# Big operators #

In this file we define products and sums indexed by finite sets (specifically, Finset).

## Notation #

We introduce the following notation.

Let s be a Finset α, and f : α → β a function.

• ∏ x ∈ s, f x is notation for Finset.prod s f (assuming β is a CommMonoid)
• ∑ x ∈ s, f x is notation for Finset.sum s f (assuming β is an AddCommMonoid)
• ∏ x, f x is notation for Finset.prod Finset.univ f (assuming α is a Fintype and β is a CommMonoid)
• ∑ x, f x is notation for Finset.sum Finset.univ f (assuming α is a Fintype and β is an AddCommMonoid)

## Implementation Notes #

The first arguments in all definitions and lemmas is the codomain of the function of the big operator. This is necessary for the heuristic in @[to_additive]. See the documentation of to_additive.attr for more information.

def Finset.sum {α : Type u_3} {β : Type u_4} [] (s : ) (f : αβ) :
β

∑ x ∈ s, f x is the sum of f x as x ranges over the elements of the finite set s.

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def Finset.prod {α : Type u_3} {β : Type u_4} [] (s : ) (f : αβ) :
β

∏ x ∈ s, f x is the product of f x as x ranges over the elements of the finite set s.

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@[simp]
theorem Finset.sum_mk {α : Type u_3} {β : Type u_4} [] (s : ) (hs : s.Nodup) (f : αβ) :
{ val := s, nodup := hs }.sum f = ().sum
@[simp]
theorem Finset.prod_mk {α : Type u_3} {β : Type u_4} [] (s : ) (hs : s.Nodup) (f : αβ) :
{ val := s, nodup := hs }.prod f = ().prod
@[simp]
theorem Finset.sum_val {α : Type u_3} [] (s : ) :
s.val.sum = s.sum id
@[simp]
theorem Finset.prod_val {α : Type u_3} [] (s : ) :
s.val.prod = s.prod id

A bigOpBinder is like an extBinder and has the form x, x : ty, or x pred where pred is a binderPred like < 2. Unlike extBinder, x is a term.

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A BigOperator binder in parentheses

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A list of parenthesized binders

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A single (unparenthesized) binder, or a list of parenthesized binders

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def BigOperators.processBigOpBinder (processed : ) (binder : Lean.TSyntax BigOperators.bigOpBinder) :

Collects additional binder/Finset pairs for the given bigOpBinder. Note: this is not extensible at the moment, unlike the usual bigOpBinder expansions.

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def BigOperators.processBigOpBinders (binders : Lean.TSyntax BigOperators.bigOpBinders) :

Collects the binder/Finset pairs for the given bigOpBinders.

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Collect the binderIdents into a ⟨...⟩ expression.

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def BigOperators.bigOpBindersProd (processed : ) :

Collect the terms into a product of sets.

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• ∑ x, f x is notation for Finset.sum Finset.univ f. It is the sum of f x, where x ranges over the finite domain of f.
• ∑ x ∈ s, f x is notation for Finset.sum s f. It is the sum of f x, where x ranges over the finite set s (either a Finset or a Set with a Fintype instance).
• ∑ x ∈ s with p x, f x is notation for Finset.sum (Finset.filter p s) f.
• ∑ (x ∈ s) (y ∈ t), f x y is notation for Finset.sum (s ×ˢ t) (fun ⟨x, y⟩ ↦ f x y).

These support destructuring, for example ∑ ⟨x, y⟩ ∈ s ×ˢ t, f x y.

Notation: "∑" bigOpBinders* ("with" term)? "," term

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• ∏ x, f x is notation for Finset.prod Finset.univ f. It is the product of f x, where x ranges over the finite domain of f.
• ∏ x ∈ s, f x is notation for Finset.prod s f. It is the product of f x, where x ranges over the finite set s (either a Finset or a Set with a Fintype instance).
• ∏ x ∈ s with p x, f x is notation for Finset.prod (Finset.filter p s) f.
• ∏ (x ∈ s) (y ∈ t), f x y is notation for Finset.prod (s ×ˢ t) (fun ⟨x, y⟩ ↦ f x y).

These support destructuring, for example ∏ ⟨x, y⟩ ∈ s ×ˢ t, f x y.

Notation: "∏" bigOpBinders* ("with" term)? "," term

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(Deprecated, use ∑ x ∈ s, f x) ∑ x in s, f x is notation for Finset.sum s f. It is the sum of f x, where x ranges over the finite set s.

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(Deprecated, use ∏ x ∈ s, f x) ∏ x in s, f x is notation for Finset.prod s f. It is the product of f x, where x ranges over the finite set s.

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Delaborator for Finset.prod. The pp.piBinderTypes option controls whether to show the domain type when the product is over Finset.univ.

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Delaborator for Finset.sum. The pp.piBinderTypes option controls whether to show the domain type when the sum is over Finset.univ.

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theorem Finset.sum_eq_multiset_sum {α : Type u_3} {β : Type u_4} [] (s : ) (f : αβ) :
xs, f x = (Multiset.map f s.val).sum
theorem Finset.prod_eq_multiset_prod {α : Type u_3} {β : Type u_4} [] (s : ) (f : αβ) :
xs, f x = (Multiset.map f s.val).prod
@[simp]
theorem Finset.sum_map_val {α : Type u_3} {β : Type u_4} [] (s : ) (f : αβ) :
(Multiset.map f s.val).sum = as, f a
@[simp]
theorem Finset.prod_map_val {α : Type u_3} {β : Type u_4} [] (s : ) (f : αβ) :
(Multiset.map f s.val).prod = as, f a
theorem Finset.sum_eq_fold {α : Type u_3} {β : Type u_4} [] (s : ) (f : αβ) :
xs, f x = Finset.fold (fun (x x_1 : β) => x + x_1) 0 f s
theorem Finset.prod_eq_fold {α : Type u_3} {β : Type u_4} [] (s : ) (f : αβ) :
xs, f x = Finset.fold (fun (x x_1 : β) => x * x_1) 1 f s
@[simp]
theorem Finset.sum_multiset_singleton {α : Type u_3} (s : ) :
xs, {x} = s.val
@[simp]
theorem map_sum {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] [] {G : Type u_6} [FunLike G β γ] [] (g : G) (f : αβ) (s : ) :
g (xs, f x) = xs, g (f x)
@[simp]
theorem map_prod {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] [] {G : Type u_6} [FunLike G β γ] [] (g : G) (f : αβ) (s : ) :
g (xs, f x) = xs, g (f x)
theorem AddMonoidHom.coe_finset_sum {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] [] (f : αβ →+ γ) (s : ) :
(xs, f x) = xs, (f x)
theorem MonoidHom.coe_finset_prod {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] [] (f : αβ →* γ) (s : ) :
(xs, f x) = xs, (f x)
@[simp]
theorem AddMonoidHom.finset_sum_apply {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] [] (f : αβ →+ γ) (s : ) (b : β) :
(xs, f x) b = xs, (f x) b

See also Finset.sum_apply, with the same conclusion but with the weaker hypothesis f : α → β → γ

@[simp]
theorem MonoidHom.finset_prod_apply {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] [] (f : αβ →* γ) (s : ) (b : β) :
(xs, f x) b = xs, (f x) b

See also Finset.prod_apply, with the same conclusion but with the weaker hypothesis f : α → β → γ

@[simp]
theorem Finset.sum_empty {α : Type u_3} {β : Type u_4} {f : αβ} [] :
x, f x = 0
@[simp]
theorem Finset.prod_empty {α : Type u_3} {β : Type u_4} {f : αβ} [] :
x, f x = 1
theorem Finset.sum_of_isEmpty {α : Type u_3} {β : Type u_4} {f : αβ} [] [] (s : ) :
is, f i = 0
theorem Finset.prod_of_isEmpty {α : Type u_3} {β : Type u_4} {f : αβ} [] [] (s : ) :
is, f i = 1
@[deprecated Finset.prod_of_isEmpty]
theorem Finset.prod_of_empty {α : Type u_3} {β : Type u_4} {f : αβ} [] [] (s : ) :
is, f i = 1

Alias of Finset.prod_of_isEmpty.

@[deprecated Finset.sum_of_isEmpty]
theorem Finset.sum_of_empty {α : Type u_3} {β : Type u_4} {f : αβ} [] [] (s : ) :
is, f i = 0

Alias of Finset.sum_of_isEmpty.

@[simp]
theorem Finset.sum_cons {α : Type u_3} {β : Type u_4} {s : } {a : α} {f : αβ} [] (h : as) :
xFinset.cons a s h, f x = f a + xs, f x
@[simp]
theorem Finset.prod_cons {α : Type u_3} {β : Type u_4} {s : } {a : α} {f : αβ} [] (h : as) :
xFinset.cons a s h, f x = f a * xs, f x
@[simp]
theorem Finset.sum_insert {α : Type u_3} {β : Type u_4} {s : } {a : α} {f : αβ} [] [] :
asxinsert a s, f x = f a + xs, f x
@[simp]
theorem Finset.prod_insert {α : Type u_3} {β : Type u_4} {s : } {a : α} {f : αβ} [] [] :
asxinsert a s, f x = f a * xs, f x
@[simp]
theorem Finset.sum_insert_of_eq_zero_if_not_mem {α : Type u_3} {β : Type u_4} {s : } {a : α} {f : αβ} [] [] (h : asf a = 0) :
xinsert a s, f x = xs, f x

The sum of f over insert a s is the same as the sum over s, as long as a is in s or f a = 0.

@[simp]
theorem Finset.prod_insert_of_eq_one_if_not_mem {α : Type u_3} {β : Type u_4} {s : } {a : α} {f : αβ} [] [] (h : asf a = 1) :
xinsert a s, f x = xs, f x

The product of f over insert a s is the same as the product over s, as long as a is in s or f a = 1.

@[simp]
theorem Finset.sum_insert_zero {α : Type u_3} {β : Type u_4} {s : } {a : α} {f : αβ} [] [] (h : f a = 0) :
xinsert a s, f x = xs, f x

The sum of f over insert a s is the same as the sum over s, as long as f a = 0.

@[simp]
theorem Finset.prod_insert_one {α : Type u_3} {β : Type u_4} {s : } {a : α} {f : αβ} [] [] (h : f a = 1) :
xinsert a s, f x = xs, f x

The product of f over insert a s is the same as the product over s, as long as f a = 1.

theorem Finset.sum_insert_sub {α : Type u_3} {s : } {a : α} {M : Type u_6} [] [] (ha : as) {f : αM} :
xinsert a s, f x - f a = xs, f x
theorem Finset.prod_insert_div {α : Type u_3} {s : } {a : α} {M : Type u_6} [] [] (ha : as) {f : αM} :
(xinsert a s, f x) / f a = xs, f x
@[simp]
theorem Finset.sum_singleton {α : Type u_3} {β : Type u_4} [] (f : αβ) (a : α) :
x{a}, f x = f a
@[simp]
theorem Finset.prod_singleton {α : Type u_3} {β : Type u_4} [] (f : αβ) (a : α) :
x{a}, f x = f a
theorem Finset.sum_pair {α : Type u_3} {β : Type u_4} {f : αβ} [] [] {a : α} {b : α} (h : a b) :
x{a, b}, f x = f a + f b
theorem Finset.prod_pair {α : Type u_3} {β : Type u_4} {f : αβ} [] [] {a : α} {b : α} (h : a b) :
x{a, b}, f x = f a * f b
@[simp]
theorem Finset.sum_const_zero {α : Type u_3} {β : Type u_4} {s : } [] :
_xs, 0 = 0
@[simp]
theorem Finset.prod_const_one {α : Type u_3} {β : Type u_4} {s : } [] :
_xs, 1 = 1
@[simp]
theorem Finset.sum_image {α : Type u_3} {β : Type u_4} {γ : Type u_5} {f : αβ} [] [] {s : } {g : γα} :
(xs, ys, g x = g yx = y)x, f x = xs, f (g x)
@[simp]
theorem Finset.prod_image {α : Type u_3} {β : Type u_4} {γ : Type u_5} {f : αβ} [] [] {s : } {g : γα} :
(xs, ys, g x = g yx = y)x, f x = xs, f (g x)
@[simp]
theorem Finset.sum_map {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] (s : ) (e : α γ) (f : γβ) :
x, f x = xs, f (e x)
@[simp]
theorem Finset.prod_map {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] (s : ) (e : α γ) (f : γβ) :
x, f x = xs, f (e x)
theorem Finset.sum_attach {α : Type u_3} {β : Type u_4} [] (s : ) (f : αβ) :
xs.attach, f x = xs, f x
theorem Finset.prod_attach {α : Type u_3} {β : Type u_4} [] (s : ) (f : αβ) :
xs.attach, f x = xs, f x
theorem Finset.sum_congr {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} {g : αβ} [] (h : s₁ = s₂) :
(xs₂, f x = g x)s₁.sum f = s₂.sum g
theorem Finset.prod_congr {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} {g : αβ} [] (h : s₁ = s₂) :
(xs₂, f x = g x)s₁.prod f = s₂.prod g
theorem Finset.sum_eq_zero {α : Type u_3} {β : Type u_4} [] {f : αβ} {s : } (h : xs, f x = 0) :
xs, f x = 0
theorem Finset.prod_eq_one {α : Type u_3} {β : Type u_4} [] {f : αβ} {s : } (h : xs, f x = 1) :
xs, f x = 1
theorem Finset.sum_disjUnion {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} [] (h : Disjoint s₁ s₂) :
xs₁.disjUnion s₂ h, f x = xs₁, f x + xs₂, f x
theorem Finset.prod_disjUnion {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} [] (h : Disjoint s₁ s₂) :
xs₁.disjUnion s₂ h, f x = (xs₁, f x) * xs₂, f x
theorem Finset.sum_disjiUnion {ι : Type u_1} {α : Type u_3} {β : Type u_4} {f : αβ} [] (s : ) (t : ι) (h : (s).PairwiseDisjoint t) :
xs.disjiUnion t h, f x = is, xt i, f x
theorem Finset.prod_disjiUnion {ι : Type u_1} {α : Type u_3} {β : Type u_4} {f : αβ} [] (s : ) (t : ι) (h : (s).PairwiseDisjoint t) :
xs.disjiUnion t h, f x = is, xt i, f x
theorem Finset.sum_union_inter {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} [] [] :
xs₁ s₂, f x + xs₁ s₂, f x = xs₁, f x + xs₂, f x
theorem Finset.prod_union_inter {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} [] [] :
(xs₁ s₂, f x) * xs₁ s₂, f x = (xs₁, f x) * xs₂, f x
theorem Finset.sum_union {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} [] [] (h : Disjoint s₁ s₂) :
xs₁ s₂, f x = xs₁, f x + xs₂, f x
theorem Finset.prod_union {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} [] [] (h : Disjoint s₁ s₂) :
xs₁ s₂, f x = (xs₁, f x) * xs₂, f x
theorem Finset.sum_filter_add_sum_filter_not {α : Type u_3} {β : Type u_4} [] (s : ) (p : αProp) [] [(x : α) → Decidable ¬p x] (f : αβ) :
x, f x + xFinset.filter (fun (x : α) => ¬p x) s, f x = xs, f x
theorem Finset.prod_filter_mul_prod_filter_not {α : Type u_3} {β : Type u_4} [] (s : ) (p : αProp) [] [(x : α) → Decidable ¬p x] (f : αβ) :
(x, f x) * xFinset.filter (fun (x : α) => ¬p x) s, f x = xs, f x
@[simp]
theorem Finset.sum_to_list {α : Type u_3} {β : Type u_4} [] (s : ) (f : αβ) :
(List.map f s.toList).sum = s.sum f
@[simp]
theorem Finset.prod_to_list {α : Type u_3} {β : Type u_4} [] (s : ) (f : αβ) :
(List.map f s.toList).prod = s.prod f
theorem Equiv.Perm.sum_comp {α : Type u_3} {β : Type u_4} [] (σ : ) (s : ) (f : αβ) (hs : {a : α | σ a a} s) :
xs, f (σ x) = xs, f x
theorem Equiv.Perm.prod_comp {α : Type u_3} {β : Type u_4} [] (σ : ) (s : ) (f : αβ) (hs : {a : α | σ a a} s) :
xs, f (σ x) = xs, f x
theorem Equiv.Perm.sum_comp' {α : Type u_3} {β : Type u_4} [] (σ : ) (s : ) (f : ααβ) (hs : {a : α | σ a a} s) :
xs, f (σ x) x = xs, f x (() x)
theorem Equiv.Perm.prod_comp' {α : Type u_3} {β : Type u_4} [] (σ : ) (s : ) (f : ααβ) (hs : {a : α | σ a a} s) :
xs, f (σ x) x = xs, f x (() x)
theorem Finset.sum_powerset_insert {α : Type u_3} {β : Type u_4} {s : } {a : α} [] [] (ha : as) (f : β) :
t(insert a s).powerset, f t = ts.powerset, f t + ts.powerset, f (insert a t)

A sum over all subsets of s ∪ {x} is obtained by summing the sum over all subsets of s, and over all subsets of s to which one adds x.

theorem Finset.prod_powerset_insert {α : Type u_3} {β : Type u_4} {s : } {a : α} [] [] (ha : as) (f : β) :
t(insert a s).powerset, f t = (ts.powerset, f t) * ts.powerset, f (insert a t)

A product over all subsets of s ∪ {x} is obtained by multiplying the product over all subsets of s, and over all subsets of s to which one adds x.

theorem Finset.sum_powerset_cons {α : Type u_3} {β : Type u_4} {s : } {a : α} [] (ha : as) (f : β) :
t(Finset.cons a s ha).powerset, f t = ts.powerset, f t + ts.powerset.attach, f (Finset.cons a t )

A sum over all subsets of s ∪ {x} is obtained by summing the sum over all subsets of s, and over all subsets of s to which one adds x.

theorem Finset.prod_powerset_cons {α : Type u_3} {β : Type u_4} {s : } {a : α} [] (ha : as) (f : β) :
t(Finset.cons a s ha).powerset, f t = (ts.powerset, f t) * ts.powerset.attach, f (Finset.cons a t )

A product over all subsets of s ∪ {x} is obtained by multiplying the product over all subsets of s, and over all subsets of s to which one adds x.

theorem Finset.sum_powerset {α : Type u_3} {β : Type u_4} [] (s : ) (f : β) :
ts.powerset, f t = jFinset.range (s.card + 1), t, f t

A sum over powerset s is equal to the double sum over sets of subsets of s with card s = k, for k = 1, ..., card s

theorem Finset.prod_powerset {α : Type u_3} {β : Type u_4} [] (s : ) (f : β) :
ts.powerset, f t = jFinset.range (s.card + 1), t, f t

A product over powerset s is equal to the double product over sets of subsets of s with card s = k, for k = 1, ..., card s.

theorem IsCompl.sum_add_sum {α : Type u_3} {β : Type u_4} [] [] {s : } {t : } (h : IsCompl s t) (f : αβ) :
is, f i + it, f i = i : α, f i
theorem IsCompl.prod_mul_prod {α : Type u_3} {β : Type u_4} [] [] {s : } {t : } (h : IsCompl s t) (f : αβ) :
(is, f i) * it, f i = i : α, f i
theorem Finset.sum_add_sum_compl {α : Type u_3} {β : Type u_4} [] [] [] (s : ) (f : αβ) :
is, f i + is, f i = i : α, f i

Adding the sums of a function over s and over sᶜ gives the whole sum. For a version expressed with subtypes, see Fintype.sum_subtype_add_sum_subtype.

theorem Finset.prod_mul_prod_compl {α : Type u_3} {β : Type u_4} [] [] [] (s : ) (f : αβ) :
(is, f i) * is, f i = i : α, f i

Multiplying the products of a function over s and over sᶜ gives the whole product. For a version expressed with subtypes, see Fintype.prod_subtype_mul_prod_subtype.

theorem Finset.sum_compl_add_sum {α : Type u_3} {β : Type u_4} [] [] [] (s : ) (f : αβ) :
is, f i + is, f i = i : α, f i
theorem Finset.prod_compl_mul_prod {α : Type u_3} {β : Type u_4} [] [] [] (s : ) (f : αβ) :
(is, f i) * is, f i = i : α, f i
theorem Finset.sum_sdiff {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} [] [] (h : s₁ s₂) :
xs₂ \ s₁, f x + xs₁, f x = xs₂, f x
theorem Finset.prod_sdiff {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} [] [] (h : s₁ s₂) :
(xs₂ \ s₁, f x) * xs₁, f x = xs₂, f x
theorem Finset.sum_subset_zero_on_sdiff {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} {g : αβ} [] [] (h : s₁ s₂) (hg : xs₂ \ s₁, g x = 0) (hfg : xs₁, f x = g x) :
is₁, f i = is₂, g i
theorem Finset.prod_subset_one_on_sdiff {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} {g : αβ} [] [] (h : s₁ s₂) (hg : xs₂ \ s₁, g x = 1) (hfg : xs₁, f x = g x) :
is₁, f i = is₂, g i
theorem Finset.sum_subset {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} [] (h : s₁ s₂) (hf : xs₂, xs₁f x = 0) :
xs₁, f x = xs₂, f x
theorem Finset.prod_subset {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} [] (h : s₁ s₂) (hf : xs₂, xs₁f x = 1) :
xs₁, f x = xs₂, f x
@[simp]
theorem Finset.sum_disj_sum {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] (s : ) (t : ) (f : α γβ) :
xs.disjSum t, f x = xs, f () + xt, f ()
@[simp]
theorem Finset.prod_disj_sum {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] (s : ) (t : ) (f : α γβ) :
xs.disjSum t, f x = (xs, f ()) * xt, f ()
theorem Finset.sum_sum_elim {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] (s : ) (t : ) (f : αβ) (g : γβ) :
xs.disjSum t, Sum.elim f g x = xs, f x + xt, g x
theorem Finset.prod_sum_elim {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] (s : ) (t : ) (f : αβ) (g : γβ) :
xs.disjSum t, Sum.elim f g x = (xs, f x) * xt, g x
theorem Finset.sum_biUnion {α : Type u_3} {β : Type u_4} {γ : Type u_5} {f : αβ} [] [] {s : } {t : γ} (hs : (s).PairwiseDisjoint t) :
xs.biUnion t, f x = xs, it x, f i
theorem Finset.prod_biUnion {α : Type u_3} {β : Type u_4} {γ : Type u_5} {f : αβ} [] [] {s : } {t : γ} (hs : (s).PairwiseDisjoint t) :
xs.biUnion t, f x = xs, it x, f i
theorem Finset.sum_sigma {α : Type u_3} {β : Type u_4} [] {σ : αType u_6} (s : ) (t : (a : α) → Finset (σ a)) (f : β) :
xs.sigma t, f x = as, st a, f a, s

Sum over a sigma type equals the sum of fiberwise sums. For rewriting in the reverse direction, use Finset.sum_sigma'

theorem Finset.prod_sigma {α : Type u_3} {β : Type u_4} [] {σ : αType u_6} (s : ) (t : (a : α) → Finset (σ a)) (f : β) :
xs.sigma t, f x = as, st a, f a, s

Product over a sigma type equals the product of fiberwise products. For rewriting in the reverse direction, use Finset.prod_sigma'.

theorem Finset.sum_sigma' {α : Type u_3} {β : Type u_4} [] {σ : αType u_6} (s : ) (t : (a : α) → Finset (σ a)) (f : (a : α) → σ aβ) :
as, st a, f a s = xs.sigma t, f x.fst x.snd
theorem Finset.prod_sigma' {α : Type u_3} {β : Type u_4} [] {σ : αType u_6} (s : ) (t : (a : α) → Finset (σ a)) (f : (a : α) → σ aβ) :
as, st a, f a s = xs.sigma t, f x.fst x.snd
theorem Finset.sum_bij {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } {f : ια} {g : κα} (i : (a : ι) → a sκ) (hi : ∀ (a : ι) (ha : a s), i a ha t) (i_inj : ∀ (a₁ : ι) (ha₁ : a₁ s) (a₂ : ι) (ha₂ : a₂ s), i a₁ ha₁ = i a₂ ha₂a₁ = a₂) (i_surj : bt, ∃ (a : ι) (ha : a s), i a ha = b) (h : ∀ (a : ι) (ha : a s), f a = g (i a ha)) :
xs, f x = xt, g x

Reorder a sum.

The difference with Finset.sum_bij' is that the bijection is specified as a surjective injection, rather than by an inverse function.

The difference with Finset.sum_nbij is that the bijection is allowed to use membership of the domain of the sum, rather than being a non-dependent function.

theorem Finset.prod_bij {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } {f : ια} {g : κα} (i : (a : ι) → a sκ) (hi : ∀ (a : ι) (ha : a s), i a ha t) (i_inj : ∀ (a₁ : ι) (ha₁ : a₁ s) (a₂ : ι) (ha₂ : a₂ s), i a₁ ha₁ = i a₂ ha₂a₁ = a₂) (i_surj : bt, ∃ (a : ι) (ha : a s), i a ha = b) (h : ∀ (a : ι) (ha : a s), f a = g (i a ha)) :
xs, f x = xt, g x

Reorder a product.

The difference with Finset.prod_bij' is that the bijection is specified as a surjective injection, rather than by an inverse function.

The difference with Finset.prod_nbij is that the bijection is allowed to use membership of the domain of the product, rather than being a non-dependent function.

theorem Finset.sum_bij' {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } {f : ια} {g : κα} (i : (a : ι) → a sκ) (j : (a : κ) → a tι) (hi : ∀ (a : ι) (ha : a s), i a ha t) (hj : ∀ (a : κ) (ha : a t), j a ha s) (left_inv : ∀ (a : ι) (ha : a s), j (i a ha) = a) (right_inv : ∀ (a : κ) (ha : a t), i (j a ha) = a) (h : ∀ (a : ι) (ha : a s), f a = g (i a ha)) :
xs, f x = xt, g x

Reorder a sum.

The difference with Finset.sum_bij is that the bijection is specified with an inverse, rather than as a surjective injection.

The difference with Finset.sum_nbij' is that the bijection and its inverse are allowed to use membership of the domains of the sums, rather than being non-dependent functions.

theorem Finset.prod_bij' {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } {f : ια} {g : κα} (i : (a : ι) → a sκ) (j : (a : κ) → a tι) (hi : ∀ (a : ι) (ha : a s), i a ha t) (hj : ∀ (a : κ) (ha : a t), j a ha s) (left_inv : ∀ (a : ι) (ha : a s), j (i a ha) = a) (right_inv : ∀ (a : κ) (ha : a t), i (j a ha) = a) (h : ∀ (a : ι) (ha : a s), f a = g (i a ha)) :
xs, f x = xt, g x

Reorder a product.

The difference with Finset.prod_bij is that the bijection is specified with an inverse, rather than as a surjective injection.

The difference with Finset.prod_nbij' is that the bijection and its inverse are allowed to use membership of the domains of the products, rather than being non-dependent functions.

theorem Finset.sum_nbij {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } {f : ια} {g : κα} (i : ικ) (hi : as, i a t) (i_inj : Set.InjOn i s) (i_surj : Set.SurjOn i s t) (h : as, f a = g (i a)) :
xs, f x = xt, g x

Reorder a sum.

The difference with Finset.sum_nbij' is that the bijection is specified as a surjective injection, rather than by an inverse function.

The difference with Finset.sum_bij is that the bijection is a non-dependent function, rather than being allowed to use membership of the domain of the sum.

theorem Finset.prod_nbij {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } {f : ια} {g : κα} (i : ικ) (hi : as, i a t) (i_inj : Set.InjOn i s) (i_surj : Set.SurjOn i s t) (h : as, f a = g (i a)) :
xs, f x = xt, g x

Reorder a product.

The difference with Finset.prod_nbij' is that the bijection is specified as a surjective injection, rather than by an inverse function.

The difference with Finset.prod_bij is that the bijection is a non-dependent function, rather than being allowed to use membership of the domain of the product.

theorem Finset.sum_nbij' {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } {f : ια} {g : κα} (i : ικ) (j : κι) (hi : as, i a t) (hj : at, j a s) (left_inv : as, j (i a) = a) (right_inv : at, i (j a) = a) (h : as, f a = g (i a)) :
xs, f x = xt, g x

Reorder a sum.

The difference with Finset.sum_nbij is that the bijection is specified with an inverse, rather than as a surjective injection.

The difference with Finset.sum_bij' is that the bijection and its inverse are non-dependent functions, rather than being allowed to use membership of the domains of the sums.

The difference with Finset.sum_equiv is that bijectivity is only required to hold on the domains of the sums, rather than on the entire types.

theorem Finset.prod_nbij' {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } {f : ια} {g : κα} (i : ικ) (j : κι) (hi : as, i a t) (hj : at, j a s) (left_inv : as, j (i a) = a) (right_inv : at, i (j a) = a) (h : as, f a = g (i a)) :
xs, f x = xt, g x

Reorder a product.

The difference with Finset.prod_nbij is that the bijection is specified with an inverse, rather than as a surjective injection.

The difference with Finset.prod_bij' is that the bijection and its inverse are non-dependent functions, rather than being allowed to use membership of the domains of the products.

The difference with Finset.prod_equiv is that bijectivity is only required to hold on the domains of the products, rather than on the entire types.

theorem Finset.sum_equiv {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } {f : ια} {g : κα} (e : ι κ) (hst : ∀ (i : ι), i s e i t) (hfg : is, f i = g (e i)) :
is, f i = it, g i

Specialization of Finset.sum_nbij' that automatically fills in most arguments.

See Fintype.sum_equiv for the version where s and t are univ.

theorem Finset.prod_equiv {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } {f : ια} {g : κα} (e : ι κ) (hst : ∀ (i : ι), i s e i t) (hfg : is, f i = g (e i)) :
is, f i = it, g i

Specialization of Finset.prod_nbij' that automatically fills in most arguments.

See Fintype.prod_equiv for the version where s and t are univ.

theorem Finset.sum_bijective {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } {f : ια} {g : κα} (e : ικ) (he : ) (hst : ∀ (i : ι), i s e i t) (hfg : is, f i = g (e i)) :
is, f i = it, g i

Specialization of Finset.sum_bij that automatically fills in most arguments.

See Fintype.sum_bijective for the version where s and t are univ.

theorem Finset.prod_bijective {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } {f : ια} {g : κα} (e : ικ) (he : ) (hst : ∀ (i : ι), i s e i t) (hfg : is, f i = g (e i)) :
is, f i = it, g i

Specialization of Finset.prod_bij that automatically fills in most arguments.

See Fintype.prod_bijective for the version where s and t are univ.

theorem Finset.sum_of_injOn {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } {f : ια} {g : κα} (e : ικ) (he : Set.InjOn e s) (hest : Set.MapsTo e s t) (h' : it, ie '' sg i = 0) (h : is, f i = g (e i)) :
is, f i = jt, g j
theorem Finset.prod_of_injOn {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } {f : ια} {g : κα} (e : ικ) (he : Set.InjOn e s) (hest : Set.MapsTo e s t) (h' : it, ie '' sg i = 1) (h : is, f i = g (e i)) :
is, f i = jt, g j
theorem Finset.sum_fiberwise_eq_sum_filter {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] [] (s : ) (t : ) (g : ικ) (f : ια) :
jt, iFinset.filter (fun (i : ι) => g i = j) s, f i = iFinset.filter (fun (i : ι) => g i t) s, f i
theorem Finset.prod_fiberwise_eq_prod_filter {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] [] (s : ) (t : ) (g : ικ) (f : ια) :
jt, iFinset.filter (fun (i : ι) => g i = j) s, f i = iFinset.filter (fun (i : ι) => g i t) s, f i
theorem Finset.sum_fiberwise_eq_sum_filter' {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] [] (s : ) (t : ) (g : ικ) (f : κα) :
jt, _iFinset.filter (fun (i : ι) => g i = j) s, f j = iFinset.filter (fun (i : ι) => g i t) s, f (g i)
theorem Finset.prod_fiberwise_eq_prod_filter' {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] [] (s : ) (t : ) (g : ικ) (f : κα) :
jt, _iFinset.filter (fun (i : ι) => g i = j) s, f j = iFinset.filter (fun (i : ι) => g i t) s, f (g i)
theorem Finset.sum_fiberwise_of_maps_to {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } [] {g : ικ} (h : is, g i t) (f : ια) :
jt, iFinset.filter (fun (i : ι) => g i = j) s, f i = is, f i
theorem Finset.prod_fiberwise_of_maps_to {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } [] {g : ικ} (h : is, g i t) (f : ια) :
jt, iFinset.filter (fun (i : ι) => g i = j) s, f i = is, f i
theorem Finset.sum_fiberwise_of_maps_to' {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } [] {g : ικ} (h : is, g i t) (f : κα) :
jt, _iFinset.filter (fun (i : ι) => g i = j) s, f j = is, f (g i)
theorem Finset.prod_fiberwise_of_maps_to' {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] {s : } {t : } [] {g : ικ} (h : is, g i t) (f : κα) :
jt, _iFinset.filter (fun (i : ι) => g i = j) s, f j = is, f (g i)
theorem Finset.sum_fiberwise {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] [] [] (s : ) (g : ικ) (f : ια) :
j : κ, iFinset.filter (fun (i : ι) => g i = j) s, f i = is, f i
theorem Finset.prod_fiberwise {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] [] [] (s : ) (g : ικ) (f : ια) :
j : κ, iFinset.filter (fun (i : ι) => g i = j) s, f i = is, f i
theorem Finset.sum_fiberwise' {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] [] [] (s : ) (g : ικ) (f : κα) :
j : κ, _iFinset.filter (fun (i : ι) => g i = j) s, f j = is, f (g i)
theorem Finset.prod_fiberwise' {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [] [] [] (s : ) (g : ικ) (f : κα) :
j : κ, _iFinset.filter (fun (i : ι) => g i = j) s, f j = is, f (g i)
theorem Finset.sum_univ_pi {ι : Type u_1} {β : Type u_4} [] [] [] {κ : ιType u_6} (t : (i : ι) → Finset (κ i)) (f : ((i : ι) → i Finset.univκ i)β) :
xFinset.univ.pi t, f x = x, f fun (a : ι) (x_1 : a Finset.univ) => x a

Taking a sum over univ.pi t is the same as taking the sum over Fintype.piFinset t. univ.pi t and Fintype.piFinset t are essentially the same Finset, but differ in the type of their element, univ.pi t is a Finset (Π a ∈ univ, t a) and Fintype.piFinset t is a Finset (Π a, t a).

theorem Finset.prod_univ_pi {ι : Type u_1} {β : Type u_4} [] [] [] {κ : ιType u_6} (t : (i : ι) → Finset (κ i)) (f : ((i : ι) → i Finset.univκ i)β) :
xFinset.univ.pi t, f x = x, f fun (a : ι) (x_1 : a Finset.univ) => x a

Taking a product over univ.pi t is the same as taking the product over Fintype.piFinset t. univ.pi t and Fintype.piFinset t are essentially the same Finset, but differ in the type of their element, univ.pi t is a Finset (Π a ∈ univ, t a) and Fintype.piFinset t is a Finset (Π a, t a).

@[simp]
theorem Finset.sum_diag {α : Type u_3} {β : Type u_4} [] [] (s : ) (f : α × αβ) :
is.diag, f i = is, f (i, i)
@[simp]
theorem Finset.prod_diag {α : Type u_3} {β : Type u_4} [] [] (s : ) (f : α × αβ) :
is.diag, f i = is, f (i, i)
theorem Finset.sum_finset_product {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] (r : Finset (γ × α)) (s : ) (t : γ) (h : ∀ (p : γ × α), p r p.1 s p.2 t p.1) {f : γ × αβ} :
pr, f p = cs, at c, f (c, a)
theorem Finset.prod_finset_product {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] (r : Finset (γ × α)) (s : ) (t : γ) (h : ∀ (p : γ × α), p r p.1 s p.2 t p.1) {f : γ × αβ} :
pr, f p = cs, at c, f (c, a)
theorem Finset.sum_finset_product' {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] (r : Finset (γ × α)) (s : ) (t : γ) (h : ∀ (p : γ × α), p r p.1 s p.2 t p.1) {f : γαβ} :
pr, f p.1 p.2 = cs, at c, f c a
theorem Finset.prod_finset_product' {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] (r : Finset (γ × α)) (s : ) (t : γ) (h : ∀ (p : γ × α), p r p.1 s p.2 t p.1) {f : γαβ} :
pr, f p.1 p.2 = cs, at c, f c a
theorem Finset.sum_finset_product_right {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] (r : Finset (α × γ)) (s : ) (t : γ) (h : ∀ (p : α × γ), p r p.2 s p.1 t p.2) {f : α × γβ} :
pr, f p = cs, at c, f (a, c)
theorem Finset.prod_finset_product_right {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] (r : Finset (α × γ)) (s : ) (t : γ) (h : ∀ (p : α × γ), p r p.2 s p.1 t p.2) {f : α × γβ} :
pr, f p = cs, at c, f (a, c)
theorem Finset.sum_finset_product_right' {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] (r : Finset (α × γ)) (s : ) (t : γ) (h : ∀ (p : α × γ), p r p.2 s p.1 t p.2) {f : αγβ} :
pr, f p.1 p.2 = cs, at c, f a c
theorem Finset.prod_finset_product_right' {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] (r : Finset (α × γ)) (s : ) (t : γ) (h : ∀ (p : α × γ), p r p.2 s p.1 t p.2) {f : αγβ} :
pr, f p.1 p.2 = cs, at c, f a c
theorem Finset.sum_image' {α : Type u_3} {β : Type u_4} {γ : Type u_5} {f : αβ} [] [] {s : } {g : γα} (h : γβ) (eq : cs, f (g c) = xFinset.filter (fun (c' : γ) => g c' = g c) s, h x) :
x, f x = xs, h x
abbrev Finset.sum_image'.match_1 {α : Type u_2} {γ : Type u_1} {s : } {g : γα} (_x : α) (motive : (as, g a = _x)Prop) :
∀ (x : as, g a = _x), (∀ (c : γ) (hcs : c s) (hc : g c = _x), motive )motive x
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theorem Finset.prod_image' {α : Type u_3} {β : Type u_4} {γ : Type u_5} {f : αβ} [] [] {s : } {g : γα} (h : γβ) (eq : cs, f (g c) = xFinset.filter (fun (c' : γ) => g c' = g c) s, h x) :
x, f x = xs, h x
theorem Finset.sum_add_distrib {α : Type u_3} {β : Type u_4} {s : } {f : αβ} {g : αβ} [] :
xs, (f x + g x) = xs, f x + xs, g x
theorem Finset.prod_mul_distrib {α : Type u_3} {β : Type u_4} {s : } {f : αβ} {g : αβ} [] :
xs, f x * g x = (xs, f x) * xs, g x
theorem Finset.sum_add_sum_comm {α : Type u_3} {β : Type u_4} {s : } [] (f : αβ) (g : αβ) (h : αβ) (i : αβ) :
as, (f a + g a) + as, (h a + i a) = as, (f a + h a) + as, (g a + i a)
theorem Finset.prod_mul_prod_comm {α : Type u_3} {β : Type u_4} {s : } [] (f : αβ) (g : αβ) (h : αβ) (i : αβ) :
(as, f a * g a) * as, h a * i a = (as, f a * h a) * as, g a * i a
theorem Finset.sum_product {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {t : } {f : γ × αβ} :
xs ×ˢ t, f x = xs, yt, f (x, y)
theorem Finset.prod_product {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {t : } {f : γ × αβ} :
xs ×ˢ t, f x = xs, yt, f (x, y)
theorem Finset.sum_product' {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {t : } {f : γαβ} :
xs ×ˢ t, f x.1 x.2 = xs, yt, f x y

An uncurried version of Finset.sum_product

theorem Finset.prod_product' {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {t : } {f : γαβ} :
xs ×ˢ t, f x.1 x.2 = xs, yt, f x y

An uncurried version of Finset.prod_product.

theorem Finset.sum_product_right {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {t : } {f : γ × αβ} :
xs ×ˢ t, f x = yt, xs, f (x, y)
theorem Finset.prod_product_right {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {t : } {f : γ × αβ} :
xs ×ˢ t, f x = yt, xs, f (x, y)
theorem Finset.sum_product_right' {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {t : } {f : γαβ} :
xs ×ˢ t, f x.1 x.2 = yt, xs, f x y

An uncurried version of Finset.sum_product_right

theorem Finset.prod_product_right' {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {t : } {f : γαβ} :
xs ×ˢ t, f x.1 x.2 = yt, xs, f x y

An uncurried version of Finset.prod_product_right.

theorem Finset.sum_comm' {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {t : γ} {t' : } {s' : α} (h : ∀ (x : γ) (y : α), x s y t x x s' y y t') {f : γαβ} :
xs, yt x, f x y = yt', xs' y, f x y

Generalization of Finset.sum_comm to the case when the inner Finsets depend on the outer variable.

abbrev Finset.sum_comm'.match_1 {α : Type u_2} {γ : Type u_1} (motive : γ × αProp) :
∀ (x : γ × α), (∀ (x : γ) (y : α), motive (x, y))motive x
Equations
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Instances For
theorem Finset.prod_comm' {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {t : γ} {t' : } {s' : α} (h : ∀ (x : γ) (y : α), x s y t x x s' y y t') {f : γαβ} :
xs, yt x, f x y = yt', xs' y, f x y

Generalization of Finset.prod_comm to the case when the inner Finsets depend on the outer variable.

theorem Finset.sum_comm {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {t : } {f : γαβ} :
xs, yt, f x y = yt, xs, f x y
theorem Finset.prod_comm {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {t : } {f : γαβ} :
xs, yt, f x y = yt, xs, f x y
theorem Finset.sum_hom_rel {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] [] {r : βγProp} {f : αβ} {g : αγ} {s : } (h₁ : r 0 0) (h₂ : ∀ (a : α) (b : β) (c : γ), r b cr (f a + b) (g a + c)) :
r (xs, f x) (xs, g x)
theorem Finset.prod_hom_rel {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] [] {r : βγProp} {f : αβ} {g : αγ} {s : } (h₁ : r 1 1) (h₂ : ∀ (a : α) (b : β) (c : γ), r b cr (f a * b) (g a * c)) :
r (xs, f x) (xs, g x)
theorem Finset.sum_filter_of_ne {α : Type u_3} {β : Type u_4} {s : } {f : αβ} [] {p : αProp} [] (hp : xs, f x 0p x) :
x, f x = xs, f x
theorem Finset.prod_filter_of_ne {α : Type u_3} {β : Type u_4} {s : } {f : αβ} [] {p : αProp} [] (hp : xs, f x 1p x) :
x, f x = xs, f x
theorem Finset.sum_filter_ne_zero {α : Type u_3} {β : Type u_4} {f : αβ} [] (s : ) [(x : α) → Decidable (f x 0)] :
xFinset.filter (fun (x : α) => f x 0) s, f x = xs, f x
theorem Finset.prod_filter_ne_one {α : Type u_3} {β : Type u_4} {f : αβ} [] (s : ) [(x : α) → Decidable (f x 1)] :
xFinset.filter (fun (x : α) => f x 1) s, f x = xs, f x
theorem Finset.sum_filter {α : Type u_3} {β : Type u_4} {s : } [] (p : αProp) [] (f : αβ) :
a, f a = as, if p a then f a else 0
theorem Finset.prod_filter {α : Type u_3} {β : Type u_4} {s : } [] (p : αProp) [] (f : αβ) :
a, f a = as, if p a then f a else 1
theorem Finset.sum_eq_single_of_mem {α : Type u_3} {β : Type u_4} [] {s : } {f : αβ} (a : α) (h : a s) (h₀ : bs, b af b = 0) :
xs, f x = f a
theorem Finset.prod_eq_single_of_mem {α : Type u_3} {β : Type u_4} [] {s : } {f : αβ} (a : α) (h : a s) (h₀ : bs, b af b = 1) :
xs, f x = f a
theorem Finset.sum_eq_single {α : Type u_3} {β : Type u_4} [] {s : } {f : αβ} (a : α) (h₀ : bs, b af b = 0) (h₁ : asf a = 0) :
xs, f x = f a
theorem Finset.prod_eq_single {α : Type u_3} {β : Type u_4} [] {s : } {f : αβ} (a : α) (h₀ : bs, b af b = 1) (h₁ : asf a = 1) :
xs, f x = f a
theorem Finset.sum_union_eq_left {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} [] [] (hs : as₂, as₁f a = 0) :
as₁ s₂, f a = as₁, f a
theorem Finset.prod_union_eq_left {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} [] [] (hs : as₂, as₁f a = 1) :
as₁ s₂, f a = as₁, f a
theorem Finset.sum_union_eq_right {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} [] [] (hs : as₁, as₂f a = 0) :
as₁ s₂, f a = as₂, f a
theorem Finset.prod_union_eq_right {α : Type u_3} {β : Type u_4} {s₁ : } {s₂ : } {f : αβ} [] [] (hs : as₁, as₂f a = 1) :
as₁ s₂, f a = as₂, f a
theorem Finset.sum_eq_add_of_mem {α : Type u_3} {β : Type u_4} [] {s : } {f : αβ} (a : α) (b : α) (ha : a s) (hb : b s) (hn : a b) (h₀ : cs, c a c bf c = 0) :
xs, f x = f a + f b
theorem Finset.prod_eq_mul_of_mem {α : Type u_3} {β : Type u_4} [] {s : } {f : αβ} (a : α) (b : α) (ha : a s) (hb : b s) (hn : a b) (h₀ : cs, c a c bf c = 1) :
xs, f x = f a * f b
theorem Finset.sum_eq_add {α : Type u_3} {β : Type u_4} [] {s : } {f : αβ} (a : α) (b : α) (hn : a b) (h₀ : cs, c a c bf c = 0) (ha : asf a = 0) (hb : bsf b = 0) :
xs, f x = f a + f b
theorem Finset.prod_eq_mul {α : Type u_3} {β : Type u_4} [] {s : } {f : αβ} (a : α) (b : α) (hn : a b) (h₀ : cs, c a c bf c = 1) (ha : asf a = 1) (hb : bsf b = 1) :
xs, f x = f a * f b
@[simp]
theorem Finset.sum_subtype_eq_sum_filter {α : Type u_3} {β : Type u_4} {s : } [] (f : αβ) {p : αProp} [] :
x, f x = x, f x

A sum over s.subtype p equals one over s.filter p.

@[simp]
theorem Finset.prod_subtype_eq_prod_filter {α : Type u_3} {β : Type u_4} {s : } [] (f : αβ) {p : αProp} [] :
x, f x = x, f x

A product over s.subtype p equals one over s.filter p.

theorem Finset.sum_subtype_of_mem {α : Type u_3} {β : Type u_4} {s : } [] (f : αβ) {p : αProp} [] (h : xs, p x) :
x, f x = xs, f x

If all elements of a Finset satisfy the predicate p, a sum over s.subtype p equals that sum over s.

theorem Finset.prod_subtype_of_mem {α : Type u_3} {β : Type u_4} {s : } [] (f : αβ) {p : αProp} [] (h : xs, p x) :
x, f x = xs, f x

If all elements of a Finset satisfy the predicate p, a product over s.subtype p equals that product over s.

theorem Finset.sum_subtype_map_embedding {α : Type u_3} {β : Type u_4} [] {p : αProp} {s : Finset { x : α // p x }} {f : { x : α // p x }β} {g : αβ} (h : xs, g x = f x) :
xFinset.map (Function.Embedding.subtype fun (x : α) => p x) s, g x = xs, f x

A sum of a function over a Finset in a subtype equals a sum in the main type of a function that agrees with the first function on that Finset.

theorem Finset.prod_subtype_map_embedding {α : Type u_3} {β : Type u_4} [] {p : αProp} {s : Finset { x : α // p x }} {f : { x : α // p x }β} {g : αβ} (h : xs, g x = f x) :
xFinset.map (Function.Embedding.subtype fun (x : α) => p x) s, g x = xs, f x

A product of a function over a Finset in a subtype equals a product in the main type of a function that agrees with the first function on that Finset.

theorem Finset.sum_coe_sort_eq_attach {α : Type u_3} {β : Type u_4} (s : ) [] (f : { x : α // x s }β) :
i : { x : α // x s }, f i = is.attach, f i
theorem Finset.prod_coe_sort_eq_attach {α : Type u_3} {β : Type u_4} (s : ) [] (f : { x : α // x s }β) :
i : { x : α // x s }, f i = is.attach, f i
theorem Finset.sum_coe_sort {α : Type u_3} {β : Type u_4} (s : ) (f : αβ) [] :
i : { x : α // x s }, f i = is, f i
theorem Finset.prod_coe_sort {α : Type u_3} {β : Type u_4} (s : ) (f : αβ) [] :
i : { x : α // x s }, f i = is, f i
theorem Finset.sum_finset_coe {α : Type u_3} {β : Type u_4} [] (f : αβ) (s : ) :
i : s, f i = is, f i
theorem Finset.prod_finset_coe {α : Type u_3} {β : Type u_4} [] (f : αβ) (s : ) :
i : s, f i = is, f i
theorem Finset.sum_subtype {α : Type u_3} {β : Type u_4} [] {p : αProp} {F : Fintype ()} (s : ) (h : ∀ (x : α), x s p x) (f : αβ) :
as, f a = a : , f a
theorem Finset.prod_subtype {α : Type u_3} {β : Type u_4} [] {p : αProp} {F : Fintype ()} (s : ) (h : ∀ (x : α), x s p x) (f : αβ) :
as, f a = a : , f a
theorem Finset.sum_preimage' {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [] (f : ικ) [DecidablePred fun (x : κ) => x ] (s : ) (hf : Set.InjOn f (f ⁻¹' s)) (g : κβ) :
xs.preimage f hf, g (f x) = xFinset.filter (fun (x : κ) => x ) s, g x
theorem Finset.prod_preimage' {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [] (f : ικ) [DecidablePred fun (x : κ) => x ] (s : ) (hf : Set.InjOn f (f ⁻¹' s)) (g : κβ) :
xs.preimage f hf, g (f x) = xFinset.filter (fun (x : κ) => x ) s, g x
theorem Finset.sum_preimage {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [] (f : ικ) (s : ) (hf : Set.InjOn f (f ⁻¹' s)) (g : κβ) (hg : xs, xg x = 0) :
xs.preimage f hf, g (f x) = xs, g x
theorem Finset.prod_preimage {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [] (f : ικ) (s : ) (hf : Set.InjOn f (f ⁻¹' s)) (g : κβ) (hg : xs, xg x = 1) :
xs.preimage f hf, g (f x) = xs, g x
theorem Finset.sum_preimage_of_bij {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [] (f : ικ) (s : ) (hf : Set.BijOn f (f ⁻¹' s) s) (g : κβ) :
xs.preimage f , g (f x) = xs, g x
theorem Finset.prod_preimage_of_bij {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [] (f : ικ) (s : ) (hf : Set.BijOn f (f ⁻¹' s) s) (g : κβ) :
xs.preimage f , g (f x) = xs, g x
theorem Finset.sum_set_coe {α : Type u_3} {β : Type u_4} {f : αβ} [] (s : Set α) [Fintype s] :
i : s, f i = is.toFinset, f i
theorem Finset.prod_set_coe {α : Type u_3} {β : Type u_4} {f : αβ} [] (s : Set α) [Fintype s] :
i : s, f i = is.toFinset, f i
theorem Finset.sum_congr_set {α : Type u_6} [] {β : Type u_7} [] (s : Set β) [DecidablePred fun (x : β) => x s] (f : βα) (g : sα) (w : ∀ (x : β) (h : x s), f x = g x, h) (w' : xs, f x = 0) :
Finset.univ.sum f = Finset.univ.sum g

The sum of a function g defined only on a set s is equal to the sum of a function f defined everywhere, as long as f and g agree on s, and f = 0 off s.

abbrev Finset.sum_congr_set.match_1 {β : Type u_1} [] (s : Set β) [DecidablePred fun (x : β) => x s] (motive : (x : { x : β // x s }) → x Finset.univProp) :
∀ (x : { x : β // x s }) (x_1 : x Finset.univ), (∀ (x : β) (h : x s) (x_2 : x, h Finset.univ), motive x, h x_2)motive x x_1
Equations
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Instances For
theorem Finset.prod_congr_set {α : Type u_6} [] {β : Type u_7} [] (s : Set β) [DecidablePred fun (x : β) => x s] (f : βα) (g : sα) (w : ∀ (x : β) (h : x s), f x = g x, h) (w' : xs, f x = 1) :
Finset.univ.prod f = Finset.univ.prod g

The product of a function g defined only on a set s is equal to the product of a function f defined everywhere, as long as f and g agree on s, and f = 1 off s.

theorem Finset.sum_apply_dite {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {p : αProp} {hp : } [DecidablePred fun (x : α) => ¬p x] (f : (x : α) → p xγ) (g : (x : α) → ¬p xγ) (h : γβ) :
xs, h (if hx : p x then f x hx else g x hx) = x().attach, h (f x ) + x(Finset.filter (fun (x : α) => ¬p x) s).attach, h (g x )
theorem Finset.prod_apply_dite {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {p : αProp} {hp : } [DecidablePred fun (x : α) => ¬p x] (f : (x : α) → p xγ) (g : (x : α) → ¬p xγ) (h : γβ) :
xs, h (if hx : p x then f x hx else g x hx) = (x().attach, h (f x )) * x(Finset.filter (fun (x : α) => ¬p x) s).attach, h (g x )
theorem Finset.sum_apply_ite {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {p : αProp} {_hp : } (f : αγ) (g : αγ) (h : γβ) :
xs, h (if p x then f x else g x) = x, h (f x) + xFinset.filter (fun (x : α) => ¬p x) s, h (g x)
theorem Finset.prod_apply_ite {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {p : αProp} {_hp : } (f : αγ) (g : αγ) (h : γβ) :
xs, h (if p x then f x else g x) = (x, h (f x)) * xFinset.filter (fun (x : α) => ¬p x) s, h (g x)
theorem Finset.sum_dite {α : Type u_3} {β : Type u_4} [] {s : } {p : αProp} {hp : } (f : (x : α) → p xβ) (g : (x : α) → ¬p xβ) :
(xs, if hx : p x then f x hx else g x hx) = x().attach, f x + x(Finset.filter (fun (x : α) => ¬p x) s).attach, g x
theorem Finset.prod_dite {α : Type u_3} {β : Type u_4} [] {s : } {p : αProp} {hp : } (f : (x : α) → p xβ) (g : (x : α) → ¬p xβ) :
(xs, if hx : p x then f x hx else g x hx) = (x().attach, f x ) * x(Finset.filter (fun (x : α) => ¬p x) s).attach, g x
theorem Finset.sum_ite {α : Type u_3} {β : Type u_4} [] {s : } {p : αProp} {hp : } (f : αβ) (g : αβ) :
(xs, if p x then f x else g x) = x, f x + xFinset.filter (fun (x : α) => ¬p x) s, g x
theorem Finset.prod_ite {α : Type u_3} {β : Type u_4} [] {s : } {p : αProp} {hp : } (f : αβ) (g : αβ) :
(xs, if p x then f x else g x) = (x, f x) * xFinset.filter (fun (x : α) => ¬p x) s, g x
theorem Finset.sum_dite_of_false {α : Type u_3} {β : Type u_4} {s : } [] {p : αProp} :
∀ {x : } (h : is, ¬p i) (f : (i : α) → p iβ) (g : (i : α) → ¬p iβ), (is, if hi : p i then f i hi else g i hi) = i : { x : α // x s }, g i
theorem Finset.prod_dite_of_false {α : Type u_3} {β : Type u_4} {s : } [] {p : αProp} :
∀ {x : } (h : is, ¬p i) (f : (i : α) → p iβ) (g : (i : α) → ¬p iβ), (is, if hi : p i then f i hi else g i hi) = i : { x : α // x s }, g i
theorem Finset.sum_ite_of_false {α : Type u_3} {β : Type u_4} {s : } [] {p : αProp} :
∀ {x : }, (xs, ¬p x)∀ (f g : αβ), (x_1s, if p x_1 then f x_1 else g x_1) = xs, g x
theorem Finset.prod_ite_of_false {α : Type u_3} {β : Type u_4} {s : } [] {p : αProp} :
∀ {x : }, (xs, ¬p x)∀ (f g : αβ), (x_1s, if p x_1 then f x_1 else g x_1) = xs, g x
theorem Finset.sum_dite_of_true {α : Type u_3} {β : Type u_4} {s : } [] {p : αProp} :
∀ {x : } (h : is, p i) (f : (i : α) → p iβ) (g : (i : α) → ¬p iβ), (is, if hi : p i then f i hi else g i hi) = i : { x : α // x s }, f i
theorem Finset.prod_dite_of_true {α : Type u_3} {β : Type u_4} {s : } [] {p : αProp} :
∀ {x : } (h : is, p i) (f : (i : α) → p iβ) (g : (i : α) → ¬p iβ), (is, if hi : p i then f i hi else g i hi) = i : { x : α // x s }, f i
theorem Finset.sum_ite_of_true {α : Type u_3} {β : Type u_4} {s : } [] {p : αProp} :
∀ {x : }, (xs, p x)∀ (f g : αβ), (x_1s, if p x_1 then f x_1 else g x_1) = xs, f x
theorem Finset.prod_ite_of_true {α : Type u_3} {β : Type u_4} {s : } [] {p : αProp} :
∀ {x : }, (xs, p x)∀ (f g : αβ), (x_1s, if p x_1 then f x_1 else g x_1) = xs, f x
theorem Finset.sum_apply_ite_of_false {α : Type u_3} {β : Type u_4} {γ : Type u_5} {s : } [] {p : αProp} {hp : } (f : αγ) (g : αγ) (k : γβ) (h : xs, ¬p x) :
xs, k (if p x then f x else g x) = xs, k (g x)
theorem Finset.prod_apply_ite_of_false {α : Type u_3} {β : Type u_4} {γ : Type u_5} {s : } [] {p : αProp} {hp : } (f : αγ) (g : αγ) (k : γβ) (h : xs, ¬p x) :
xs, k (if p x then f x else g x) = xs, k (g x)
theorem Finset.sum_apply_ite_of_true {α : Type u_3} {β : Type u_4} {γ : Type u_5} {s : } [] {p : αProp} {hp : } (f : αγ) (g : αγ) (k : γβ) (h : xs, p x) :
xs, k (if p x then f x else g x) = xs, k (f x)
theorem Finset.prod_apply_ite_of_true {α : Type u_3} {β : Type u_4} {γ : Type u_5} {s : } [] {p : αProp} {hp : } (f : αγ) (g : αγ) (k : γβ) (h : xs, p x) :
xs, k (if p x then f x else g x) = xs, k (f x)
theorem Finset.sum_extend_by_zero {α : Type u_3} {β : Type u_4} [] [] (s : ) (f : αβ) :
(is, if i s then f i else 0) = is, f i
theorem Finset.prod_extend_by_one {α : Type u_3} {β : Type u_4} [] [] (s : ) (f : αβ) :
(is, if i s then f i else 1) = is, f i
@[simp]
theorem Finset.sum_ite_mem {α : Type u_3} {β : Type u_4} [] [] (s : ) (t : ) (f : αβ) :
(is, if i t then f i else 0) = is t, f i
@[simp]
theorem Finset.prod_ite_mem {α : Type u_3} {β : Type u_4} [] [] (s : ) (t : ) (f : αβ) :
(is, if i t then f i else 1) = is t, f i
@[simp]
theorem Finset.sum_dite_eq {α : Type u_3} {β : Type u_4} [] [] (s : ) (a : α) (b : (x : α) → a = xβ) :
(xs, if h : a = x then b x h else 0) = if a s then b a else 0
@[simp]
theorem Finset.prod_dite_eq {α : Type u_3} {β : Type u_4} [] [] (s : ) (a : α) (b : (x : α) → a = xβ) :
(xs, if h : a = x then b x h else 1) = if a s then b a else 1
@[simp]
theorem Finset.sum_dite_eq' {α : Type u_3} {β : Type u_4} [] [] (s : ) (a : α) (b : (x : α) → x = aβ) :
(xs, if h : x = a then b x h else 0) = if a s then b a else 0
@[simp]
theorem Finset.prod_dite_eq' {α : Type u_3} {β : Type u_4} [] [] (s : ) (a : α) (b : (x : α) → x = aβ) :
(xs, if h : x = a then b x h else 1) = if a s then b a else 1
@[simp]
theorem Finset.sum_ite_eq {α : Type u_3} {β : Type u_4} [] [] (s : ) (a : α) (b : αβ) :
(xs, if a = x then b x else 0) = if a s then b a else 0
@[simp]
theorem Finset.prod_ite_eq {α : Type u_3} {β : Type u_4} [] [] (s : ) (a : α) (b : αβ) :
(xs, if a = x then b x else 1) = if a s then b a else 1
@[simp]
theorem Finset.sum_ite_eq' {α : Type u_3} {β : Type u_4} [] [] (s : ) (a : α) (b : αβ) :
(xs, if x = a then b x else 0) = if a s then b a else 0

A sum taken over a conditional whose condition is an equality test on the index and whose alternative is 0 has value either the term at that index or 0.

The difference with Finset.sum_ite_eq is that the arguments to Eq are swapped.

@[simp]
theorem Finset.prod_ite_eq' {α : Type u_3} {β : Type u_4} [] [] (s : ) (a : α) (b : αβ) :
(xs, if x = a then b x else 1) = if a s then b a else 1

A product taken over a conditional whose condition is an equality test on the index and whose alternative is 1 has value either the term at that index or 1.

The difference with Finset.prod_ite_eq is that the arguments to Eq are swapped.

theorem Finset.sum_ite_index {α : Type u_3} {β : Type u_4} [] (p : Prop) [] (s : ) (t : ) (f : αβ) :
xif p then s else t, f x = if p then xs, f x else xt, f x
theorem Finset.prod_ite_index {α : Type u_3} {β : Type u_4} [] (p : Prop) [] (s : ) (t : ) (f : αβ) :
xif p then s else t, f x = if p then xs, f x else xt, f x
@[simp]
theorem Finset.sum_ite_irrel {α : Type u_3} {β : Type u_4} [] (p : Prop) [] (s : ) (f : αβ) (g : αβ) :
(xs, if p then f x else g x) = if p then xs, f x else xs, g x
@[simp]
theorem Finset.prod_ite_irrel {α : Type u_3} {β : Type u_4} [] (p : Prop) [] (s : ) (f : αβ) (g : αβ) :
(xs, if p then f x else g x) = if p then xs, f x else xs, g x
@[simp]
theorem Finset.sum_dite_irrel {α : Type u_3} {β : Type u_4} [] (p : Prop) [] (s : ) (f : pαβ) (g : ¬pαβ) :
(xs, if h : p then f h x else g h x) = if h : p then xs, f h x else xs, g h x
@[simp]
theorem Finset.prod_dite_irrel {α : Type u_3} {β : Type u_4} [] (p : Prop) [] (s : ) (f : pαβ) (g : ¬pαβ) :
(xs, if h : p then f h x else g h x) = if h : p then xs, f h x else xs, g h x
@[simp]
theorem Finset.sum_pi_single' {α : Type u_3} {β : Type u_4} [] [] (a : α) (x : β) (s : ) :
a's, Pi.single a x a' = if a s then x else 0
@[simp]
theorem Finset.prod_pi_mulSingle' {α : Type u_3} {β : Type u_4} [] [] (a : α) (x : β) (s : ) :
a's, Pi.mulSingle a x a' = if a s then x else 1
@[simp]
theorem Finset.sum_pi_single {α : Type u_3} {β : αType u_6} [] [(a : α) → AddCommMonoid (β a)] (a : α) (f : (a : α) → β a) (s : ) :
a's, Pi.single a' (f a') a = if a s then f a else 0
@[simp]
theorem Finset.prod_pi_mulSingle {α : Type u_3} {β : αType u_6} [] [(a : α) → CommMonoid (β a)] (a : α) (f : (a : α) → β a) (s : ) :
a's, Pi.mulSingle a' (f a') a = if a s then f a else 1
theorem Finset.support_sum {ι : Type u_1} {α : Type u_3} {β : Type u_4} [] (s : ) (f : ιαβ) :
(Function.support fun (x : α) => is, f i x) is, Function.support (f i)
theorem Finset.mulSupport_prod {ι : Type u_1} {α : Type u_3} {β : Type u_4} [] (s : ) (f : ιαβ) :
(Function.mulSupport fun (x : α) => is, f i x) is, Function.mulSupport (f i)
theorem Finset.sum_indicator_subset_of_eq_zero {ι : Type u_1} {α : Type u_3} {β : Type u_4} [] [Zero α] (f : ια) (g : ιαβ) {s : } {t : } (h : s t) (hg : ∀ (a : ι), g a 0 = 0) :
it, g i ((s).indicator f i) = is, g i (f i)

Consider a sum of g i (f i) over a finset. Suppose g is a function such as n ↦ (n • ·), which maps a second argument of 0 to 0 (or a weighted sum of f i * h i or f i • h i, where f gives the weights that are multiplied by some other function h). Then if f is replaced by the corresponding indicator function, the finset may be replaced by a possibly larger finset without changing the value of the sum.

theorem Finset.prod_mulIndicator_subset_of_eq_one {ι : Type u_1} {α : Type u_3} {β : Type u_4} [] [One α] (f : ια) (g : ιαβ) {s : } {t : } (h : s t) (hg : ∀ (a : ι), g a 1 = 1) :
it, g i ((s).mulIndicator f i) = is, g i (f i)

Consider a product of g i (f i) over a finset. Suppose g is a function such as n ↦ (· ^ n), which maps a second argument of 1 to 1. Then if f is replaced by the corresponding multiplicative indicator function, the finset may be replaced by a possibly larger finset without changing the value of the product.

theorem Finset.sum_indicator_subset {ι : Type u_1} {β : Type u_4} [] (f : ιβ) {s : } {t : } (h : s t) :
it, (s).indicator f i = is, f i

Summing an indicator function over a possibly larger Finset is the same as summing the original function over the original finset.

theorem Finset.prod_mulIndicator_subset {ι : Type u_1} {β : Type u_4} [] (f : ιβ) {s : } {t : } (h : s t) :
it, (s).mulIndicator f i = is, f i

Taking the product of an indicator function over a possibly larger finset is the same as taking the original function over the original finset.

theorem Finset.sum_indicator_eq_sum_filter {ι : Type u_1} {β : Type u_4} [] {κ : Type u_6} (s : ) (f : ικβ) (t : ιSet κ) (g : ικ) [DecidablePred fun (i : ι) => g i t i] :
is, (t i).indicator (f i) (g i) = iFinset.filter (fun (i : ι) => g i t i) s, f i (g i)
theorem Finset.prod_mulIndicator_eq_prod_filter {ι : Type u_1} {β : Type u_4} [] {κ : Type u_6} (s : ) (f : ικβ) (t : ιSet κ) (g : ικ) [DecidablePred fun (i : ι) => g i t i] :
is, (t i).mulIndicator (f i) (g i) = iFinset.filter (fun (i : ι) => g i t i) s, f i (g i)
theorem Finset.sum_indicator_eq_sum_inter {ι : Type u_1} {β : Type u_4} [] [] (s : ) (t : ) (f : ιβ) :
is, (t).indicator f i = is t, f i
theorem Finset.prod_mulIndicator_eq_prod_inter {ι : Type u_1} {β : Type u_4} [] [] (s : ) (t : ) (f : ιβ) :
is, (t).mulIndicator f i = is t, f i
theorem Finset.indicator_sum {ι : Type u_1} {β : Type u_4} [] {κ : Type u_6} (s : ) (t : Set κ) (f : ικβ) :
t.indicator (is, f i) = is, t.indicator (f i)
theorem Finset.mulIndicator_prod {ι : Type u_1} {β : Type u_4} [] {κ : Type u_6} (s : ) (t : Set κ) (f : ικβ) :
t.mulIndicator (is, f i) = is, t.mulIndicator (f i)
theorem Finset.indicator_biUnion {ι : Type u_1} {β : Type u_4} [] {κ : Type u_7} (s : ) (t : ιSet κ) {f : κβ} :
(s).PairwiseDisjoint t(is, t i).indicator f = fun (a : κ) => is, (t i).indicator f a
theorem Finset.mulIndicator_biUnion {ι : Type u_1} {β : Type u_4} [] {κ : Type u_7} (s : ) (t : ιSet κ) {f : κβ} :
(s).PairwiseDisjoint t(is, t i).mulIndicator f = fun (a : κ) => is, (t i).mulIndicator f a
theorem Finset.indicator_biUnion_apply {ι : Type u_1} {β : Type u_4} [] {κ : Type u_7} (s : ) (t : ιSet κ) {f : κβ} (h : (s).PairwiseDisjoint t) (x : κ) :
(is, t i).indicator f x = is, (t i).indicator f x
theorem Finset.mulIndicator_biUnion_apply {ι : Type u_1} {β : Type u_4} [] {κ : Type u_7} (s : ) (t : ιSet κ) {f : κβ} (h : (s).PairwiseDisjoint t) (x : κ) :
(is, t i).mulIndicator f x = is, (t i).mulIndicator f x
theorem Finset.sum_bij_ne_zero {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {t : } {f : αβ} {g : γβ} (i : (a : α) → a sf a 0γ) (hi : ∀ (a : α) (h₁ : a s) (h₂ : f a 0), i a h₁ h₂ t) (i_inj : ∀ (a₁ : α) (h₁₁ : a₁ s) (h₁₂ : f a₁ 0) (a₂ : α) (h₂₁ : a₂ s) (h₂₂ : f a₂ 0), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂a₁ = a₂) (i_surj : bt, g b 0∃ (a : α) (h₁ : a s) (h₂ : f a 0), i a h₁ h₂ = b) (h : ∀ (a : α) (h₁ : a s) (h₂ : f a 0), f a = g (i a h₁ h₂)) :
xs, f x = xt, g x
theorem Finset.prod_bij_ne_one {α : Type u_3} {β : Type u_4} {γ : Type u_5} [] {s : } {t : } {f : αβ} {g : γβ} (i : (a : α) → a sf a 1γ) (hi : ∀ (a : α) (h₁ : a s) (h₂ : f a 1), i a h₁ h₂ t) (i_inj : ∀ (a₁ : α) (h₁₁ : a₁ s) (h₁₂ : f a₁ 1) (a₂ : α) (h₂₁ : a₂ s) (h₂₂ : f a₂ 1), i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂a₁ = a₂) (i_surj : bt, g b 1∃ (a : α) (h₁ : a s) (h₂ : f a 1), i a h₁ h₂ = b) (h : ∀ (a : α) (h₁ : a s) (h₂ : f a 1), f a = g (i a h₁ h₂)) :
xs, f x = xt, g x
theorem Finset.nonempty_of_sum_ne_zero {α : Type u_3} {β : Type u_4} {s : } {f : αβ} [] (h : xs, f x 0) :
s.Nonempty
theorem Finset.nonempty_of_prod_ne_one {α : Type u_3} {β : Type u_4} {s : } {f : αβ} [] (h : xs, f x 1) :
s.Nonempty
theorem Finset.exists_ne_zero_of_sum_ne_zero {α : Type u_3} {β : Type u_4} {s : } {f : αβ} [] (h : xs, f x 0) :
as, f a 0
theorem Finset.exists_ne_one_of_prod_ne_one {α : Type u_3} {β : Type u_4} {s : } {f : αβ} [] (h : xs, f x 1) :
as, f a 1
theorem Finset.sum_range_succ_comm {β : Type u_4} [] (f : β) (n : ) :
xFinset.range (n + 1), f x = f n + x, f x
theorem Finset.prod_range_succ_comm {β : Type u_4} [] (f : β) (n : ) :
xFinset.range (n + 1), f x = f n * x, f x
theorem Finset.sum_range_succ {β : Type u_4} [] (f : β) (n : ) :
xFinset.range (n + 1), f x = x, f x + f n
theorem Finset.prod_range_succ {β : Type u_4} [] (f : β) (n : ) :
xFinset.range (n + 1), f x = (x, f x) * f n
theorem Finset.sum_range_succ' {β : Type u_4} [] (f : β) (n : ) :
kFinset.range (n + 1), f k = k, f (k + 1) + f 0
abbrev Finset.sum_range_succ'.match_1 (motive : ) :
∀ (x : ), (Unitmotive 0)(∀ (n : ), motive n.succ)motive x
Equations
• =
Instances For
theorem Finset.prod_range_succ' {β : Type u_4} [] (f : β) (n : ) :
kFinset.range (n + 1), f k = (k, f (k + 1)) * f 0
theorem Finset.eventually_constant_sum {β : Type u_4} [] {u : β} {N : } (hu : nN, u n = 0) {n : } (hn : N n) :
k, u k = k, u k
theorem Finset.eventually_constant_prod {β : Type u_4} [] {u : β} {N : } (hu : nN, u n = 1) {n : } (hn : N n) :
k, u k = k, u k
theorem Finset.sum_range_add {β : Type u_4} [] (f : β) (n : ) (m : ) :
xFinset.range (n + m), f x = x, f x + x, f (n + x)
theorem Finset.prod_range_add {β : Type u_4} [] (f : β) (n : ) (m : ) :
xFinset.range (n + m), f x = (x, f x) * x, f (n + x)
theorem Finset.sum_range_add_sub_sum_range {α : Type u_6} [] (f : α) (n : ) (m : ) :
kFinset.range (n + m), f k - k, f k = k, f (n + k)
theorem Finset.prod_range_add_div_prod_range {α : Type u_6} [] (f : α) (n : ) (m : ) :
(kFinset.range (n + m), f k) / k, f k = k, f (n + k)
theorem Finset.sum_range_zero {β : Type u_4} [] (f : β) :
k, f k = 0
theorem Finset.prod_range_zero {β : Type u_4} [] (f : β) :
k, f k = 1
theorem Finset.sum_range_one {β : Type u_4} [] (f : β) :
k, f k = f 0
theorem Finset.prod_range_one {β : Type u_4} [] (f : β) :
k, f k = f 0
theorem Finset.sum_list_map_count {α : Type u_3} [] (l : List α) {M : Type u_6} [] (f : αM) :
(List.map f l).sum = ml.toFinset, f m
theorem Finset.prod_list_map_count {α : Type u_3} [] (l : List α) {M : Type u_6} [] (f : αM) :
(List.map f l).prod = ml.toFinset, f m ^
theorem Finset.sum_list_count {α : Type u_3} [] [] (s : List α) :
s.sum = ms.toFinset, m
theorem Finset.prod_list_count {α : Type u_3} [] [] (s : List α) :
s.prod = ms.toFinset, m ^
theorem Finset.sum_list_count_of_subset {α : Type u_3} [] [] (m : List α) (s : ) (hs : m.toFinset s) :
m.sum = is, i
theorem Finset.prod_list_count_of_subset {α : Type u_3} [] [] (m : List α) (s : ) (hs : m.toFinset s) :
m.prod = is, i ^
theorem Finset.sum_filter_count_eq_countP {α : Type u_3} [] (p : αProp) [] (l : List α) :
xFinset.filter p l.toFinset, = List.countP (fun (b : α) => decide (p b)) l
theorem Finset.sum_multiset_map_count {α : Type u_3} [] (s : ) {M : Type u_6} [] (f : αM) :
().sum = ms.toFinset, f m
theorem Finset.prod_multiset_map_count {α : Type u_3} [] (s : ) {M : Type u_6} [] (f : αM) :
().prod = ms.toFinset, f m ^
theorem Finset.sum_multiset_count {α : Type u_3} [] [] (s : ) :
s.sum = ms.toFinset, m
theorem Finset.prod_multiset_count {α : Type u_3} [] [] (s : ) :
s.prod = ms.toFinset, m ^
theorem Finset.sum_multiset_count_of_subset {α : Type u_3} [] [] (m : ) (s : ) (hs : m.toFinset s) :
m.sum = is, i
theorem Finset.prod_multiset_count_of_subset {α : Type u_3} [] [] (m : ) (s : ) (hs : m.toFinset s) :
m.prod = is, i ^
theorem Finset.sum_mem_multiset {α : Type u_3} {β : Type u_4} [] [] (m : ) (f : { x : α // x m }β) (g : αβ) (hfg : ∀ (x : { x : α // x m }), f x = g x) :
x : { x : α // x m }, f x = xm.toFinset, g x
theorem Finset.prod_mem_multiset {α : Type u_3} {β : Type u_4} [] [] (m : ) (f : { x : α // x m }β) (g : αβ) (hfg : ∀ (x : { x : α // x m }), f x = g x) :
x : { x : α // x m }, f x = xm.toFinset, g x
theorem Finset.sum_induction {α : Type u_3} {s : } {M : Type u_6} [] (f : αM) (p : MProp) (hom : ∀ (a b : M), p ap bp (a + b)) (unit : p 0) (base : xs, p (f x)) :
p (xs, f x)

To prove a property of a sum, it suffices to prove that the property is additive and holds on summands.

theorem Finset.prod_induction {α : Type u_3} {s : } {M : Type u_6} [] (f : αM) (p : MProp) (hom : ∀ (a b : M), p ap bp (a * b)) (unit : p 1) (base : xs, p (f x)) :
p (xs, f x)

To prove a property of a product, it suffices to prove that the property is multiplicative and holds on factors.

theorem Finset.sum_induction_nonempty {α : Type u_3} {s : } {M : Type u_6} [] (f : αM) (p : MProp) (hom : ∀ (a b : M), p ap bp (a + b)) (nonempty : s.Nonempty) (base : xs, p (f x)) :
p (xs, f x)

To prove a property of a sum, it suffices to prove that the property is additive and holds on summands.

theorem Finset.prod_induction_nonempty {α : Type u_3} {s : } {M : Type u_6} [] (f : αM) (p : MProp) (hom : ∀ (a b : M), p ap bp (a * b)) (nonempty : s.Nonempty) (base : xs, p (f x)) :
p (xs, f x)

To prove a property of a product, it suffices to prove that the property is multiplicative and holds on factors.

theorem Finset.sum_range_induction {β : Type u_4} [] (f : β) (s : β) (base : s 0 = 0) (step : ∀ (n : ), s (n + 1) = s n + f n) (n : ) :
k, f k = s n

For any sum along {0, ..., n - 1} of a commutative-monoid-valued function, we can verify that it's equal to a different function just by checking differences of adjacent terms.

This is a discrete analogue of the fundamental theorem of calculus.

theorem Finset.prod_range_induction {β : Type u_4} [] (f : β) (s : β) (base : s 0 = 1) (step : ∀ (n : ), s (n + 1) = s n * f n) (n : ) :
k, f k = s n

For any product along {0, ..., n - 1} of a commutative-monoid-valued function, we can verify that it's equal to a different function just by checking ratios of adjacent terms.

This is a multiplicative discrete analogue of the fundamental theorem of calculus.

theorem Finset.sum_range_sub {M : Type u_6} [] (f : M) (n : ) :
i, (f (i + 1) - f i) = f n - f 0

A telescoping sum along {0, ..., n - 1} of an additive commutative group valued function reduces to the difference of the last and first terms.

theorem Finset.prod_range_div {M : Type u_6} [] (f : M) (n : ) :
i, f (i + 1) / f i = f n / f 0

A telescoping product along {0, ..., n - 1} of a commutative group valued function reduces to the ratio of the last and first factors.

theorem Finset.sum_range_sub' {M : Type u_6} [] (f : M) (n : ) :
i, (f i - f (i + 1)) = f 0 - f n
theorem Finset.prod_range_div' {M : Type u_6} [] (f : M) (n : ) :
i, f i / f (i + 1) = f 0 / f n
theorem Finset.eq_sum_range_sub {M : Type u_6} [] (f : M) (n : ) :
f n = f 0 + i, (f (i + 1) - f i)
theorem Finset.eq_prod_range_div {M : Type u_6} [] (f : M) (n : ) :
f n = f 0 * i, f (i + 1) / f i
theorem Finset.eq_sum_range_sub' {M : Type u_6} [] (f : M) (n : ) :
f n = iFinset.range (n + 1), if i = 0 then f 0 else f i - f (i - 1)
theorem Finset.eq_prod_range_div' {M : Type u_6} [] (f : M) (n : ) :
f n = iFinset.range (n + 1), if i = 0 then f 0 else f i / f (i - 1)
theorem Finset.sum_range_tsub {α : Type u_3} [Sub α] [] [ContravariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] {f : α} (h : ) (n : ) :
i, (f (i + 1) - f i) = f n - f 0

A telescoping sum along {0, ..., n-1} of an ℕ-valued function reduces to the difference of the last and first terms when the function we are summing is monotone.

@[simp]
theorem Finset.sum_const {α : Type u_3} {β : Type u_4} {s : } [] (b : β) :
_xs, b = s.card b
@[simp]
theorem Finset.prod_const {α : Type u_3} {β : Type u_4} {s : } [] (b : β) :
_xs, b = b ^ s.card
theorem Finset.sum_eq_card_nsmul {α : Type u_3} {β : Type u_4} {s : } {f : αβ} [] {b : β} (hf : as, f a = b) :
as, f a = s.card b
theorem Finset.prod_eq_pow_card {α : Type u_3} {β : Type u_4} {s : } {f : αβ} [] {b : β} (hf : as, f a = b) :
as, f a = b ^ s.card
theorem Finset.card_nsmul_add_sum {α : Type u_3} {β : Type u_4} {s : } {f : αβ} [] {b : β} :
s.card b + as, f a = as, (b + f a)
theorem Finset.pow_card_mul_prod {α : Type u_3} {β : Type u_4} {s : } {f : αβ} [] {b : β} :
b ^ s.card * as, f a = as, b * f a
theorem Finset.sum_add_card_nsmul {α : Type u_3} {β : Type u_4} {s : } {f : αβ} [] {b : β} :
as, f a + s.card b = as, (f a + b)
theorem Finset.prod_mul_pow_card {α : Type u_3} {β : Type u_4} {s : } {f : αβ} [] {b : β} :
(as, f a) * b ^ s.card = as, f a * b
theorem Finset.nsmul_eq_sum_const {β : Type u_4} [] (b : β) (n : ) :
n b =