| {"state":"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns : Finset γ\nf : γ → α\ng : γ → β\nthis : DecidableEq γ\n⊢ ∀ ⦃a : γ⦄ {s : Finset γ},\n a ∉ s →\n (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) →\n (∏ x in insert a s, f x, ∏ x in insert a s, g x) = ∏ x in insert a s, (f x, g x)","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n haveI := Classical.decEq γ\n Finset.induction_on s rfl (by ","nextTactic":"simp (config := { contextual := true }) [Prod.ext_iff]","declUpToTactic":"@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n haveI := Classical.decEq γ\n Finset.induction_on s rfl (by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.63_0.f38FU5HrzKIlXWd","decl":"@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) "} | |
| {"state":"ι : Type u_1\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\nR : Type u_2\ninst✝ : Semiring R\nx : ι → R\n⊢ x = ∑ i : ι, x i • fun j => if i = j then 1 else 0","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n haveI := Classical.decEq γ\n Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n ","nextTactic":"ext","declUpToTactic":"/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.71_0.f38FU5HrzKIlXWd","decl":"/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 "} | |
| {"state":"case h\nι : Type u_1\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\nR : Type u_2\ninst✝ : Semiring R\nx : ι → R\nx✝ : ι\n⊢ x x✝ = Finset.sum Finset.univ (fun i => x i • fun j => if i = j then 1 else 0) x✝","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n haveI := Classical.decEq γ\n Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n ext\n ","nextTactic":"simp","declUpToTactic":"/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n ext\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.71_0.f38FU5HrzKIlXWd","decl":"/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 "} | |
| {"state":"I : Type u_1\ninst✝² : DecidableEq I\nZ : I → Type u_2\ninst✝¹ : (i : I) → CommMonoid (Z i)\ninst✝ : Fintype I\nf : (i : I) → Z i\n⊢ ∏ i : I, Pi.mulSingle i (f i) = f","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n haveI := Classical.decEq γ\n Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n ext\n simp\n#align pi_eq_sum_univ pi_eq_sum_univ\n\nsection MulSingle\n\nvariable {I : Type*} [DecidableEq I] {Z : I → Type*}\n\nvariable [∀ i, CommMonoid (Z i)]\n\n@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n (∏ i, Pi.mulSingle i (f i)) = f := by\n ","nextTactic":"ext a","declUpToTactic":"@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n (∏ i, Pi.mulSingle i (f i)) = f := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.84_0.f38FU5HrzKIlXWd","decl":"@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n (∏ i, Pi.mulSingle i (f i)) = f "} | |
| {"state":"case h\nI : Type u_1\ninst✝² : DecidableEq I\nZ : I → Type u_2\ninst✝¹ : (i : I) → CommMonoid (Z i)\ninst✝ : Fintype I\nf : (i : I) → Z i\na : I\n⊢ Finset.prod univ (fun i => Pi.mulSingle i (f i)) a = f a","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n haveI := Classical.decEq γ\n Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n ext\n simp\n#align pi_eq_sum_univ pi_eq_sum_univ\n\nsection MulSingle\n\nvariable {I : Type*} [DecidableEq I] {Z : I → Type*}\n\nvariable [∀ i, CommMonoid (Z i)]\n\n@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n (∏ i, Pi.mulSingle i (f i)) = f := by\n ext a\n ","nextTactic":"simp","declUpToTactic":"@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n (∏ i, Pi.mulSingle i (f i)) = f := by\n ext a\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.84_0.f38FU5HrzKIlXWd","decl":"@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n (∏ i, Pi.mulSingle i (f i)) = f "} | |
| {"state":"I : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_2\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_3\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), g (Pi.mulSingle i x) = h (Pi.mulSingle i x)\n⊢ g = h","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n haveI := Classical.decEq γ\n Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n ext\n simp\n#align pi_eq_sum_univ pi_eq_sum_univ\n\nsection MulSingle\n\nvariable {I : Type*} [DecidableEq I] {Z : I → Type*}\n\nvariable [∀ i, CommMonoid (Z i)]\n\n@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n (∏ i, Pi.mulSingle i (f i)) = f := by\n ext a\n simp\n#align finset.univ_prod_mul_single Finset.univ_prod_mulSingle\n#align finset.univ_sum_single Finset.univ_sum_single\n\n@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n ","nextTactic":"cases nonempty_fintype I","declUpToTactic":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.92_0.f38FU5HrzKIlXWd","decl":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h "} | |
| {"state":"case intro\nI : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_2\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_3\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), g (Pi.mulSingle i x) = h (Pi.mulSingle i x)\nval✝ : Fintype I\n⊢ g = h","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n haveI := Classical.decEq γ\n Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n ext\n simp\n#align pi_eq_sum_univ pi_eq_sum_univ\n\nsection MulSingle\n\nvariable {I : Type*} [DecidableEq I] {Z : I → Type*}\n\nvariable [∀ i, CommMonoid (Z i)]\n\n@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n (∏ i, Pi.mulSingle i (f i)) = f := by\n ext a\n simp\n#align finset.univ_prod_mul_single Finset.univ_prod_mulSingle\n#align finset.univ_sum_single Finset.univ_sum_single\n\n@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n cases nonempty_fintype I\n ","nextTactic":"ext k","declUpToTactic":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n cases nonempty_fintype I\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.92_0.f38FU5HrzKIlXWd","decl":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h "} | |
| {"state":"case intro.h\nI : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_2\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_3\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), g (Pi.mulSingle i x) = h (Pi.mulSingle i x)\nval✝ : Fintype I\nk : (i : I) → Z i\n⊢ g k = h k","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n haveI := Classical.decEq γ\n Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n ext\n simp\n#align pi_eq_sum_univ pi_eq_sum_univ\n\nsection MulSingle\n\nvariable {I : Type*} [DecidableEq I] {Z : I → Type*}\n\nvariable [∀ i, CommMonoid (Z i)]\n\n@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n (∏ i, Pi.mulSingle i (f i)) = f := by\n ext a\n simp\n#align finset.univ_prod_mul_single Finset.univ_prod_mulSingle\n#align finset.univ_sum_single Finset.univ_sum_single\n\n@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n cases nonempty_fintype I\n ext k\n ","nextTactic":"rw [← Finset.univ_prod_mulSingle k]","declUpToTactic":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n cases nonempty_fintype I\n ext k\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.92_0.f38FU5HrzKIlXWd","decl":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h "} | |
| {"state":"case intro.h\nI : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_2\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_3\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), g (Pi.mulSingle i x) = h (Pi.mulSingle i x)\nval✝ : Fintype I\nk : (i : I) → Z i\n⊢ g (∏ i : I, Pi.mulSingle i (k i)) = h (∏ i : I, Pi.mulSingle i (k i))","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n haveI := Classical.decEq γ\n Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n ext\n simp\n#align pi_eq_sum_univ pi_eq_sum_univ\n\nsection MulSingle\n\nvariable {I : Type*} [DecidableEq I] {Z : I → Type*}\n\nvariable [∀ i, CommMonoid (Z i)]\n\n@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n (∏ i, Pi.mulSingle i (f i)) = f := by\n ext a\n simp\n#align finset.univ_prod_mul_single Finset.univ_prod_mulSingle\n#align finset.univ_sum_single Finset.univ_sum_single\n\n@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n cases nonempty_fintype I\n ext k\n rw [← Finset.univ_prod_mulSingle k]\n ","nextTactic":"rw [map_prod]","declUpToTactic":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n cases nonempty_fintype I\n ext k\n rw [← Finset.univ_prod_mulSingle k]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.92_0.f38FU5HrzKIlXWd","decl":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h "} | |
| {"state":"case intro.h\nI : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_2\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_3\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), g (Pi.mulSingle i x) = h (Pi.mulSingle i x)\nval✝ : Fintype I\nk : (i : I) → Z i\n⊢ ∏ x : I, g (Pi.mulSingle x (k x)) = h (∏ i : I, Pi.mulSingle i (k i))","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n haveI := Classical.decEq γ\n Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n ext\n simp\n#align pi_eq_sum_univ pi_eq_sum_univ\n\nsection MulSingle\n\nvariable {I : Type*} [DecidableEq I] {Z : I → Type*}\n\nvariable [∀ i, CommMonoid (Z i)]\n\n@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n (∏ i, Pi.mulSingle i (f i)) = f := by\n ext a\n simp\n#align finset.univ_prod_mul_single Finset.univ_prod_mulSingle\n#align finset.univ_sum_single Finset.univ_sum_single\n\n@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n cases nonempty_fintype I\n ext k\n rw [← Finset.univ_prod_mulSingle k]\n rw [map_prod]\n ","nextTactic":"rw [map_prod]","declUpToTactic":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n cases nonempty_fintype I\n ext k\n rw [← Finset.univ_prod_mulSingle k]\n rw [map_prod]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.92_0.f38FU5HrzKIlXWd","decl":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h "} | |
| {"state":"case intro.h\nI : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_2\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_3\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), g (Pi.mulSingle i x) = h (Pi.mulSingle i x)\nval✝ : Fintype I\nk : (i : I) → Z i\n⊢ ∏ x : I, g (Pi.mulSingle x (k x)) = ∏ x : I, h (Pi.mulSingle x (k x))","srcUpToTactic":"/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Algebra.Ring.Pi\n\n#align_import algebra.big_operators.pi from \"leanprover-community/mathlib\"@\"fa2309577c7009ea243cffdf990cd6c84f0ad497\"\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\n\nopen BigOperators\n\nnamespace Pi\n\n@[to_additive]\ntheorem list_prod_apply {α : Type*} {β : α → Type*} [∀ a, Monoid (β a)] (a : α)\n (l : List (∀ a, β a)) : l.prod a = (l.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_list_prod _\n#align pi.list_prod_apply Pi.list_prod_apply\n#align pi.list_sum_apply Pi.list_sum_apply\n\n@[to_additive]\ntheorem multiset_prod_apply {α : Type*} {β : α → Type*} [∀ a, CommMonoid (β a)] (a : α)\n (s : Multiset (∀ a, β a)) : s.prod a = (s.map fun f : ∀ a, β a ↦ f a).prod :=\n (evalMonoidHom β a).map_multiset_prod _\n#align pi.multiset_prod_apply Pi.multiset_prod_apply\n#align pi.multiset_sum_apply Pi.multiset_sum_apply\n\nend Pi\n\n@[to_additive (attr := simp)]\ntheorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)\n (s : Finset γ) (g : γ → ∀ a, β a) : (∏ c in s, g c) a = ∏ c in s, g c a :=\n map_prod (Pi.evalMonoidHom β a) _ _\n#align finset.prod_apply Finset.prod_apply\n#align finset.sum_apply Finset.sum_apply\n\n/-- An 'unapplied' analogue of `Finset.prod_apply`. -/\n@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)\n (g : γ → ∀ a, β a) : ∏ c in s, g c = fun a ↦ ∏ c in s, g c a :=\n funext fun _ ↦ Finset.prod_apply _ _ _\n#align finset.prod_fn Finset.prod_fn\n#align finset.sum_fn Finset.sum_fn\n\n@[to_additive]\ntheorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]\n [∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=\n Finset.prod_apply a Finset.univ g\n#align fintype.prod_apply Fintype.prod_apply\n#align fintype.sum_apply Fintype.sum_apply\n\n@[to_additive prod_mk_sum]\ntheorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)\n (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) :=\n haveI := Classical.decEq γ\n Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])\n#align prod_mk_prod prod_mk_prod\n#align prod_mk_sum prod_mk_sum\n\n/-- decomposing `x : ι → R` as a sum along the canonical basis -/\ntheorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]\n (x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by\n ext\n simp\n#align pi_eq_sum_univ pi_eq_sum_univ\n\nsection MulSingle\n\nvariable {I : Type*} [DecidableEq I] {Z : I → Type*}\n\nvariable [∀ i, CommMonoid (Z i)]\n\n@[to_additive]\ntheorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :\n (∏ i, Pi.mulSingle i (f i)) = f := by\n ext a\n simp\n#align finset.univ_prod_mul_single Finset.univ_prod_mulSingle\n#align finset.univ_sum_single Finset.univ_sum_single\n\n@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n cases nonempty_fintype I\n ext k\n rw [← Finset.univ_prod_mulSingle k]\n rw [map_prod]\n rw [map_prod]\n ","nextTactic":"simp only [H]","declUpToTactic":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by\n cases nonempty_fintype I\n ext k\n rw [← Finset.univ_prod_mulSingle k]\n rw [map_prod]\n rw [map_prod]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_BigOperators_Pi.92_0.f38FU5HrzKIlXWd","decl":"@[to_additive]\ntheorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)\n (H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h "} | |