢ ∏ x in Ico 0 (succ n✝), f (n✝ - x) = ∏ x in Ico 0 (succ n✝), f x","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n ","nextTactic":"rw [prod_Ico_reflect _ _ le_rfl]","declUpToTactic":"theorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.183_0.jTt8VzqpIRePtoS","decl":"theorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j "} +{"state":"case succ\nα : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nn✝ : ℕ\n⊢ ∏ j in Ico (n✝ + 1 - (n✝ + 1)) (n✝ + 1 - 0), f j = ∏ x in Ico 0 (succ n✝), f x","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n ","nextTactic":"simp","declUpToTactic":"theorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.183_0.jTt8VzqpIRePtoS","decl":"theorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f : α → β\ninst✝ : CommMonoid β\nn : ℕ\n⊢ ∏ x in Ico 1 (n + 1 + 1), x = (n + 1)!","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n ","nextTactic":"rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]","declUpToTactic":"@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.197_0.jTt8VzqpIRePtoS","decl":"@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]"} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f : α → β\ninst✝ : CommMonoid β\nn : ℕ\n⊢ ∏ x in range (n + 1), (x + 1) = (n + 1)!","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by ","nextTactic":"simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]","declUpToTactic":"@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.205_0.jTt8VzqpIRePtoS","decl":"@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]"} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f : α → β\ninst✝ : CommMonoid β\nn : ℕ\n⊢ (∑ i in range n, i) * 2 = ∑ i in range n, i + ∑ i in range n, (n - 1 - i)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n ","nextTactic":"rw [sum_range_reflect (fun i => i) n]","declUpToTactic":"/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.213_0.jTt8VzqpIRePtoS","decl":"/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f : α → β\ninst✝ : CommMonoid β\nn : ℕ\n⊢ (∑ i in range n, i) * 2 = ∑ i in range n, i + ∑ j in range n, j","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n ","nextTactic":"rw [mul_two]","declUpToTactic":"/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.213_0.jTt8VzqpIRePtoS","decl":"/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f : α → β\ninst✝ : CommMonoid β\nn : ℕ\n⊢ ∑ i in range n, (n - 1) = n * (n - 1)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by ","nextTactic":"rw [sum_const, card_range, Nat.nsmul_eq_mul]","declUpToTactic":"/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.213_0.jTt8VzqpIRePtoS","decl":"/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f : α → β\ninst✝ : CommMonoid β\nn : ℕ\n⊢ ∑ i in range n, i = n * (n - 1) / 2","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n ","nextTactic":"rw [← sum_range_id_mul_two n]","declUpToTactic":"/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.225_0.jTt8VzqpIRePtoS","decl":"/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 "} +{"state":"α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f : α → β\ninst✝ : CommMonoid β\nn : ℕ\n⊢ ∑ i in range n, i = (∑ i in range n, i) * 2 / 2","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n ","nextTactic":"rw [Nat.mul_div_cancel _ zero_lt_two]","declUpToTactic":"/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.225_0.jTt8VzqpIRePtoS","decl":"/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\n⊢ ∑ i in Ico m n, f i • g i =\n f (n - 1) • ∑ i in range n, g i - f m • ∑ i in range m, g i -\n ∑ i in Ico m (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n ","nextTactic":"have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\n⊢ ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n ","nextTactic":"rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\n⊢ ∑ i in Ico (m + 1) (n - 1 + 1), f i • ∑ i in range i, g i =\n ∑ i in Ico m (n - 1 + 1 - 1), f (i + 1) • ∑ i in range (i + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n ","nextTactic":"rw [← sum_Ico_add']","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\n⊢ ∑ x in Ico m (n - 1), f (x + 1) • ∑ i in range (x + 1), g i =\n ∑ i in Ico m (n - 1 + 1 - 1), f (i + 1) • ∑ i in range (i + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n ","nextTactic":"simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\n⊢ ∑ i in Ico m n, f i • g i =\n f (n - 1) • ∑ i in range n, g i - f m • ∑ i in range m, g i -\n ∑ i in Ico m (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n ","nextTactic":"have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) �� ∑ i in range (i + 1), g i\n⊢ ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n ","nextTactic":"rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\n⊢ ∑ i in Ico m n, f i • g i =\n f (n - 1) • ∑ i in range n, g i - f m • ∑ i in range m, g i -\n ∑ i in Ico m (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n ","nextTactic":"rw [sum_eq_sum_Ico_succ_bot hmn]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\n⊢ f m • g m + ∑ k in Ico (m + 1) n, f k • g k =\n f (n - 1) • ∑ i in range n, g i - f m • ∑ i in range m, g i -\n ∑ i in Ico m (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n ","nextTactic":"have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\n⊢ ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n ","nextTactic":"congr","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"case e_f\nR : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\n⊢ (fun i => f i • g i) = fun i => f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; ","nextTactic":"funext","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"case e_f.h\nR : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nx✝ : ℕ\n⊢ f x✝ • g x✝ = f x✝ • (Finset.sum (range (x✝ + 1)) g - Finset.sum (range x✝) g)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ���) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; ","nextTactic":"rw [← sum_range_succ_sub_sum g]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nh₃ :\n ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\n⊢ f m • g m + ∑ k in Ico (m + 1) n, f k • g k =\n f (n - 1) • ∑ i in range n, g i - f m • ∑ i in range m, g i -\n ∑ i in Ico m (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n ","nextTactic":"rw [h₃]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nh₃ :\n ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\n⊢ f m • g m + ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g) =\n f (n - 1) • ∑ i in range n, g i - f m • ∑ i in range m, g i -\n ∑ i in Ico m (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n ","nextTactic":"simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nh₃ :\n ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\n⊢ f m • g m +\n (∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i -\n ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i) =\n f (n - 1) • ∑ i in range n, g i - f m ��� ∑ i in range m, g i -\n ∑ i in Ico m (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n ","nextTactic":"have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nh₃ :\n ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\n⊢ ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i -\n ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i =\n f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +\n ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n ","nextTactic":"rw [← add_sub]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nh₃ :\n ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\n⊢ ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i +\n (f (n - 1) • ∑ i in range n, g i - f m • ∑ i in range (m + 1), g i) -\n ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i =\n f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +\n ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n ","nextTactic":"rw [add_comm]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nh₃ :\n ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\n⊢ f (n - 1) • ∑ i in range n, g i - f m • ∑ i in range (m + 1), g i +\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i -\n ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i =\n f (n - 1) ��� Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +\n ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n ","nextTactic":"rw [← add_sub]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nh₃ :\n ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\n⊢ f (n - 1) • ∑ i in range n, g i - f m • ∑ i in range (m + 1), g i +\n (∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i -\n ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i) =\n f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +\n ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n ","nextTactic":"rw [← sum_sub_distrib]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nh₃ :\n ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh₄ :\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i -\n ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i =\n f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +\n ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)\n⊢ f m • g m +\n (∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i -\n ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i) =\n f (n - 1) • ∑ i in range n, g i - f m • ∑ i in range m, g i -\n ∑ i in Ico m (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n ","nextTactic":"rw [h₄]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nh₃ :\n ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh₄ :\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i -\n ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i =\n f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +\n ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)\n⊢ f m • g m +\n (f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +\n ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)) =\n f (n - 1) • ∑ i in range n, g i - f m • ∑ i in range m, g i -\n ∑ i in Ico m (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n ","nextTactic":"have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n intro i\n rw [sub_smul]\n abel","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nh₃ :\n ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh₄ :\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i -\n ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i =\n f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +\n ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)\n⊢ ∀ (i : ℕ),\n f i • ∑ i in range (i + 1), g i - f (i + 1) • ∑ i in range (i + 1), g i =\n -((f (i + 1) - f i) • ∑ i in range (i + 1), g i)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ��⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n ","nextTactic":"intro i","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nh₃ :\n ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh₄ :\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i -\n ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i =\n f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +\n ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)\ni : ℕ\n⊢ f i • ∑ i in range (i + 1), g i - f (i + 1) • ∑ i in range (i + 1), g i =\n -((f (i + 1) - f i) • ∑ i in range (i + 1), g i)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n intro i\n ","nextTactic":"rw [sub_smul]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n intro i\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nh₃ :\n ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh₄ :\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i -\n ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i =\n f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +\n ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)\ni : ℕ\n⊢ f i • ∑ i in range (i + 1), g i - f (i + 1) • ∑ i in range (i + 1), g i =\n -(f (i + 1) • ∑ i in range (i + 1), g i - f i • ∑ i in range (i + 1), g i)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n intro i\n rw [sub_smul]\n ","nextTactic":"abel","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n intro i\n rw [sub_smul]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nh₃ :\n ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh₄ :\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i -\n ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i =\n f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +\n ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)\ni : ℕ\n⊢ f i • ∑ i in range (i + 1), g i - f (i + 1) • ∑ i in range (i + 1), g i =\n -(f (i + 1) • ∑ i in range (i + 1), g i - f i • ∑ i in range (i + 1), g i)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n intro i\n rw [sub_smul]\n ","nextTactic":"abel","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n intro i\n rw [sub_smul]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nh₃ :\n ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh₄ :\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i -\n ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i =\n f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +\n ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)\nthis :\n ∀ (i : ℕ),\n f i • ∑ i in range (i + 1), g i - f (i + 1) • ∑ i in range (i + 1), g i =\n -((f (i + 1) - f i) • ∑ i in range (i + 1), g i)\n⊢ f m • g m +\n (f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +\n ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)) =\n f (n - 1) • ∑ i in range n, g i - f m • ∑ i in range m, g i -\n ∑ i in Ico m (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n intro i\n rw [sub_smul]\n abel\n ","nextTactic":"simp_rw [this, sum_neg_distrib, sum_range_succ, smul_add]","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n intro i\n rw [sub_smul]\n abel\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nh₃ :\n ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh₄ :\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i -\n ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i =\n f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +\n ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)\nthis :\n ∀ (i : ℕ),\n f i • ∑ i in range (i + 1), g i - f (i + 1) • ∑ i in range (i + 1), g i =\n -((f (i + 1) - f i) • ∑ i in range (i + 1), g i)\n⊢ f m • g m +\n (f (n - 1) • Finset.sum (range n) g - (f m • ∑ x in range m, g x + f m • g m) +\n -∑ x in Ico m (n - 1), ((f (x + 1) - f x) • ∑ x in range x, g x + (f (x + 1) - f x) • g x)) =\n f (n - 1) • ∑ x in range n, g x - f m • ∑ x in range m, g x -\n ∑ x in Ico m (n - 1), ((f (x + 1) - f x) • ∑ x in range x, g x + (f (x + 1) - f x) • g x)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n intro i\n rw [sub_smul]\n abel\n simp_rw [this, sum_neg_distrib, sum_range_succ, smul_add]\n ","nextTactic":"abel","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n intro i\n rw [sub_smul]\n abel\n simp_rw [this, sum_neg_distrib, sum_range_succ, smul_add]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i\nh₂ :\n ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i\nh₃ :\n ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh₄ :\n ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -\n f m • ∑ i in range (m + 1), g i -\n ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i =\n f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +\n ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)\nthis :\n ∀ (i : ℕ),\n f i • ∑ i in range (i + 1), g i - f (i + 1) • ∑ i in range (i + 1), g i =\n -((f (i + 1) - f i) • ∑ i in range (i + 1), g i)\n⊢ f m • g m +\n (f (n - 1) • Finset.sum (range n) g - (f m • ∑ x in range m, g x + f m • g m) +\n -∑ x in Ico m (n - 1), ((f (x + 1) - f x) • ∑ x in range x, g x + (f (x + 1) - f x) • g x)) =\n f (n - 1) • ∑ x in range n, g x - f m • ∑ x in range m, g x -\n ∑ x in Ico m (n - 1), ((f (x + 1) - f x) • ∑ x in range x, g x + (f (x + 1) - f x) • g x)","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n intro i\n rw [sub_smul]\n abel\n simp_rw [this, sum_neg_distrib, sum_range_succ, smul_add]\n ","nextTactic":"abel","declUpToTactic":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n intro i\n rw [sub_smul]\n abel\n simp_rw [this, sum_neg_distrib, sum_range_succ, smul_add]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.283_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\n⊢ ∑ i in range n, f i • g i =\n f (n - 1) • ∑ i in range n, g i - ∑ i in range (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n intro i\n rw [sub_smul]\n abel\n simp_rw [this, sum_neg_distrib, sum_range_succ, smul_add]\n abel\n#align finset.sum_Ico_by_parts Finset.sum_Ico_by_parts\n\nvariable (n)\n\n/-- **Summation by parts** for ranges -/\ntheorem sum_range_by_parts :\n ∑ i in range n, f i • g i =\n f (n - 1) • G n - ∑ i in range (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n ","nextTactic":"by_cases hn : n = 0","declUpToTactic":"/-- **Summation by parts** for ranges -/\ntheorem sum_range_by_parts :\n ∑ i in range n, f i • g i =\n f (n - 1) • G n - ∑ i in range (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.329_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts** for ranges -/\ntheorem sum_range_by_parts :\n ∑ i in range n, f i • g i =\n f (n - 1) • G n - ∑ i in range (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"case pos\nR : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhn : n = 0\n⊢ ∑ i in range n, f i • g i =\n f (n - 1) • ∑ i in range n, g i - ∑ i in range (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n intro i\n rw [sub_smul]\n abel\n simp_rw [this, sum_neg_distrib, sum_range_succ, smul_add]\n abel\n#align finset.sum_Ico_by_parts Finset.sum_Ico_by_parts\n\nvariable (n)\n\n/-- **Summation by parts** for ranges -/\ntheorem sum_range_by_parts :\n ∑ i in range n, f i • g i =\n f (n - 1) • G n - ∑ i in range (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n by_cases hn : n = 0\n · ","nextTactic":"simp [hn]","declUpToTactic":"/-- **Summation by parts** for ranges -/\ntheorem sum_range_by_parts :\n ∑ i in range n, f i • g i =\n f (n - 1) • G n - ∑ i in range (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n by_cases hn : n = 0\n · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.329_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts** for ranges -/\ntheorem sum_range_by_parts :\n ∑ i in range n, f i • g i =\n f (n - 1) • G n - ∑ i in range (n - 1), (f (i + 1) - f i) • G (i + 1) "} +{"state":"case neg\nR : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhn : ¬n = 0\n⊢ ∑ i in range n, f i • g i =\n f (n - 1) • ∑ i in range n, g i - ∑ i in range (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i","srcUpToTactic":"/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Tactic.Linarith\n\n#align_import algebra.big_operators.intervals from \"leanprover-community/mathlib\"@\"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c\"\n\n/-!\n# Results about big operators over intervals\n\nWe prove results about big operators over intervals (mostly the `ℕ`-valued `Ico m n`).\n-/\n\n\nuniverse u v w\n\nopen BigOperators\nopen Nat\n\nnamespace Finset\n\nsection Generic\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {s₂ s₁ s : Finset α} {a : α} {g f : α → β}\n\nvariable [CommMonoid β]\n\n@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (x + c)) = ∏ x in Ico (a + c) (b + c), f x := by\n rw [← map_add_right_Ico]\n rw [prod_map]\n rfl\n#align finset.prod_Ico_add' Finset.prod_Ico_add'\n#align finset.sum_Ico_add' Finset.sum_Ico_add'\n\n@[to_additive]\ntheorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]\n (f : α → β) (a b c : α) : (∏ x in Ico a b, f (c + x)) = ∏ x in Ico (a + c) (b + c), f x := by\n convert prod_Ico_add' f a b c using 2\n rw [add_comm]\n#align finset.prod_Ico_add Finset.prod_Ico_add\n#align finset.sum_Ico_add Finset.sum_Ico_add\n\n@[to_additive]\ntheorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ico a (b + 1), f k) = (∏ k in Ico a b, f k) * f b := by\n rw [Nat.Ico_succ_right_eq_insert_Ico hab]\n rw [prod_insert right_not_mem_Ico]\n rw [mul_comm]\n#align finset.prod_Ico_succ_top Finset.prod_Ico_succ_top\n#align finset.sum_Ico_succ_top Finset.sum_Ico_succ_top\n\n@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → β) :\n ∏ k in Ico a b, f k = f a * ∏ k in Ico (a + 1) b, f k := by\n have ha : a ∉ Ico (a + 1) b := by simp\n rw [← prod_insert ha]\n rw [Nat.Ico_insert_succ_left hab]\n#align finset.prod_eq_prod_Ico_succ_bot Finset.prod_eq_prod_Ico_succ_bot\n#align finset.sum_eq_sum_Ico_succ_bot Finset.sum_eq_sum_Ico_succ_bot\n\n@[to_additive]\ntheorem prod_Ico_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ico m n, f i) * ∏ i in Ico n k, f i) = ∏ i in Ico m k, f i :=\n Ico_union_Ico_eq_Ico hmn hnk ▸ Eq.symm (prod_union (Ico_disjoint_Ico_consecutive m n k))\n#align finset.prod_Ico_consecutive Finset.prod_Ico_consecutive\n#align finset.sum_Ico_consecutive Finset.sum_Ico_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_consecutive (f : ℕ → β) {m n k : ℕ} (hmn : m ≤ n) (hnk : n ≤ k) :\n ((∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i) = ∏ i in Ioc m k, f i := by\n rw [← Ioc_union_Ioc_eq_Ioc hmn hnk]\n rw [prod_union]\n apply disjoint_left.2 fun x hx h'x => _\n intros x hx h'x\n exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n#align finset.prod_Ioc_consecutive Finset.prod_Ioc_consecutive\n#align finset.sum_Ioc_consecutive Finset.sum_Ioc_consecutive\n\n@[to_additive]\ntheorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → β) :\n (∏ k in Ioc a (b + 1), f k) = (∏ k in Ioc a b, f k) * f (b + 1) := by\n rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b)]\n rw [Nat.Ioc_succ_singleton]\n rw [prod_singleton]\n#align finset.prod_Ioc_succ_top Finset.prod_Ioc_succ_top\n#align finset.sum_Ioc_succ_top Finset.sum_Ioc_succ_top\n\n@[to_additive]\ntheorem prod_range_mul_prod_Ico (f : ℕ → β) {m n : ℕ} (h : m ≤ n) :\n ((∏ k in range m, f k) * ∏ k in Ico m n, f k) = ∏ k in range n, f k :=\n Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h\n#align finset.prod_range_mul_prod_Ico Finset.prod_range_mul_prod_Ico\n#align finset.sum_range_add_sum_Ico Finset.sum_range_add_sum_Ico\n\n@[to_additive]\ntheorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) * (∏ k in range m, f k)⁻¹ :=\n eq_mul_inv_iff_mul_eq.2 <| by (rw [mul_comm]; exact prod_range_mul_prod_Ico f h)\n#align finset.prod_Ico_eq_mul_inv Finset.prod_Ico_eq_mul_inv\n#align finset.sum_Ico_eq_add_neg Finset.sum_Ico_eq_add_neg\n\n@[to_additive]\ntheorem prod_Ico_eq_div {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :\n ∏ k in Ico m n, f k = (∏ k in range n, f k) / ∏ k in range m, f k := by\n simpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n#align finset.prod_Ico_eq_div Finset.prod_Ico_eq_div\n#align finset.sum_Ico_eq_sub Finset.sum_Ico_eq_sub\n\n@[to_additive]\ntheorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :\n ((∏ k in range m, f k) / ∏ k in range n, f k) =\n ∏ k in (range m).filter fun k => n ≤ k, f k := by\n rw [← prod_Ico_eq_div f hnm]\n congr\n apply Finset.ext\n simp only [mem_Ico, mem_filter, mem_range, *]\n tauto\n#align finset.prod_range_sub_prod_range Finset.prod_range_div_prod_range\n#align finset.sum_range_sub_sum_range Finset.sum_range_sub_sum_range\n\n/-- The two ways of summing over `(i,j)` in the range `a<=i<=j (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (fun _ _ => rfl)\n (fun (x : Σ _ : ℕ, ℕ) _ => (⟨x.2, x.1⟩ : Σ _ : ℕ, ℕ)) _ (by (rintro ⟨⟩ _; rfl))\n (by (rintro ⟨⟩ _; rfl)) <;>\n simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>\n rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>\n refine' ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>\n linarith\n#align finset.sum_Ico_Ico_comm Finset.sum_Ico_Ico_comm\n\n@[to_additive]\ntheorem prod_Ico_eq_prod_range (f : ℕ → β) (m n : ℕ) :\n ∏ k in Ico m n, f k = ∏ k in range (n - m), f (m + k) := by\n by_cases h : m ≤ n\n · rw [← Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n · replace h : n ≤ m := le_of_not_ge h\n rw [Ico_eq_empty_of_le h]\n rw [tsub_eq_zero_iff_le.mpr h]\n rw [range_zero]\n rw [prod_empty]\n rw [prod_empty]\n#align finset.prod_Ico_eq_prod_range Finset.prod_Ico_eq_prod_range\n#align finset.sum_Ico_eq_sum_range Finset.sum_Ico_eq_sum_range\n\ntheorem prod_Ico_reflect (f : ℕ → β) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :\n (∏ j in Ico k m, f (n - j)) = ∏ j in Ico (n + 1 - m) (n + 1 - k), f j := by\n have : ∀ i < m, i ≤ n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n cases' lt_or_le k m with hkm hkm\n · rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n refine' (prod_image _).symm\n simp only [mem_Ico]\n rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij\n rw [← tsub_tsub_cancel_of_le (this _ im)]\n rw [Hij]\n rw [tsub_tsub_cancel_of_le (this _ jm)]\n · have : n + 1 - k ≤ n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm\n simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]\n#align finset.prod_Ico_reflect Finset.prod_Ico_reflect\n\ntheorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}\n (h : m ≤ n + 1) : (∑ j in Ico k m, f (n - j)) = ∑ j in Ico (n + 1 - m) (n + 1 - k), f j :=\n @prod_Ico_reflect (Multiplicative δ) _ f k m n h\n#align finset.sum_Ico_reflect Finset.sum_Ico_reflect\n\ntheorem prod_range_reflect (f : ℕ → β) (n : ℕ) :\n (∏ j in range n, f (n - 1 - j)) = ∏ j in range n, f j := by\n cases n\n · simp\n · simp only [← Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n rw [prod_Ico_reflect _ _ le_rfl]\n simp\n#align finset.prod_range_reflect Finset.prod_range_reflect\n\ntheorem sum_range_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (n : ℕ) :\n (∑ j in range n, f (n - 1 - j)) = ∑ j in range n, f j :=\n @prod_range_reflect (Multiplicative δ) _ f n\n#align finset.sum_range_reflect Finset.sum_range_reflect\n\n@[simp]\ntheorem prod_Ico_id_eq_factorial : ∀ n : ℕ, (∏ x in Ico 1 (n + 1), x) = n !\n | 0 => rfl\n | n + 1 => by\n rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,\n prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm]\n#align finset.prod_Ico_id_eq_factorial Finset.prod_Ico_id_eq_factorial\n\n@[simp]\ntheorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x in range n, (x + 1)) = n !\n | 0 => rfl\n | n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]\n#align finset.prod_range_add_one_eq_factorial Finset.prod_range_add_one_eq_factorial\n\nsection GaussSum\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id_mul_two (n : ℕ) : (∑ i in range n, i) * 2 = n * (n - 1) :=\n calc\n (∑ i in range n, i) * 2 = (∑ i in range n, i) + ∑ i in range n, (n - 1 - i) := by\n rw [sum_range_reflect (fun i => i) n]\n rw [mul_two]\n _ = ∑ i in range n, (i + (n - 1 - i)) := sum_add_distrib.symm\n _ = ∑ i in range n, (n - 1) :=\n sum_congr rfl fun i hi => add_tsub_cancel_of_le <| Nat.le_sub_one_of_lt <| mem_range.1 hi\n _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]\n#align finset.sum_range_id_mul_two Finset.sum_range_id_mul_two\n\n/-- Gauss' summation formula -/\ntheorem sum_range_id (n : ℕ) : ∑ i in range n, i = n * (n - 1) / 2 := by\n rw [← sum_range_id_mul_two n]\n rw [Nat.mul_div_cancel _ zero_lt_two]\n#align finset.sum_range_id Finset.sum_range_id\n\nend GaussSum\n\nend Generic\n\nsection Nat\n\nvariable {β : Type*}\n\nvariable (f g : ℕ → β) {m n : ℕ}\n\nsection Group\n\nvariable [CommGroup β]\n\n@[to_additive]\ntheorem prod_range_succ_div_prod : ((∏ i in range (n + 1), f i) / ∏ i in range n, f i) = f n :=\n div_eq_iff_eq_mul'.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_prod Finset.prod_range_succ_div_prod\n#align finset.sum_range_succ_sub_sum Finset.sum_range_succ_sub_sum\n\n@[to_additive]\ntheorem prod_range_succ_div_top : (∏ i in range (n + 1), f i) / f n = ∏ i in range n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_range_succ f n\n#align finset.prod_range_succ_div_top Finset.prod_range_succ_div_top\n#align finset.sum_range_succ_sub_top Finset.sum_range_succ_sub_top\n\n@[to_additive]\ntheorem prod_Ico_div_bot (hmn : m < n) : (∏ i in Ico m n, f i) / f m = ∏ i in Ico (m + 1) n, f i :=\n div_eq_iff_eq_mul'.mpr <| prod_eq_prod_Ico_succ_bot hmn _\n#align finset.prod_Ico_div_bot Finset.prod_Ico_div_bot\n#align finset.sum_Ico_sub_bot Finset.sum_Ico_sub_bot\n\n@[to_additive]\ntheorem prod_Ico_succ_div_top (hmn : m ≤ n) :\n (∏ i in Ico m (n + 1), f i) / f n = ∏ i in Ico m n, f i :=\n div_eq_iff_eq_mul.mpr <| prod_Ico_succ_top hmn _\n#align finset.prod_Ico_succ_div_top Finset.prod_Ico_succ_div_top\n#align finset.sum_Ico_succ_sub_top Finset.sum_Ico_succ_sub_top\n\nend Group\n\nend Nat\n\nsection Module\n\nvariable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}\n\nopen Finset\n\n-- The partial sum of `g`, starting from zero\nlocal notation \"G \" n:80 => ∑ i in range n, g i\n\n/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/\ntheorem sum_Ico_by_parts (hmn : m < n) :\n ∑ i in Ico m n, f i • g i =\n f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by\n rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn)]\n rw [← sum_Ico_add']\n simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,\n tsub_eq_zero_iff_le, add_tsub_cancel_right]\n have h₂ :\n (∑ i in Ico (m + 1) n, f i • G (i + 1)) =\n (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by\n rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),\n Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]\n rw [sum_eq_sum_Ico_succ_bot hmn]\n -- porting note: the following used to be done with `conv`\n have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =\n (Finset.sum (Ico (m + 1) n) fun i =>\n f i • ((Finset.sum (Finset.range (i + 1)) g) -\n (Finset.sum (Finset.range i) g))) := by\n congr; funext; rw [← sum_range_succ_sub_sum g]\n rw [h₃]\n simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]\n -- porting note: the following used to be done with `conv`\n have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +\n f (n - 1) • Finset.sum (range n) fun i => g i) -\n f m • Finset.sum (range (m + 1)) fun i => g i) -\n Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =\n f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +\n Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -\n f (i + 1) • (range (i + 1)).sum g) := by\n rw [← add_sub]\n rw [add_comm]\n rw [← add_sub]\n rw [← sum_sub_distrib]\n rw [h₄]\n have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by\n intro i\n rw [sub_smul]\n abel\n simp_rw [this, sum_neg_distrib, sum_range_succ, smul_add]\n abel\n#align finset.sum_Ico_by_parts Finset.sum_Ico_by_parts\n\nvariable (n)\n\n/-- **Summation by parts** for ranges -/\ntheorem sum_range_by_parts :\n ∑ i in range n, f i • g i =\n f (n - 1) • G n - ∑ i in range (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n by_cases hn : n = 0\n · simp [hn]\n · ","nextTactic":"rw [range_eq_Ico, sum_Ico_by_parts f g (Nat.pos_of_ne_zero hn), sum_range_zero, smul_zero,\n sub_zero, range_eq_Ico]","declUpToTactic":"/-- **Summation by parts** for ranges -/\ntheorem sum_range_by_parts :\n ∑ i in range n, f i • g i =\n f (n - 1) • G n - ∑ i in range (n - 1), (f (i + 1) - f i) • G (i + 1) := by\n by_cases hn : n = 0\n · simp [hn]\n · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Intervals.329_0.jTt8VzqpIRePtoS","decl":"/-- **Summation by parts** for ranges -/\ntheorem sum_range_by_parts :\n ∑ i in range n, f i • g i =\n f (n - 1) • G n - ∑ i in range (n - 1), (f (i + 1) - f i) • G (i + 1) "}