Documentation

Mathlib.Algebra.BigOperators.Pi

Big operators for Pi Types #

This file contains theorems relevant to big operators in binary and arbitrary product of monoids and groups

theorem Pi.list_sum_apply {α : Type u_1} {β : αType u_2} [(a : α) → AddMonoid (β a)] (a : α) (l : List ((a : α) → β a)) :
l.sum a = (List.map (fun (f : (a : α) → β a) => f a) l).sum
theorem Pi.list_prod_apply {α : Type u_1} {β : αType u_2} [(a : α) → Monoid (β a)] (a : α) (l : List ((a : α) → β a)) :
l.prod a = (List.map (fun (f : (a : α) → β a) => f a) l).prod
theorem Pi.multiset_sum_apply {α : Type u_1} {β : αType u_2} [(a : α) → AddCommMonoid (β a)] (a : α) (s : Multiset ((a : α) → β a)) :
s.sum a = (Multiset.map (fun (f : (a : α) → β a) => f a) s).sum
theorem Pi.multiset_prod_apply {α : Type u_1} {β : αType u_2} [(a : α) → CommMonoid (β a)] (a : α) (s : Multiset ((a : α) → β a)) :
s.prod a = (Multiset.map (fun (f : (a : α) → β a) => f a) s).prod
@[simp]
theorem Finset.sum_apply {α : Type u_1} {β : αType u_2} {γ : Type u_3} [(a : α) → AddCommMonoid (β a)] (a : α) (s : Finset γ) (g : γ(a : α) → β a) :
(∑ cs, g c) a = cs, g c a
@[simp]
theorem Finset.prod_apply {α : Type u_1} {β : αType u_2} {γ : Type u_3} [(a : α) → CommMonoid (β a)] (a : α) (s : Finset γ) (g : γ(a : α) → β a) :
(∏ cs, g c) a = cs, g c a
theorem Finset.sum_fn {α : Type u_1} {β : αType u_2} {γ : Type u_3} [(a : α) → AddCommMonoid (β a)] (s : Finset γ) (g : γ(a : α) → β a) :
cs, g c = fun (a : α) => cs, g c a

An 'unapplied' analogue of Finset.sum_apply.

theorem Finset.prod_fn {α : Type u_1} {β : αType u_2} {γ : Type u_3} [(a : α) → CommMonoid (β a)] (s : Finset γ) (g : γ(a : α) → β a) :
cs, g c = fun (a : α) => cs, g c a

An 'unapplied' analogue of Finset.prod_apply.

theorem Fintype.sum_apply {α : Type u_1} {β : αType u_2} {γ : Type u_3} [Fintype γ] [(a : α) → AddCommMonoid (β a)] (a : α) (g : γ(a : α) → β a) :
(∑ c : γ, g c) a = c : γ, g c a
theorem Fintype.prod_apply {α : Type u_1} {β : αType u_2} {γ : Type u_3} [Fintype γ] [(a : α) → CommMonoid (β a)] (a : α) (g : γ(a : α) → β a) :
(∏ c : γ, g c) a = c : γ, g c a
theorem prod_mk_sum {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [AddCommMonoid β] (s : Finset γ) (f : γα) (g : γβ) :
(xs, f x, xs, g x) = xs, (f x, g x)
theorem prod_mk_prod {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γα) (g : γβ) :
(xs, f x, xs, g x) = xs, (f x, g x)
theorem pi_eq_sum_univ {ι : Type u_1} [Fintype ι] [DecidableEq ι] {R : Type u_2} [Semiring R] (x : ιR) :
x = i : ι, x i fun (j : ι) => if i = j then 1 else 0

decomposing x : ι → R as a sum along the canonical basis

theorem Finset.univ_sum_single {I : Type u_1} [DecidableEq I] {Z : IType u_2} [(i : I) → AddCommMonoid (Z i)] [Fintype I] (f : (i : I) → Z i) :
i : I, Pi.single i (f i) = f
theorem Finset.univ_prod_mulSingle {I : Type u_1} [DecidableEq I] {Z : IType u_2} [(i : I) → CommMonoid (Z i)] [Fintype I] (f : (i : I) → Z i) :
i : I, Pi.mulSingle i (f i) = f
theorem AddMonoidHom.functions_ext {I : Type u_1} [DecidableEq I] {Z : IType u_2} [(i : I) → AddCommMonoid (Z i)] [Finite I] (G : Type u_3) [AddCommMonoid G] (g : ((i : I) → Z i) →+ G) (h : ((i : I) → Z i) →+ G) (H : ∀ (i : I) (x : Z i), g (Pi.single i x) = h (Pi.single i x)) :
g = h
theorem MonoidHom.functions_ext {I : Type u_1} [DecidableEq I] {Z : IType u_2} [(i : I) → CommMonoid (Z i)] [Finite I] (G : Type u_3) [CommMonoid G] (g : ((i : I) → Z i) →* G) (h : ((i : I) → Z i) →* G) (H : ∀ (i : I) (x : Z i), g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) :
g = h
theorem AddMonoidHom.functions_ext' {I : Type u_1} [DecidableEq I] {Z : IType u_2} [(i : I) → AddCommMonoid (Z i)] [Finite I] (M : Type u_3) [AddCommMonoid M] (g : ((i : I) → Z i) →+ M) (h : ((i : I) → Z i) →+ M) (H : ∀ (i : I), g.comp (AddMonoidHom.single Z i) = h.comp (AddMonoidHom.single Z i)) :
g = h

This is used as the ext lemma instead of AddMonoidHom.functions_ext for reasons explained in note [partially-applied ext lemmas].

theorem AddMonoidHom.functions_ext'_iff {I : Type u_1} [DecidableEq I] {Z : IType u_2} [(i : I) → AddCommMonoid (Z i)] [Finite I] {M : Type u_3} [AddCommMonoid M] {g : ((i : I) → Z i) →+ M} {h : ((i : I) → Z i) →+ M} :
g = h ∀ (i : I), g.comp (AddMonoidHom.single Z i) = h.comp (AddMonoidHom.single Z i)
theorem MonoidHom.functions_ext'_iff {I : Type u_1} [DecidableEq I] {Z : IType u_2} [(i : I) → CommMonoid (Z i)] [Finite I] {M : Type u_3} [CommMonoid M] {g : ((i : I) → Z i) →* M} {h : ((i : I) → Z i) →* M} :
g = h ∀ (i : I), g.comp (MonoidHom.mulSingle Z i) = h.comp (MonoidHom.mulSingle Z i)
theorem MonoidHom.functions_ext' {I : Type u_1} [DecidableEq I] {Z : IType u_2} [(i : I) → CommMonoid (Z i)] [Finite I] (M : Type u_3) [CommMonoid M] (g : ((i : I) → Z i) →* M) (h : ((i : I) → Z i) →* M) (H : ∀ (i : I), g.comp (MonoidHom.mulSingle Z i) = h.comp (MonoidHom.mulSingle Z i)) :
g = h

This is used as the ext lemma instead of MonoidHom.functions_ext for reasons explained in note [partially-applied ext lemmas].

theorem RingHom.functions_ext_iff {I : Type u_1} [DecidableEq I] {f : IType u_2} [(i : I) → NonAssocSemiring (f i)] [Finite I] {G : Type u_3} [NonAssocSemiring G] {g : ((i : I) → f i) →+* G} {h : ((i : I) → f i) →+* G} :
g = h ∀ (i : I) (x : f i), g (Pi.single i x) = h (Pi.single i x)
theorem RingHom.functions_ext {I : Type u_1} [DecidableEq I] {f : IType u_2} [(i : I) → NonAssocSemiring (f i)] [Finite I] (G : Type u_3) [NonAssocSemiring G] (g : ((i : I) → f i) →+* G) (h : ((i : I) → f i) →+* G) (H : ∀ (i : I) (x : f i), g (Pi.single i x) = h (Pi.single i x)) :
g = h
theorem Prod.fst_sum {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [AddCommMonoid β] {s : Finset γ} {f : γα × β} :
(∑ cs, f c).1 = cs, (f c).1
theorem Prod.fst_prod {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [CommMonoid β] {s : Finset γ} {f : γα × β} :
(∏ cs, f c).1 = cs, (f c).1
theorem Prod.snd_sum {α : Type u_1} {β : Type u_2} {γ : Type u_3} [AddCommMonoid α] [AddCommMonoid β] {s : Finset γ} {f : γα × β} :
(∑ cs, f c).2 = cs, (f c).2
theorem Prod.snd_prod {α : Type u_1} {β : Type u_2} {γ : Type u_3} [CommMonoid α] [CommMonoid β] {s : Finset γ} {f : γα × β} :
(∏ cs, f c).2 = cs, (f c).2