Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_BigOperators_Option.jsonl

{"state":"α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nf : Option α → M\ns : Finset α\n⊢ ∏ x in insertNone s, f x = f none * ∏ x in s, f (some x)","srcUpToTactic":"/-\nCopyright (c) 2021 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Data.Finset.Option\n\n#align_import algebra.big_operators.option from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Lemmas about products and sums over finite sets in `Option α`\n\nIn this file we prove formulas for products and sums over `Finset.insertNone s` and\n`Finset.eraseNone s`.\n-/\n\nopen BigOperators\n\nopen Function\n\nnamespace Finset\n\nvariable {α M : Type*} [CommMonoid M]\n\n@[to_additive (attr := simp)]\ntheorem prod_insertNone (f : Option α → M) (s : Finset α) :\n    ∏ x in insertNone s, f x = f none * ∏ x in s, f (some x) := by ","nextTactic":"simp [insertNone]","declUpToTactic":"@[to_additive (attr := simp)]\ntheorem prod_insertNone (f : Option α → M) (s : Finset α) :\n    ∏ x in insertNone s, f x = f none * ∏ x in s, f (some x) := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Option.26_0.TnTVWDXfJXrB22D","decl":"@[to_additive (attr "}
{"state":"α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nf : α → M\ns : Finset (Option α)\n⊢ ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x","srcUpToTactic":"/-\nCopyright (c) 2021 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Data.Finset.Option\n\n#align_import algebra.big_operators.option from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Lemmas about products and sums over finite sets in `Option α`\n\nIn this file we prove formulas for products and sums over `Finset.insertNone s` and\n`Finset.eraseNone s`.\n-/\n\nopen BigOperators\n\nopen Function\n\nnamespace Finset\n\nvariable {α M : Type*} [CommMonoid M]\n\n@[to_additive (attr := simp)]\ntheorem prod_insertNone (f : Option α → M) (s : Finset α) :\n    ∏ x in insertNone s, f x = f none * ∏ x in s, f (some x) := by simp [insertNone]\n#align finset.prod_insert_none Finset.prod_insertNone\n#align finset.sum_insert_none Finset.sum_insertNone\n\n@[to_additive]\ntheorem mul_prod_eq_prod_insertNone (f : α → M) (x : M) (s : Finset α) :\n    x * ∏ i in s, f i = ∏ i in insertNone s, i.elim x f :=\n  (prod_insertNone (fun i => i.elim x f) _).symm\n\n@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x := by\n  ","nextTactic":"classical calc\n      ∏ x in eraseNone s, f x = ∏ x in (eraseNone s).map Embedding.some, Option.elim' 1 f x :=\n        (prod_map (eraseNone s) Embedding.some <| Option.elim' 1 f).symm\n      _ = ∏ x in s.erase none, Option.elim' 1 f x := by rw [map_some_eraseNone]\n      _ = ∏ x in s, Option.elim' 1 f x := prod_erase _ rfl","declUpToTactic":"@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Option.37_0.TnTVWDXfJXrB22D","decl":"@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x "}
{"state":"α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nf : α → M\ns : Finset (Option α)\n⊢ ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x","srcUpToTactic":"/-\nCopyright (c) 2021 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Data.Finset.Option\n\n#align_import algebra.big_operators.option from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Lemmas about products and sums over finite sets in `Option α`\n\nIn this file we prove formulas for products and sums over `Finset.insertNone s` and\n`Finset.eraseNone s`.\n-/\n\nopen BigOperators\n\nopen Function\n\nnamespace Finset\n\nvariable {α M : Type*} [CommMonoid M]\n\n@[to_additive (attr := simp)]\ntheorem prod_insertNone (f : Option α → M) (s : Finset α) :\n    ∏ x in insertNone s, f x = f none * ∏ x in s, f (some x) := by simp [insertNone]\n#align finset.prod_insert_none Finset.prod_insertNone\n#align finset.sum_insert_none Finset.sum_insertNone\n\n@[to_additive]\ntheorem mul_prod_eq_prod_insertNone (f : α → M) (x : M) (s : Finset α) :\n    x * ∏ i in s, f i = ∏ i in insertNone s, i.elim x f :=\n  (prod_insertNone (fun i => i.elim x f) _).symm\n\n@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x := by\n  classical ","nextTactic":"calc\n      ∏ x in eraseNone s, f x = ∏ x in (eraseNone s).map Embedding.some, Option.elim' 1 f x :=\n        (prod_map (eraseNone s) Embedding.some <| Option.elim' 1 f).symm\n      _ = ∏ x in s.erase none, Option.elim' 1 f x := by rw [map_some_eraseNone]\n      _ = ∏ x in s, Option.elim' 1 f x := prod_erase _ rfl","declUpToTactic":"@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x := by\n  classical ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Option.37_0.TnTVWDXfJXrB22D","decl":"@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x "}
{"state":"α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nf : α → M\ns : Finset (Option α)\n⊢ ∏ x in map Embedding.some (eraseNone s), Option.elim' 1 f x = ∏ x in erase s none, Option.elim' 1 f x","srcUpToTactic":"/-\nCopyright (c) 2021 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Data.Finset.Option\n\n#align_import algebra.big_operators.option from \"leanprover-community/mathlib\"@\"008205aa645b3f194c1da47025c5f110c8406eab\"\n\n/-!\n# Lemmas about products and sums over finite sets in `Option α`\n\nIn this file we prove formulas for products and sums over `Finset.insertNone s` and\n`Finset.eraseNone s`.\n-/\n\nopen BigOperators\n\nopen Function\n\nnamespace Finset\n\nvariable {α M : Type*} [CommMonoid M]\n\n@[to_additive (attr := simp)]\ntheorem prod_insertNone (f : Option α → M) (s : Finset α) :\n    ∏ x in insertNone s, f x = f none * ∏ x in s, f (some x) := by simp [insertNone]\n#align finset.prod_insert_none Finset.prod_insertNone\n#align finset.sum_insert_none Finset.sum_insertNone\n\n@[to_additive]\ntheorem mul_prod_eq_prod_insertNone (f : α → M) (x : M) (s : Finset α) :\n    x * ∏ i in s, f i = ∏ i in insertNone s, i.elim x f :=\n  (prod_insertNone (fun i => i.elim x f) _).symm\n\n@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x := by\n  classical calc\n      ∏ x in eraseNone s, f x = ∏ x in (eraseNone s).map Embedding.some, Option.elim' 1 f x :=\n        (prod_map (eraseNone s) Embedding.some <| Option.elim' 1 f).symm\n      _ = ∏ x in s.erase none, Option.elim' 1 f x := by ","nextTactic":"rw [map_some_eraseNone]","declUpToTactic":"@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x := by\n  classical calc\n      ∏ x in eraseNone s, f x = ∏ x in (eraseNone s).map Embedding.some, Option.elim' 1 f x :=\n        (prod_map (eraseNone s) Embedding.some <| Option.elim' 1 f).symm\n      _ = ∏ x in s.erase none, Option.elim' 1 f x := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Option.37_0.TnTVWDXfJXrB22D","decl":"@[to_additive]\ntheorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :\n    ∏ x in eraseNone s, f x = ∏ x in s, Option.elim' 1 f x "}