Documentation

Mathlib.Algebra.BigOperators.Group.Finset

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.

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} [AddCommMonoid β] (s : Finset α) (f : αβ) :
β

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

When the index type is a Fintype, the notation ∑ x, f x, is a shorthand for ∑ x ∈ Finset.univ, f x.

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

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

    When the index type is a Fintype, the notation ∏ x, f x, is a shorthand for ∏ x ∈ Finset.univ, f x.

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      @[simp]
      theorem Finset.sum_mk {α : Type u_3} {β : Type u_4} [AddCommMonoid β] (s : Multiset α) (hs : s.Nodup) (f : αβ) :
      { val := s, nodup := hs }.sum f = (Multiset.map f s).sum
      @[simp]
      theorem Finset.prod_mk {α : Type u_3} {β : Type u_4} [CommMonoid β] (s : Multiset α) (hs : s.Nodup) (f : αβ) :
      { val := s, nodup := hs }.prod f = (Multiset.map f s).prod
      @[simp]
      theorem Finset.sum_val {α : Type u_3} [AddCommMonoid α] (s : Finset α) :
      s.val.sum = s.sum id
      @[simp]
      theorem Finset.prod_val {α : Type u_3} [CommMonoid α] (s : Finset α) :
      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 : Array (Lean.Term × Lean.Term)) (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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                    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} [AddCommMonoid β] (s : Finset α) (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} [CommMonoid β] (s : Finset α) (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} [AddCommMonoid β] (s : Finset α) (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} [CommMonoid β] (s : Finset α) (f : αβ) :
                                  (Multiset.map f s.val).prod = as, f a
                                  theorem Finset.sum_eq_fold {α : Type u_3} {β : Type u_4} [AddCommMonoid β] (s : Finset α) (f : αβ) :
                                  xs, f x = Finset.fold (fun (x1 x2 : β) => x1 + x2) 0 f s
                                  theorem Finset.prod_eq_fold {α : Type u_3} {β : Type u_4} [CommMonoid β] (s : Finset α) (f : αβ) :
                                  xs, f x = Finset.fold (fun (x1 x2 : β) => x1 * x2) 1 f s
                                  @[simp]
                                  theorem Finset.sum_multiset_singleton {α : Type u_3} (s : Finset α) :
                                  xs, {x} = s.val
                                  @[simp]
                                  theorem map_sum {α : Type u_3} {β : Type u_4} {γ : Type u_5} [AddCommMonoid β] [AddCommMonoid γ] {G : Type u_6} [FunLike G β γ] [AddMonoidHomClass G β γ] (g : G) (f : αβ) (s : Finset α) :
                                  g (∑ xs, f x) = xs, g (f x)
                                  @[simp]
                                  theorem map_prod {α : Type u_3} {β : Type u_4} {γ : Type u_5} [CommMonoid β] [CommMonoid γ] {G : Type u_6} [FunLike G β γ] [MonoidHomClass G β γ] (g : G) (f : αβ) (s : Finset α) :
                                  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} [AddZeroClass β] [AddCommMonoid γ] (f : αβ →+ γ) (s : Finset α) :
                                  (∑ xs, f x) = xs, (f x)
                                  theorem MonoidHom.coe_finset_prod {α : Type u_3} {β : Type u_4} {γ : Type u_5} [MulOneClass β] [CommMonoid γ] (f : αβ →* γ) (s : Finset α) :
                                  (∏ xs, f x) = xs, (f x)
                                  @[simp]
                                  theorem AddMonoidHom.finset_sum_apply {α : Type u_3} {β : Type u_4} {γ : Type u_5} [AddZeroClass β] [AddCommMonoid γ] (f : αβ →+ γ) (s : Finset α) (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} [MulOneClass β] [CommMonoid γ] (f : αβ →* γ) (s : Finset α) (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} [AddCommMonoid β] :
                                  x, f x = 0
                                  @[simp]
                                  theorem Finset.prod_empty {α : Type u_3} {β : Type u_4} {f : αβ} [CommMonoid β] :
                                  x, f x = 1
                                  theorem Finset.sum_of_isEmpty {α : Type u_3} {β : Type u_4} {f : αβ} [AddCommMonoid β] [IsEmpty α] (s : Finset α) :
                                  is, f i = 0
                                  theorem Finset.prod_of_isEmpty {α : Type u_3} {β : Type u_4} {f : αβ} [CommMonoid β] [IsEmpty α] (s : Finset α) :
                                  is, f i = 1
                                  @[deprecated Finset.prod_of_isEmpty]
                                  theorem Finset.prod_of_empty {α : Type u_3} {β : Type u_4} {f : αβ} [CommMonoid β] [IsEmpty α] (s : Finset α) :
                                  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} [AddCommMonoid β] [IsEmpty α] (s : Finset α) :
                                  is, f i = 0

                                  Alias of Finset.sum_of_isEmpty.

                                  @[simp]
                                  theorem Finset.sum_cons {α : Type u_3} {β : Type u_4} {s : Finset α} {a : α} {f : αβ} [AddCommMonoid β] (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 : Finset α} {a : α} {f : αβ} [CommMonoid β] (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 : Finset α} {a : α} {f : αβ} [AddCommMonoid β] [DecidableEq α] :
                                  asxinsert a s, f x = f a + xs, f x
                                  @[simp]
                                  theorem Finset.prod_insert {α : Type u_3} {β : Type u_4} {s : Finset α} {a : α} {f : αβ} [CommMonoid β] [DecidableEq α] :
                                  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 : Finset α} {a : α} {f : αβ} [AddCommMonoid β] [DecidableEq α] (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 : Finset α} {a : α} {f : αβ} [CommMonoid β] [DecidableEq α] (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 : Finset α} {a : α} {f : αβ} [AddCommMonoid β] [DecidableEq α] (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 : Finset α} {a : α} {f : αβ} [CommMonoid β] [DecidableEq α] (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 : Finset α} {a : α} {M : Type u_6} [AddCommGroup M] [DecidableEq α] (ha : as) {f : αM} :
                                  xinsert a s, f x - f a = xs, f x
                                  theorem Finset.prod_insert_div {α : Type u_3} {s : Finset α} {a : α} {M : Type u_6} [CommGroup M] [DecidableEq α] (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} [AddCommMonoid β] (f : αβ) (a : α) :
                                  x{a}, f x = f a
                                  @[simp]
                                  theorem Finset.prod_singleton {α : Type u_3} {β : Type u_4} [CommMonoid β] (f : αβ) (a : α) :
                                  x{a}, f x = f a
                                  theorem Finset.sum_pair {α : Type u_3} {β : Type u_4} {f : αβ} [AddCommMonoid β] [DecidableEq α] {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} [CommMonoid β] [DecidableEq α] {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 : Finset α} [AddCommMonoid β] :
                                  _xs, 0 = 0
                                  @[simp]
                                  theorem Finset.prod_const_one {α : Type u_3} {β : Type u_4} {s : Finset α} [CommMonoid β] :
                                  _xs, 1 = 1
                                  @[simp]
                                  theorem Finset.sum_image {α : Type u_3} {β : Type u_4} {γ : Type u_5} {f : αβ} [AddCommMonoid β] [DecidableEq α] {s : Finset γ} {g : γα} :
                                  (∀ xs, ys, g x = g yx = y)xFinset.image g s, 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} [CommMonoid β] [DecidableEq α] {s : Finset γ} {g : γα} :
                                  (∀ xs, ys, g x = g yx = y)xFinset.image g s, f x = xs, f (g x)
                                  @[simp]
                                  theorem Finset.sum_map {α : Type u_3} {β : Type u_4} {γ : Type u_5} [AddCommMonoid β] (s : Finset α) (e : α γ) (f : γβ) :
                                  xFinset.map e s, f x = xs, f (e x)
                                  @[simp]
                                  theorem Finset.prod_map {α : Type u_3} {β : Type u_4} {γ : Type u_5} [CommMonoid β] (s : Finset α) (e : α γ) (f : γβ) :
                                  xFinset.map e s, f x = xs, f (e x)
                                  theorem Finset.sum_attach {α : Type u_3} {β : Type u_4} [AddCommMonoid β] (s : Finset α) (f : αβ) :
                                  xs.attach, f x = xs, f x
                                  theorem Finset.prod_attach {α : Type u_3} {β : Type u_4} [CommMonoid β] (s : Finset α) (f : αβ) :
                                  xs.attach, f x = xs, f x
                                  theorem Finset.sum_congr {α : Type u_3} {β : Type u_4} {s₁ : Finset α} {s₂ : Finset α} {f : αβ} {g : αβ} [AddCommMonoid β] (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 : Finset α} {s₂ : Finset α} {f : αβ} {g : αβ} [CommMonoid β] (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} [AddCommMonoid β] {f : αβ} {s : Finset α} (h : xs, f x = 0) :
                                  xs, f x = 0
                                  theorem Finset.prod_eq_one {α : Type u_3} {β : Type u_4} [CommMonoid β] {f : αβ} {s : Finset α} (h : xs, f x = 1) :
                                  xs, f x = 1
                                  theorem Finset.sum_disjUnion {α : Type u_3} {β : Type u_4} {s₁ : Finset α} {s₂ : Finset α} {f : αβ} [AddCommMonoid β] (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 : Finset α} {s₂ : Finset α} {f : αβ} [CommMonoid β] (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} [AddCommMonoid β] (s : Finset ι) (t : ιFinset α) (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} [CommMonoid β] (s : Finset ι) (t : ιFinset α) (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 : Finset α} {s₂ : Finset α} {f : αβ} [AddCommMonoid β] [DecidableEq α] :
                                  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 : Finset α} {s₂ : Finset α} {f : αβ} [CommMonoid β] [DecidableEq α] :
                                  (∏ 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 : Finset α} {s₂ : Finset α} {f : αβ} [AddCommMonoid β] [DecidableEq α] (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 : Finset α} {s₂ : Finset α} {f : αβ} [CommMonoid β] [DecidableEq α] (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} [AddCommMonoid β] (s : Finset α) (p : αProp) [DecidablePred p] [(x : α) → Decidable ¬p x] (f : αβ) :
                                  xFinset.filter p s, 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} [CommMonoid β] (s : Finset α) (p : αProp) [DecidablePred p] [(x : α) → Decidable ¬p x] (f : αβ) :
                                  (∏ xFinset.filter p s, 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} [AddCommMonoid β] (s : Finset α) (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} [CommMonoid β] (s : Finset α) (f : αβ) :
                                  (List.map f s.toList).prod = s.prod f
                                  theorem Equiv.Perm.sum_comp {α : Type u_3} {β : Type u_4} [AddCommMonoid β] (σ : Equiv.Perm α) (s : Finset α) (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} [CommMonoid β] (σ : Equiv.Perm α) (s : Finset α) (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} [AddCommMonoid β] (σ : Equiv.Perm α) (s : Finset α) (f : ααβ) (hs : {a : α | σ a a} s) :
                                  xs, f (σ x) x = xs, f x ((Equiv.symm σ) x)
                                  theorem Equiv.Perm.prod_comp' {α : Type u_3} {β : Type u_4} [CommMonoid β] (σ : Equiv.Perm α) (s : Finset α) (f : ααβ) (hs : {a : α | σ a a} s) :
                                  xs, f (σ x) x = xs, f x ((Equiv.symm σ) x)
                                  theorem Finset.sum_powerset_insert {α : Type u_3} {β : Type u_4} {s : Finset α} {a : α} [AddCommMonoid β] [DecidableEq α] (ha : as) (f : Finset αβ) :
                                  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 : Finset α} {a : α} [CommMonoid β] [DecidableEq α] (ha : as) (f : Finset αβ) :
                                  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 : Finset α} {a : α} [AddCommMonoid β] (ha : as) (f : Finset αβ) :
                                  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 : Finset α} {a : α} [CommMonoid β] (ha : as) (f : Finset αβ) :
                                  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} [AddCommMonoid β] (s : Finset α) (f : Finset αβ) :
                                  ts.powerset, f t = jFinset.range (s.card + 1), tFinset.powersetCard j s, 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} [CommMonoid β] (s : Finset α) (f : Finset αβ) :
                                  ts.powerset, f t = jFinset.range (s.card + 1), tFinset.powersetCard j s, 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} [Fintype α] [AddCommMonoid β] {s : Finset α} {t : Finset α} (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} [Fintype α] [CommMonoid β] {s : Finset α} {t : Finset α} (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} [AddCommMonoid β] [Fintype α] [DecidableEq α] (s : Finset α) (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} [CommMonoid β] [Fintype α] [DecidableEq α] (s : Finset α) (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} [AddCommMonoid β] [Fintype α] [DecidableEq α] (s : Finset α) (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} [CommMonoid β] [Fintype α] [DecidableEq α] (s : Finset α) (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 : Finset α} {s₂ : Finset α} {f : αβ} [AddCommMonoid β] [DecidableEq α] (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 : Finset α} {s₂ : Finset α} {f : αβ} [CommMonoid β] [DecidableEq α] (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 : Finset α} {s₂ : Finset α} {f : αβ} {g : αβ} [AddCommMonoid β] [DecidableEq α] (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 : Finset α} {s₂ : Finset α} {f : αβ} {g : αβ} [CommMonoid β] [DecidableEq α] (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 : Finset α} {s₂ : Finset α} {f : αβ} [AddCommMonoid β] (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 : Finset α} {s₂ : Finset α} {f : αβ} [CommMonoid β] (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} [AddCommMonoid β] (s : Finset α) (t : Finset γ) (f : α γβ) :
                                  xs.disjSum t, f x = xs, f (Sum.inl x) + xt, f (Sum.inr x)
                                  @[simp]
                                  theorem Finset.prod_disj_sum {α : Type u_3} {β : Type u_4} {γ : Type u_5} [CommMonoid β] (s : Finset α) (t : Finset γ) (f : α γβ) :
                                  xs.disjSum t, f x = (∏ xs, f (Sum.inl x)) * xt, f (Sum.inr x)
                                  theorem Finset.sum_sum_elim {α : Type u_3} {β : Type u_4} {γ : Type u_5} [AddCommMonoid β] (s : Finset α) (t : Finset γ) (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} [CommMonoid β] (s : Finset α) (t : Finset γ) (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} [AddCommMonoid β] [DecidableEq α] {s : Finset γ} {t : γFinset α} (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} [CommMonoid β] [DecidableEq α] {s : Finset γ} {t : γFinset α} (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} [AddCommMonoid β] {σ : αType u_6} (s : Finset α) (t : (a : α) → Finset (σ a)) (f : Sigma σβ) :
                                  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} [CommMonoid β] {σ : αType u_6} (s : Finset α) (t : (a : α) → Finset (σ a)) (f : Sigma σβ) :
                                  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} [AddCommMonoid β] {σ : αType u_6} (s : Finset α) (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} [CommMonoid β] {σ : αType u_6} (s : Finset α) (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} [AddCommMonoid α] {s : Finset ι} {t : Finset κ} {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} [CommMonoid α] {s : Finset ι} {t : Finset κ} {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} [AddCommMonoid α] {s : Finset ι} {t : Finset κ} {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} [CommMonoid α] {s : Finset ι} {t : Finset κ} {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} [AddCommMonoid α] {s : Finset ι} {t : Finset κ} {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} [CommMonoid α] {s : Finset ι} {t : Finset κ} {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} [AddCommMonoid α] {s : Finset ι} {t : Finset κ} {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} [CommMonoid α] {s : Finset ι} {t : Finset κ} {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} [AddCommMonoid α] {s : Finset ι} {t : Finset κ} {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} [CommMonoid α] {s : Finset ι} {t : Finset κ} {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} [AddCommMonoid α] {s : Finset ι} {t : Finset κ} {f : ια} {g : κα} (e : ικ) (he : Function.Bijective e) (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} [CommMonoid α] {s : Finset ι} {t : Finset κ} {f : ια} {g : κα} (e : ικ) (he : Function.Bijective e) (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} [AddCommMonoid α] {s : Finset ι} {t : Finset κ} {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} [CommMonoid α] {s : Finset ι} {t : Finset κ} {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} [AddCommMonoid α] [DecidableEq κ] (s : Finset ι) (t : Finset κ) (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} [CommMonoid α] [DecidableEq κ] (s : Finset ι) (t : Finset κ) (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} [AddCommMonoid α] [DecidableEq κ] (s : Finset ι) (t : Finset κ) (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} [CommMonoid α] [DecidableEq κ] (s : Finset ι) (t : Finset κ) (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} [AddCommMonoid α] {s : Finset ι} {t : Finset κ} [DecidableEq κ] {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} [CommMonoid α] {s : Finset ι} {t : Finset κ} [DecidableEq κ] {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} [AddCommMonoid α] {s : Finset ι} {t : Finset κ} [DecidableEq κ] {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} [CommMonoid α] {s : Finset ι} {t : Finset κ} [DecidableEq κ] {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} [AddCommMonoid α] [DecidableEq κ] [Fintype κ] (s : Finset ι) (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} [CommMonoid α] [DecidableEq κ] [Fintype κ] (s : Finset ι) (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} [AddCommMonoid α] [DecidableEq κ] [Fintype κ] (s : Finset ι) (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} [CommMonoid α] [DecidableEq κ] [Fintype κ] (s : Finset ι) (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} [AddCommMonoid β] [DecidableEq ι] [Fintype ι] {κ : ιType u_6} (t : (i : ι) → Finset (κ i)) (f : ((i : ι) → i Finset.univκ i)β) :
                                  xFinset.univ.pi t, f x = xFintype.piFinset t, 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} [CommMonoid β] [DecidableEq ι] [Fintype ι] {κ : ιType u_6} (t : (i : ι) → Finset (κ i)) (f : ((i : ι) → i Finset.univκ i)β) :
                                  xFinset.univ.pi t, f x = xFintype.piFinset t, 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} [AddCommMonoid β] [DecidableEq α] (s : Finset α) (f : α × αβ) :
                                  is.diag, f i = is, f (i, i)
                                  @[simp]
                                  theorem Finset.prod_diag {α : Type u_3} {β : Type u_4} [CommMonoid β] [DecidableEq α] (s : Finset α) (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} [AddCommMonoid β] (r : Finset (γ × α)) (s : Finset γ) (t : γFinset α) (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} [CommMonoid β] (r : Finset (γ × α)) (s : Finset γ) (t : γFinset α) (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} [AddCommMonoid β] (r : Finset (γ × α)) (s : Finset γ) (t : γFinset α) (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} [CommMonoid β] (r : Finset (γ × α)) (s : Finset γ) (t : γFinset α) (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} [AddCommMonoid β] (r : Finset (α × γ)) (s : Finset γ) (t : γFinset α) (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} [CommMonoid β] (r : Finset (α × γ)) (s : Finset γ) (t : γFinset α) (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} [AddCommMonoid β] (r : Finset (α × γ)) (s : Finset γ) (t : γFinset α) (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} [CommMonoid β] (r : Finset (α × γ)) (s : Finset γ) (t : γFinset α) (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} [AddCommMonoid β] [DecidableEq α] {s : Finset γ} {g : γα} (h : γβ) (eq : cs, f (g c) = xFinset.filter (fun (c' : γ) => g c' = g c) s, h x) :
                                  xFinset.image g s, f x = xs, h x
                                  theorem Finset.prod_image' {α : Type u_3} {β : Type u_4} {γ : Type u_5} {f : αβ} [CommMonoid β] [DecidableEq α] {s : Finset γ} {g : γα} (h : γβ) (eq : cs, f (g c) = xFinset.filter (fun (c' : γ) => g c' = g c) s, h x) :
                                  xFinset.image g s, f x = xs, h x
                                  theorem Finset.sum_add_distrib {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} {g : αβ} [AddCommMonoid β] :
                                  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 : Finset α} {f : αβ} {g : αβ} [CommMonoid β] :
                                  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 : Finset α} [AddCommMonoid β] (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 : Finset α} [CommMonoid β] (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} [AddCommMonoid β] {s : Finset γ} {t : Finset α} {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} [CommMonoid β] {s : Finset γ} {t : Finset α} {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} [AddCommMonoid β] {s : Finset γ} {t : Finset α} {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} [CommMonoid β] {s : Finset γ} {t : Finset α} {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} [AddCommMonoid β] {s : Finset γ} {t : Finset α} {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} [CommMonoid β] {s : Finset γ} {t : Finset α} {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} [AddCommMonoid β] {s : Finset γ} {t : Finset α} {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} [CommMonoid β] {s : Finset γ} {t : Finset α} {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} [AddCommMonoid β] {s : Finset γ} {t : γFinset α} {t' : Finset α} {s' : αFinset γ} (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.

                                  theorem Finset.prod_comm' {α : Type u_3} {β : Type u_4} {γ : Type u_5} [CommMonoid β] {s : Finset γ} {t : γFinset α} {t' : Finset α} {s' : αFinset γ} (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} [AddCommMonoid β] {s : Finset γ} {t : Finset α} {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} [CommMonoid β] {s : Finset γ} {t : Finset α} {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} [AddCommMonoid β] [AddCommMonoid γ] {r : βγProp} {f : αβ} {g : αγ} {s : Finset α} (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} [CommMonoid β] [CommMonoid γ] {r : βγProp} {f : αβ} {g : αγ} {s : Finset α} (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 : Finset α} {f : αβ} [AddCommMonoid β] {p : αProp} [DecidablePred p] (hp : xs, f x 0p x) :
                                  xFinset.filter p s, f x = xs, f x
                                  theorem Finset.prod_filter_of_ne {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} [CommMonoid β] {p : αProp} [DecidablePred p] (hp : xs, f x 1p x) :
                                  xFinset.filter p s, f x = xs, f x
                                  theorem Finset.sum_filter_ne_zero {α : Type u_3} {β : Type u_4} {f : αβ} [AddCommMonoid β] (s : Finset α) [(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} [CommMonoid β] (s : Finset α) [(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 : Finset α} [AddCommMonoid β] (p : αProp) [DecidablePred p] (f : αβ) :
                                  aFinset.filter p s, f a = as, if p a then f a else 0
                                  theorem Finset.prod_filter {α : Type u_3} {β : Type u_4} {s : Finset α} [CommMonoid β] (p : αProp) [DecidablePred p] (f : αβ) :
                                  aFinset.filter p s, 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} [AddCommMonoid β] {s : Finset α} {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} [CommMonoid β] {s : Finset α} {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} [AddCommMonoid β] {s : Finset α} {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} [CommMonoid β] {s : Finset α} {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 : Finset α} {s₂ : Finset α} {f : αβ} [AddCommMonoid β] [DecidableEq α] (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 : Finset α} {s₂ : Finset α} {f : αβ} [CommMonoid β] [DecidableEq α] (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 : Finset α} {s₂ : Finset α} {f : αβ} [AddCommMonoid β] [DecidableEq α] (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 : Finset α} {s₂ : Finset α} {f : αβ} [CommMonoid β] [DecidableEq α] (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} [AddCommMonoid β] {s : Finset α} {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} [CommMonoid β] {s : Finset α} {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} [AddCommMonoid β] {s : Finset α} {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} [CommMonoid β] {s : Finset α} {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 : Finset α} [AddCommMonoid β] (f : αβ) {p : αProp} [DecidablePred p] :
                                  xFinset.subtype p s, f x = xFinset.filter p s, 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 : Finset α} [CommMonoid β] (f : αβ) {p : αProp} [DecidablePred p] :
                                  xFinset.subtype p s, f x = xFinset.filter p s, 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 : Finset α} [AddCommMonoid β] (f : αβ) {p : αProp} [DecidablePred p] (h : xs, p x) :
                                  xFinset.subtype p s, 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 : Finset α} [CommMonoid β] (f : αβ) {p : αProp} [DecidablePred p] (h : xs, p x) :
                                  xFinset.subtype p s, 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} [AddCommMonoid β] {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} [CommMonoid β] {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 : Finset α) [AddCommMonoid β] (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 : Finset α) [CommMonoid β] (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 : Finset α) (f : αβ) [AddCommMonoid β] :
                                  i : { x : α // x s }, f i = is, f i
                                  theorem Finset.prod_coe_sort {α : Type u_3} {β : Type u_4} (s : Finset α) (f : αβ) [CommMonoid β] :
                                  i : { x : α // x s }, f i = is, f i
                                  theorem Finset.sum_finset_coe {α : Type u_3} {β : Type u_4} [AddCommMonoid β] (f : αβ) (s : Finset α) :
                                  i : s, f i = is, f i
                                  theorem Finset.prod_finset_coe {α : Type u_3} {β : Type u_4} [CommMonoid β] (f : αβ) (s : Finset α) :
                                  i : s, f i = is, f i
                                  theorem Finset.sum_subtype {α : Type u_3} {β : Type u_4} [AddCommMonoid β] {p : αProp} {F : Fintype (Subtype p)} (s : Finset α) (h : ∀ (x : α), x s p x) (f : αβ) :
                                  as, f a = a : Subtype p, f a
                                  theorem Finset.prod_subtype {α : Type u_3} {β : Type u_4} [CommMonoid β] {p : αProp} {F : Fintype (Subtype p)} (s : Finset α) (h : ∀ (x : α), x s p x) (f : αβ) :
                                  as, f a = a : Subtype p, f a
                                  theorem Finset.sum_preimage' {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [AddCommMonoid β] (f : ικ) [DecidablePred fun (x : κ) => x Set.range f] (s : Finset κ) (hf : Set.InjOn f (f ⁻¹' s)) (g : κβ) :
                                  xs.preimage f hf, g (f x) = xFinset.filter (fun (x : κ) => x Set.range f) s, g x
                                  theorem Finset.prod_preimage' {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [CommMonoid β] (f : ικ) [DecidablePred fun (x : κ) => x Set.range f] (s : Finset κ) (hf : Set.InjOn f (f ⁻¹' s)) (g : κβ) :
                                  xs.preimage f hf, g (f x) = xFinset.filter (fun (x : κ) => x Set.range f) s, g x
                                  theorem Finset.sum_preimage {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [AddCommMonoid β] (f : ικ) (s : Finset κ) (hf : Set.InjOn f (f ⁻¹' s)) (g : κβ) (hg : xs, xSet.range fg 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} [CommMonoid β] (f : ικ) (s : Finset κ) (hf : Set.InjOn f (f ⁻¹' s)) (g : κβ) (hg : xs, xSet.range fg 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} [AddCommMonoid β] (f : ικ) (s : Finset κ) (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} [CommMonoid β] (f : ικ) (s : Finset κ) (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} [AddCommMonoid β] (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} [CommMonoid β] (s : Set α) [Fintype s] :
                                  i : s, f i = is.toFinset, f i
                                  theorem Finset.sum_congr_set {α : Type u_6} [AddCommMonoid α] {β : Type u_7} [Fintype β] (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.

                                  theorem Finset.prod_congr_set {α : Type u_6} [CommMonoid α] {β : Type u_7} [Fintype β] (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} [AddCommMonoid β] {s : Finset α} {p : αProp} {hp : DecidablePred p} [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(Finset.filter p s).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} [CommMonoid β] {s : Finset α} {p : αProp} {hp : DecidablePred p} [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(Finset.filter p s).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} [AddCommMonoid β] {s : Finset α} {p : αProp} {_hp : DecidablePred p} (f : αγ) (g : αγ) (h : γβ) :
                                  xs, h (if p x then f x else g x) = xFinset.filter p s, 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} [CommMonoid β] {s : Finset α} {p : αProp} {_hp : DecidablePred p} (f : αγ) (g : αγ) (h : γβ) :
                                  xs, h (if p x then f x else g x) = (∏ xFinset.filter p s, 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} [AddCommMonoid β] {s : Finset α} {p : αProp} {hp : DecidablePred p} (f : (x : α) → p xβ) (g : (x : α) → ¬p xβ) :
                                  (∑ xs, if hx : p x then f x hx else g x hx) = x(Finset.filter p s).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} [CommMonoid β] {s : Finset α} {p : αProp} {hp : DecidablePred p} (f : (x : α) → p xβ) (g : (x : α) → ¬p xβ) :
                                  (∏ xs, if hx : p x then f x hx else g x hx) = (∏ x(Finset.filter p s).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} [AddCommMonoid β] {s : Finset α} {p : αProp} {hp : DecidablePred p} (f : αβ) (g : αβ) :
                                  (∑ xs, if p x then f x else g x) = xFinset.filter p s, f x + xFinset.filter (fun (x : α) => ¬p x) s, g x
                                  theorem Finset.prod_ite {α : Type u_3} {β : Type u_4} [CommMonoid β] {s : Finset α} {p : αProp} {hp : DecidablePred p} (f : αβ) (g : αβ) :
                                  (∏ xs, if p x then f x else g x) = (∏ xFinset.filter p s, 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 : Finset α} [AddCommMonoid β] {p : αProp} :
                                  ∀ {x : DecidablePred p} (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 : Finset α} [CommMonoid β] {p : αProp} :
                                  ∀ {x : DecidablePred p} (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 : Finset α} [AddCommMonoid β] {p : αProp} :
                                  ∀ {x : DecidablePred p}, (∀ 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 : Finset α} [CommMonoid β] {p : αProp} :
                                  ∀ {x : DecidablePred p}, (∀ 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 : Finset α} [AddCommMonoid β] {p : αProp} :
                                  ∀ {x : DecidablePred p} (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 : Finset α} [CommMonoid β] {p : αProp} :
                                  ∀ {x : DecidablePred p} (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 : Finset α} [AddCommMonoid β] {p : αProp} :
                                  ∀ {x : DecidablePred p}, (∀ 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 : Finset α} [CommMonoid β] {p : αProp} :
                                  ∀ {x : DecidablePred p}, (∀ 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 : Finset α} [AddCommMonoid β] {p : αProp} {hp : DecidablePred p} (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 : Finset α} [CommMonoid β] {p : αProp} {hp : DecidablePred p} (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 : Finset α} [AddCommMonoid β] {p : αProp} {hp : DecidablePred p} (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 : Finset α} [CommMonoid β] {p : αProp} {hp : DecidablePred p} (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} [AddCommMonoid β] [DecidableEq α] (s : Finset α) (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} [CommMonoid β] [DecidableEq α] (s : Finset α) (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} [AddCommMonoid β] [DecidableEq α] (s : Finset α) (t : Finset α) (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} [CommMonoid β] [DecidableEq α] (s : Finset α) (t : Finset α) (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} [AddCommMonoid β] [DecidableEq α] (s : Finset α) (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} [CommMonoid β] [DecidableEq α] (s : Finset α) (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} [AddCommMonoid β] [DecidableEq α] (s : Finset α) (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} [CommMonoid β] [DecidableEq α] (s : Finset α) (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} [AddCommMonoid β] [DecidableEq α] (s : Finset α) (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} [CommMonoid β] [DecidableEq α] (s : Finset α) (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} [AddCommMonoid β] [DecidableEq α] (s : Finset α) (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} [CommMonoid β] [DecidableEq α] (s : Finset α) (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} [AddCommMonoid β] (p : Prop) [Decidable p] (s : Finset α) (t : Finset α) (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} [CommMonoid β] (p : Prop) [Decidable p] (s : Finset α) (t : Finset α) (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} [AddCommMonoid β] (p : Prop) [Decidable p] (s : Finset α) (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} [CommMonoid β] (p : Prop) [Decidable p] (s : Finset α) (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} [AddCommMonoid β] (p : Prop) [Decidable p] (s : Finset α) (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} [CommMonoid β] (p : Prop) [Decidable p] (s : Finset α) (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} [AddCommMonoid β] [DecidableEq α] (a : α) (x : β) (s : Finset α) :
                                  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} [CommMonoid β] [DecidableEq α] (a : α) (x : β) (s : Finset α) :
                                  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} [DecidableEq α] [(a : α) → AddCommMonoid (β a)] (a : α) (f : (a : α) → β a) (s : Finset α) :
                                  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} [DecidableEq α] [(a : α) → CommMonoid (β a)] (a : α) (f : (a : α) → β a) (s : Finset α) :
                                  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} [AddCommMonoid β] (s : Finset ι) (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} [CommMonoid β] (s : Finset ι) (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} [AddCommMonoid β] [Zero α] (f : ια) (g : ιαβ) {s : Finset ι} {t : Finset ι} (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} [CommMonoid β] [One α] (f : ια) (g : ιαβ) {s : Finset ι} {t : Finset ι} (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} [AddCommMonoid β] (f : ιβ) {s : Finset ι} {t : Finset ι} (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} [CommMonoid β] (f : ιβ) {s : Finset ι} {t : Finset ι} (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} [AddCommMonoid β] {κ : Type u_6} (s : Finset ι) (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} [CommMonoid β] {κ : Type u_6} (s : Finset ι) (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} [AddCommMonoid β] [DecidableEq ι] (s : Finset ι) (t : Finset ι) (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} [CommMonoid β] [DecidableEq ι] (s : Finset ι) (t : Finset ι) (f : ιβ) :
                                  is, (↑t).mulIndicator f i = is t, f i
                                  theorem Finset.indicator_sum {ι : Type u_1} {β : Type u_4} [AddCommMonoid β] {κ : Type u_6} (s : Finset ι) (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} [CommMonoid β] {κ : Type u_6} (s : Finset ι) (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} [AddCommMonoid β] {κ : Type u_7} (s : Finset ι) (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} [CommMonoid β] {κ : Type u_7} (s : Finset ι) (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} [AddCommMonoid β] {κ : Type u_7} (s : Finset ι) (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} [CommMonoid β] {κ : Type u_7} (s : Finset ι) (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} [AddCommMonoid β] {s : Finset α} {t : Finset γ} {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} [CommMonoid β] {s : Finset α} {t : Finset γ} {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 : Finset α} {f : αβ} [AddCommMonoid β] (h : xs, f x 0) :
                                  s.Nonempty
                                  theorem Finset.nonempty_of_prod_ne_one {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} [CommMonoid β] (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 : Finset α} {f : αβ} [AddCommMonoid β] (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 : Finset α} {f : αβ} [CommMonoid β] (h : xs, f x 1) :
                                  as, f a 1
                                  theorem Finset.sum_range_succ_comm {β : Type u_4} [AddCommMonoid β] (f : β) (n : ) :
                                  xFinset.range (n + 1), f x = f n + xFinset.range n, f x
                                  theorem Finset.prod_range_succ_comm {β : Type u_4} [CommMonoid β] (f : β) (n : ) :
                                  xFinset.range (n + 1), f x = f n * xFinset.range n, f x
                                  theorem Finset.sum_range_succ {β : Type u_4} [AddCommMonoid β] (f : β) (n : ) :
                                  xFinset.range (n + 1), f x = xFinset.range n, f x + f n
                                  theorem Finset.prod_range_succ {β : Type u_4} [CommMonoid β] (f : β) (n : ) :
                                  xFinset.range (n + 1), f x = (∏ xFinset.range n, f x) * f n
                                  theorem Finset.sum_range_succ' {β : Type u_4} [AddCommMonoid β] (f : β) (n : ) :
                                  kFinset.range (n + 1), f k = kFinset.range n, f (k + 1) + f 0
                                  theorem Finset.prod_range_succ' {β : Type u_4} [CommMonoid β] (f : β) (n : ) :
                                  kFinset.range (n + 1), f k = (∏ kFinset.range n, f (k + 1)) * f 0
                                  theorem Finset.eventually_constant_sum {β : Type u_4} [AddCommMonoid β] {u : β} {N : } (hu : nN, u n = 0) {n : } (hn : N n) :
                                  kFinset.range n, u k = kFinset.range N, u k
                                  theorem Finset.eventually_constant_prod {β : Type u_4} [CommMonoid β] {u : β} {N : } (hu : nN, u n = 1) {n : } (hn : N n) :
                                  kFinset.range n, u k = kFinset.range N, u k
                                  theorem Finset.sum_range_add {β : Type u_4} [AddCommMonoid β] (f : β) (n : ) (m : ) :
                                  xFinset.range (n + m), f x = xFinset.range n, f x + xFinset.range m, f (n + x)
                                  theorem Finset.prod_range_add {β : Type u_4} [CommMonoid β] (f : β) (n : ) (m : ) :
                                  xFinset.range (n + m), f x = (∏ xFinset.range n, f x) * xFinset.range m, f (n + x)
                                  theorem Finset.sum_range_add_sub_sum_range {α : Type u_6} [AddCommGroup α] (f : α) (n : ) (m : ) :
                                  kFinset.range (n + m), f k - kFinset.range n, f k = kFinset.range m, f (n + k)
                                  theorem Finset.prod_range_add_div_prod_range {α : Type u_6} [CommGroup α] (f : α) (n : ) (m : ) :
                                  (∏ kFinset.range (n + m), f k) / kFinset.range n, f k = kFinset.range m, f (n + k)
                                  theorem Finset.sum_range_zero {β : Type u_4} [AddCommMonoid β] (f : β) :
                                  kFinset.range 0, f k = 0
                                  theorem Finset.prod_range_zero {β : Type u_4} [CommMonoid β] (f : β) :
                                  kFinset.range 0, f k = 1
                                  theorem Finset.sum_range_one {β : Type u_4} [AddCommMonoid β] (f : β) :
                                  kFinset.range 1, f k = f 0
                                  theorem Finset.prod_range_one {β : Type u_4} [CommMonoid β] (f : β) :
                                  kFinset.range 1, f k = f 0
                                  theorem Finset.sum_list_map_count {α : Type u_3} [DecidableEq α] (l : List α) {M : Type u_6} [AddCommMonoid M] (f : αM) :
                                  (List.map f l).sum = ml.toFinset, List.count m l f m
                                  theorem Finset.prod_list_map_count {α : Type u_3} [DecidableEq α] (l : List α) {M : Type u_6} [CommMonoid M] (f : αM) :
                                  (List.map f l).prod = ml.toFinset, f m ^ List.count m l
                                  theorem Finset.sum_list_count {α : Type u_3} [DecidableEq α] [AddCommMonoid α] (s : List α) :
                                  s.sum = ms.toFinset, List.count m s m
                                  theorem Finset.prod_list_count {α : Type u_3} [DecidableEq α] [CommMonoid α] (s : List α) :
                                  s.prod = ms.toFinset, m ^ List.count m s
                                  theorem Finset.sum_list_count_of_subset {α : Type u_3} [DecidableEq α] [AddCommMonoid α] (m : List α) (s : Finset α) (hs : m.toFinset s) :
                                  m.sum = is, List.count i m i
                                  theorem Finset.prod_list_count_of_subset {α : Type u_3} [DecidableEq α] [CommMonoid α] (m : List α) (s : Finset α) (hs : m.toFinset s) :
                                  m.prod = is, i ^ List.count i m
                                  theorem Finset.sum_filter_count_eq_countP {α : Type u_3} [DecidableEq α] (p : αProp) [DecidablePred p] (l : List α) :
                                  xFinset.filter p l.toFinset, List.count x l = List.countP (fun (b : α) => decide (p b)) l
                                  theorem Finset.sum_multiset_map_count {α : Type u_3} [DecidableEq α] (s : Multiset α) {M : Type u_6} [AddCommMonoid M] (f : αM) :
                                  (Multiset.map f s).sum = ms.toFinset, Multiset.count m s f m
                                  theorem Finset.prod_multiset_map_count {α : Type u_3} [DecidableEq α] (s : Multiset α) {M : Type u_6} [CommMonoid M] (f : αM) :
                                  (Multiset.map f s).prod = ms.toFinset, f m ^ Multiset.count m s
                                  theorem Finset.sum_multiset_count {α : Type u_3} [DecidableEq α] [AddCommMonoid α] (s : Multiset α) :
                                  s.sum = ms.toFinset, Multiset.count m s m
                                  theorem Finset.prod_multiset_count {α : Type u_3} [DecidableEq α] [CommMonoid α] (s : Multiset α) :
                                  s.prod = ms.toFinset, m ^ Multiset.count m s
                                  theorem Finset.sum_multiset_count_of_subset {α : Type u_3} [DecidableEq α] [AddCommMonoid α] (m : Multiset α) (s : Finset α) (hs : m.toFinset s) :
                                  m.sum = is, Multiset.count i m i
                                  theorem Finset.prod_multiset_count_of_subset {α : Type u_3} [DecidableEq α] [CommMonoid α] (m : Multiset α) (s : Finset α) (hs : m.toFinset s) :
                                  m.prod = is, i ^ Multiset.count i m
                                  theorem Finset.sum_mem_multiset {α : Type u_3} {β : Type u_4} [AddCommMonoid β] [DecidableEq α] (m : Multiset α) (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} [CommMonoid β] [DecidableEq α] (m : Multiset α) (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 : Finset α} {M : Type u_6} [AddCommMonoid M] (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 : Finset α} {M : Type u_6} [CommMonoid M] (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 : Finset α} {M : Type u_6} [AddCommMonoid M] (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 : Finset α} {M : Type u_6} [CommMonoid M] (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} [AddCommMonoid β] (f : β) (s : β) (base : s 0 = 0) (step : ∀ (n : ), s (n + 1) = s n + f n) (n : ) :
                                  kFinset.range n, 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} [CommMonoid β] (f : β) (s : β) (base : s 0 = 1) (step : ∀ (n : ), s (n + 1) = s n * f n) (n : ) :
                                  kFinset.range n, 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} [AddCommGroup M] (f : M) (n : ) :
                                  iFinset.range n, (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} [CommGroup M] (f : M) (n : ) :
                                  iFinset.range n, 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} [AddCommGroup M] (f : M) (n : ) :
                                  iFinset.range n, (f i - f (i + 1)) = f 0 - f n
                                  theorem Finset.prod_range_div' {M : Type u_6} [CommGroup M] (f : M) (n : ) :
                                  iFinset.range n, f i / f (i + 1) = f 0 / f n
                                  theorem Finset.eq_sum_range_sub {M : Type u_6} [AddCommGroup M] (f : M) (n : ) :
                                  f n = f 0 + iFinset.range n, (f (i + 1) - f i)
                                  theorem Finset.eq_prod_range_div {M : Type u_6} [CommGroup M] (f : M) (n : ) :
                                  f n = f 0 * iFinset.range n, f (i + 1) / f i
                                  theorem Finset.eq_sum_range_sub' {M : Type u_6} [AddCommGroup M] (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} [CommGroup M] (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} [AddCommMonoid α] [PartialOrder α] [Sub α] [OrderedSub α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 x2] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 x2] [ExistsAddOfLE α] {f : α} (h : Monotone f) (n : ) :
                                  iFinset.range n, (f (i + 1) - f i) = f n - f 0

                                  A telescoping sum along {0, ..., n-1} of an -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 : Finset α} [AddCommMonoid β] (b : β) :
                                  _xs, b = s.card b
                                  @[simp]
                                  theorem Finset.prod_const {α : Type u_3} {β : Type u_4} {s : Finset α} [CommMonoid β] (b : β) :
                                  _xs, b = b ^ s.card
                                  theorem Finset.sum_eq_card_nsmul {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} [AddCommMonoid β] {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 : Finset α} {f : αβ} [CommMonoid β] {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 : Finset α} {f : αβ} [AddCommMonoid β] {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 : Finset α} {f : αβ} [CommMonoid β] {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 : Finset α} {f : αβ} [AddCommMonoid β] {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 : Finset α} {f : αβ} [CommMonoid β] {b : β} :
                                  (∏ as, f a) * b ^ s.card = as, f a * b
                                  theorem Finset.nsmul_eq_sum_const {β : Type u_4} [AddCommMonoid β] (b : β) (n : ) :
                                  n b = _kFinset.range n, b
                                  theorem Finset.pow_eq_prod_const {β : Type u_4} [CommMonoid β] (b : β) (n : ) :
                                  b ^ n = _kFinset.range n, b
                                  theorem Finset.sum_nsmul {α : Type u_3} {β : Type u_4} [AddCommMonoid β] (s : Finset α) (n : ) (f : αβ) :
                                  xs, n f x = n xs, f x
                                  theorem Finset.prod_pow {α : Type u_3} {β : Type u_4} [CommMonoid β] (s : Finset α) (n : ) (f : αβ) :
                                  xs, f x ^ n = (∏ xs, f x) ^ n
                                  theorem Finset.sum_nsmul_assoc {ι : Type u_1} {β : Type u_4} [AddCommMonoid β] (s : Finset ι) (f : ι) (a : β) :
                                  is, f i a = (∑ is, f i) a
                                  theorem Finset.prod_pow_eq_pow_sum {ι : Type u_1} {β : Type u_4} [CommMonoid β] (s : Finset ι) (f : ι) (a : β) :
                                  is, a ^ f i = a ^ is, f i
                                  theorem Finset.sum_powersetCard {α : Type u_3} {β : Type u_4} [AddCommMonoid β] (n : ) (s : Finset α) (f : β) :
                                  tFinset.powersetCard n s, f t.card = s.card.choose n f n

                                  A sum over Finset.powersetCard which only depends on the size of the sets is constant.

                                  theorem Finset.prod_powersetCard {α : Type u_3} {β : Type u_4} [CommMonoid β] (n : ) (s : Finset α) (f : β) :
                                  tFinset.powersetCard n s, f t.card = f n ^ s.card.choose n

                                  A product over Finset.powersetCard which only depends on the size of the sets is constant.

                                  theorem Finset.sum_flip {β : Type u_4} [AddCommMonoid β] {n : } (f : β) :
                                  rFinset.range (n + 1), f (n - r) = kFinset.range (n + 1), f k
                                  theorem Finset.prod_flip {β : Type u_4} [CommMonoid β] {n : } (f : β) :
                                  rFinset.range (n + 1), f (n - r) = kFinset.range (n + 1), f k
                                  theorem Finset.sum_involution {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} [AddCommMonoid β] (g : (a : α) → a sα) (hg₁ : ∀ (a : α) (ha : a s), f a + f (g a ha) = 0) (hg₃ : ∀ (a : α) (ha : a s), f a 0g a ha a) (g_mem : ∀ (a : α) (ha : a s), g a ha s) (hg₄ : ∀ (a : α) (ha : a s), g (g a ha) = a) :
                                  xs, f x = 0

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

                                  theorem Finset.prod_involution {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} [CommMonoid β] (g : (a : α) → a sα) (hg₁ : ∀ (a : α) (ha : a s), f a * f (g a ha) = 1) (hg₃ : ∀ (a : α) (ha : a s), f a 1g a ha a) (g_mem : ∀ (a : α) (ha : a s), g a ha s) (hg₄ : ∀ (a : α) (ha : a s), g (g a ha) = a) :
                                  xs, f x = 1

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

                                  theorem Finset.sum_ninvolution {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} [AddCommMonoid β] (g : αα) (hg₁ : ∀ (a : α), f a + f (g a) = 0) (hg₂ : ∀ (a : α), f a 0g a a) (g_mem : ∀ (a : α), g a s) (hg₃ : ∀ (a : α), g (g a) = a) :
                                  xs, f x = 0

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

                                  theorem Finset.prod_ninvolution {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} [CommMonoid β] (g : αα) (hg₁ : ∀ (a : α), f a * f (g a) = 1) (hg₂ : ∀ (a : α), f a 1g a a) (g_mem : ∀ (a : α), g a s) (hg₃ : ∀ (a : α), g (g a) = a) :
                                  xs, f x = 1

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

                                  theorem Finset.sum_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} {s : Finset α} [AddCommMonoid β] [DecidableEq γ] (f : γβ) (g : αγ) :
                                  as, f (g a) = bFinset.image g s, (Finset.filter (fun (a : α) => g a = b) s).card f b

                                  The sum of the composition of functions f and g, is the sum over b ∈ s.image g of f b times of the cardinality of the fibre of b. See also Finset.sum_image.

                                  theorem Finset.prod_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} {s : Finset α} [CommMonoid β] [DecidableEq γ] (f : γβ) (g : αγ) :
                                  as, f (g a) = bFinset.image g s, f b ^ (Finset.filter (fun (a : α) => g a = b) s).card

                                  The product of the composition of functions f and g, is the product over b ∈ s.image g of f b to the power of the cardinality of the fibre of b. See also Finset.prod_image.

                                  theorem Finset.sum_piecewise {α : Type u_3} {β : Type u_4} [AddCommMonoid β] [DecidableEq α] (s : Finset α) (t : Finset α) (f : αβ) (g : αβ) :
                                  xs, t.piecewise f g x = xs t, f x + xs \ t, g x
                                  theorem Finset.prod_piecewise {α : Type u_3} {β : Type u_4} [CommMonoid β] [DecidableEq α] (s : Finset α) (t : Finset α) (f : αβ) (g : αβ) :
                                  xs, t.piecewise f g x = (∏ xs t, f x) * xs \ t, g x
                                  theorem Finset.sum_inter_add_sum_diff {α : Type u_3} {β : Type u_4} [AddCommMonoid β] [DecidableEq α] (s : Finset α) (t : Finset α) (f : αβ) :
                                  xs t, f x + xs \ t, f x = xs, f x
                                  theorem Finset.prod_inter_mul_prod_diff {α : Type u_3} {β : Type u_4} [CommMonoid β] [DecidableEq α] (s : Finset α) (t : Finset α) (f : αβ) :
                                  (∏ xs t, f x) * xs \ t, f x = xs, f x
                                  theorem Finset.sum_eq_add_sum_diff_singleton {α : Type u_3} {β : Type u_4} [AddCommMonoid β] [DecidableEq α] {s : Finset α} {i : α} (h : i s) (f : αβ) :
                                  xs, f x = f i + xs \ {i}, f x
                                  theorem Finset.prod_eq_mul_prod_diff_singleton {α : Type u_3} {β : Type u_4} [CommMonoid β] [DecidableEq α] {s : Finset α} {i : α} (h : i s) (f : αβ) :
                                  xs, f x = f i * xs \ {i}, f x
                                  theorem Finset.sum_eq_sum_diff_singleton_add {α : Type u_3} {β : Type u_4} [AddCommMonoid β] [DecidableEq α] {s : Finset α} {i : α} (h : i s) (f : αβ) :
                                  xs, f x = xs \ {i}, f x + f i
                                  theorem Finset.prod_eq_prod_diff_singleton_mul {α : Type u_3} {β : Type u_4} [CommMonoid β] [DecidableEq α] {s : Finset α} {i : α} (h : i s) (f : αβ) :
                                  xs, f x = (∏ xs \ {i}, f x) * f i
                                  theorem Fintype.sum_eq_add_sum_compl {α : Type u_3} {β : Type u_4} [AddCommMonoid β] [DecidableEq α] [Fintype α] (a : α) (f : αβ) :
                                  i : α, f i = f a + i{a}, f i
                                  theorem Fintype.prod_eq_mul_prod_compl {α : Type u_3} {β : Type u_4} [CommMonoid β] [DecidableEq α] [Fintype α] (a : α) (f : αβ) :
                                  i : α, f i = f a * i{a}, f i
                                  theorem Fintype.sum_eq_sum_compl_add {α : Type u_3} {β : Type u_4} [AddCommMonoid β] [DecidableEq α] [Fintype α] (a : α) (f : αβ) :
                                  i : α, f i = i{a}, f i + f a
                                  theorem Fintype.prod_eq_prod_compl_mul {α : Type u_3} {β : Type u_4} [CommMonoid β] [DecidableEq α] [Fintype α] (a : α) (f : αβ) :
                                  i : α, f i = (∏ i{a}, f i) * f a
                                  theorem Finset.dvd_prod_of_mem {α : Type u_3} {β : Type u_4} [CommMonoid β] (f : αβ) {a : α} {s : Finset α} (ha : a s) :
                                  f a is, f i
                                  theorem Finset.sum_partition {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} [AddCommMonoid β] (R : Setoid α) [DecidableRel Setoid.r] :
                                  xs, f x = xbarFinset.image Quotient.mk'' s, yFinset.filter (fun (x : α) => x = xbar) s, f y

                                  A sum can be partitioned into a sum of sums, each equivalent under a setoid.

                                  theorem Finset.prod_partition {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} [CommMonoid β] (R : Setoid α) [DecidableRel Setoid.r] :
                                  xs, f x = xbarFinset.image Quotient.mk'' s, yFinset.filter (fun (x : α) => x = xbar) s, f y

                                  A product can be partitioned into a product of products, each equivalent under a setoid.

                                  theorem Finset.sum_cancels_of_partition_cancels {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} [AddCommMonoid β] (R : Setoid α) [DecidableRel Setoid.r] (h : xs, aFinset.filter (fun (y : α) => y x) s, f a = 0) :
                                  xs, f x = 0

                                  If we can partition a sum into subsets that cancel out, then the whole sum cancels.

                                  theorem Finset.prod_cancels_of_partition_cancels {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} [CommMonoid β] (R : Setoid α) [DecidableRel Setoid.r] (h : xs, aFinset.filter (fun (y : α) => y x) s, f a = 1) :
                                  xs, f x = 1

                                  If we can partition a product into subsets that cancel out, then the whole product cancels.

                                  theorem Finset.sum_update_of_not_mem {α : Type u_3} {β : Type u_4} [AddCommMonoid β] [DecidableEq α] {s : Finset α} {i : α} (h : is) (f : αβ) (b : β) :
                                  xs, Function.update f i b x = xs, f x
                                  theorem Finset.prod_update_of_not_mem {α : Type u_3} {β : Type u_4} [CommMonoid β] [DecidableEq α] {s : Finset α} {i : α} (h : is) (f : αβ) (b : β) :
                                  xs, Function.update f i b x = xs, f x
                                  theorem Finset.sum_update_of_mem {α : Type u_3} {β : Type u_4} [AddCommMonoid β] [DecidableEq α] {s : Finset α} {i : α} (h : i s) (f : αβ) (b : β) :
                                  xs, Function.update f i b x = b + xs \ {i}, f x
                                  theorem Finset.prod_update_of_mem {α : Type u_3} {β : Type u_4} [CommMonoid β] [DecidableEq α] {s : Finset α} {i : α} (h : i s) (f : αβ) (b : β) :
                                  xs, Function.update f i b x = b * xs \ {i}, f x
                                  theorem Finset.eq_of_card_le_one_of_sum_eq {α : Type u_3} {β : Type u_4} [AddCommMonoid β] {s : Finset α} (hc : s.card 1) {f : αβ} {b : β} (h : xs, f x = b) (x : α) :
                                  x sf x = b

                                  If a sum of a Finset of size at most 1 has a given value, so do the terms in that sum.

                                  theorem Finset.eq_of_card_le_one_of_prod_eq {α : Type u_3} {β : Type u_4} [CommMonoid β] {s : Finset α} (hc : s.card 1) {f : αβ} {b : β} (h : xs, f x = b) (x : α) :
                                  x sf x = b

                                  If a product of a Finset of size at most 1 has a given value, so do the terms in that product.

                                  theorem Finset.add_sum_erase {α : Type u_3} {β : Type u_4} [AddCommMonoid β] [DecidableEq α] (s : Finset α) (f : αβ) {a : α} (h : a s) :
                                  f a + xs.erase a, f x = xs, f x

                                  Taking a sum over s : Finset α is the same as adding the value on a single element f a to the sum over s.erase a.

                                  See Multiset.sum_map_erase for the Multiset version.

                                  theorem Finset.mul_prod_erase {α : Type u_3} {β : Type u_4} [CommMonoid β] [DecidableEq α] (s : Finset α) (f : αβ) {a : α} (h : a s) :
                                  f a * xs.erase a, f x = xs, f x

                                  Taking a product over s : Finset α is the same as multiplying the value on a single element f a by the product of s.erase a.

                                  See Multiset.prod_map_erase for the Multiset version.

                                  theorem Finset.sum_erase_add {α : Type u_3} {β : Type u_4} [AddCommMonoid β] [DecidableEq α] (s : Finset α) (f : αβ) {a : α} (h : a s) :
                                  xs.erase a, f x + f a = xs, f x

                                  A variant of Finset.add_sum_erase with the addition swapped.

                                  theorem Finset.prod_erase_mul {α : Type u_3} {β : Type u_4} [CommMonoid β] [DecidableEq α] (s : Finset α) (f : αβ) {a : α} (h : a s) :
                                  (∏ xs.erase a, f x) * f a = xs, f x

                                  A variant of Finset.mul_prod_erase with the multiplication swapped.

                                  theorem Finset.sum_erase {α : Type u_3} {β : Type u_4} [AddCommMonoid β] [DecidableEq α] (s : Finset α) {f : αβ} {a : α} (h : f a = 0) :
                                  xs.erase a, f x = xs, f x

                                  If a function applied at a point is 0, a sum is unchanged by removing that point, if present, from a Finset.

                                  theorem Finset.prod_erase {α : Type u_3} {β : Type u_4} [CommMonoid β] [DecidableEq α] (s : Finset α) {f : αβ} {a : α} (h : f a = 1) :
                                  xs.erase a, f x = xs, f x

                                  If a function applied at a point is 1, a product is unchanged by removing that point, if present, from a Finset.

                                  theorem Finset.sum_ite_zero {α : Type u_3} {β : Type u_4} [AddCommMonoid β] (s : Finset α) (p : αProp) [DecidablePred p] (h : is, js, p ip ji = j) (a : β) :
                                  (∑ is, if p i then a else 0) = if is, p i then a else 0

                                  See also Finset.sum_boole.

                                  theorem Finset.prod_ite_one {α : Type u_3} {β : Type u_4} [CommMonoid β] (s : Finset α) (p : αProp) [DecidablePred p] (h : is, js, p ip ji = j) (a : β) :
                                  (∏ is, if p i then a else 1) = if is, p i then a else 1

                                  See also Finset.prod_ite_zero.

                                  theorem Finset.sum_erase_lt_of_pos {α : Type u_3} {γ : Type u_6} [DecidableEq α] [AddCommMonoid γ] [Preorder γ] [CovariantClass γ γ (fun (x1 x2 : γ) => x1 + x2) fun (x1 x2 : γ) => x1 < x2] {s : Finset α} {d : α} (hd : d s) {f : αγ} (hdf : 0 < f d) :
                                  ms.erase d, f m < ms, f m
                                  theorem Finset.prod_erase_lt_of_one_lt {α : Type u_3} {γ : Type u_6} [DecidableEq α] [CommMonoid γ] [Preorder γ] [CovariantClass γ γ (fun (x1 x2 : γ) => x1 * x2) fun (x1 x2 : γ) => x1 < x2] {s : Finset α} {d : α} (hd : d s) {f : αγ} (hdf : 1 < f d) :
                                  ms.erase d, f m < ms, f m
                                  theorem Finset.eq_zero_of_sum_eq_zero {α : Type u_3} {β : Type u_4} [AddCommMonoid β] {s : Finset α} {f : αβ} {a : α} (hp : xs, f x = 0) (h1 : xs, x af x = 0) (x : α) :
                                  x sf x = 0

                                  If a sum is 0 and the function is 0 except possibly at one point, it is 0 everywhere on the Finset.

                                  theorem Finset.eq_one_of_prod_eq_one {α : Type u_3} {β : Type u_4} [CommMonoid β] {s : Finset α} {f : αβ} {a : α} (hp : xs, f x = 1) (h1 : xs, x af x = 1) (x : α) :
                                  x sf x = 1

                                  If a product is 1 and the function is 1 except possibly at one point, it is 1 everywhere on the Finset.

                                  theorem Finset.sum_boole_nsmul {α : Type u_3} {β : Type u_4} [AddCommMonoid β] [DecidableEq α] (s : Finset α) (f : αβ) (a : α) :
                                  xs, (if a = x then 1 else 0) f x = if a s then f a else 0
                                  theorem Finset.prod_pow_boole {α : Type u_3} {β : Type u_4} [CommMonoid β] [DecidableEq α] (s : Finset α) (f : αβ) (a : α) :
                                  (∏ xs, f x ^ if a = x then 1 else 0) = if a s then f a else 1
                                  theorem Finset.prod_dvd_prod_of_dvd {α : Type u_3} {β : Type u_4} [CommMonoid β] {S : Finset α} (g1 : αβ) (g2 : αβ) (h : aS, g1 a g2 a) :
                                  S.prod g1 S.prod g2
                                  theorem Finset.prod_dvd_prod_of_subset {ι : Type u_6} {M : Type u_7} [CommMonoid M] (s : Finset ι) (t : Finset ι) (f : ιM) (h : s t) :
                                  is, f i it, f i
                                  theorem Finset.sum_add_eq_sum_add_of_exists {α : Type u_3} {β : Type u_4} [AddCommMonoid β] {s : Finset α} {f : αβ} {b₁ : β} {b₂ : β} (a : α) (ha : a s) (h : f a + b₁ = f a + b₂) :
                                  as, f a + b₁ = as, f a + b₂
                                  theorem Finset.prod_mul_eq_prod_mul_of_exists {α : Type u_3} {β : Type u_4} [CommMonoid β] {s : Finset α} {f : αβ} {b₁ : β} {b₂ : β} (a : α) (ha : a s) (h : f a * b₁ = f a * b₂) :
                                  (∏ as, f a) * b₁ = (∏ as, f a) * b₂
                                  theorem Finset.even_sum {ι : Type u_1} {α : Type u_3} {s : Finset ι} [AddCommMonoid α] (f : ια) (h : cs, Even (f c)) :
                                  Even (∑ is, f i)
                                  theorem Finset.isSquare_prod {ι : Type u_1} {α : Type u_3} {s : Finset ι} [CommMonoid α] (f : ια) (h : cs, IsSquare (f c)) :
                                  IsSquare (∏ is, f i)
                                  theorem Finset.sum_sdiff_eq_sum_sdiff_iff {ι : Type u_1} {α : Type u_3} [DecidableEq ι] [AddCancelCommMonoid α] {s : Finset ι} {t : Finset ι} {f : ια} :
                                  is \ t, f i = it \ s, f i is, f i = it, f i
                                  theorem Finset.prod_sdiff_eq_prod_sdiff_iff {ι : Type u_1} {α : Type u_3} [DecidableEq ι] [CancelCommMonoid α] {s : Finset ι} {t : Finset ι} {f : ια} :
                                  is \ t, f i = it \ s, f i is, f i = it, f i
                                  theorem Finset.sum_sdiff_ne_sum_sdiff_iff {ι : Type u_1} {α : Type u_3} [DecidableEq ι] [AddCancelCommMonoid α] {s : Finset ι} {t : Finset ι} {f : ια} :
                                  is \ t, f i it \ s, f i is, f i it, f i
                                  theorem Finset.prod_sdiff_ne_prod_sdiff_iff {ι : Type u_1} {α : Type u_3} [DecidableEq ι] [CancelCommMonoid α] {s : Finset ι} {t : Finset ι} {f : ια} :
                                  is \ t, f i it \ s, f i is, f i it, f i
                                  theorem Finset.card_eq_sum_ones {α : Type u_3} (s : Finset α) :
                                  s.card = xs, 1
                                  theorem Finset.sum_const_nat {α : Type u_3} {s : Finset α} {m : } {f : α} (h₁ : xs, f x = m) :
                                  xs, f x = s.card * m
                                  theorem Finset.sum_card_fiberwise_eq_card_filter {ι : Type u_1} {κ : Type u_6} [DecidableEq κ] (s : Finset ι) (t : Finset κ) (g : ικ) :
                                  jt, (Finset.filter (fun (i : ι) => g i = j) s).card = (Finset.filter (fun (i : ι) => g i t) s).card
                                  theorem Finset.card_filter {α : Type u_3} (p : αProp) [DecidablePred p] (s : Finset α) :
                                  (Finset.filter p s).card = as, if p a then 1 else 0
                                  @[simp]
                                  theorem Finset.op_sum {α : Type u_3} {β : Type u_4} [AddCommMonoid β] {s : Finset α} (f : αβ) :
                                  MulOpposite.op (∑ xs, f x) = xs, MulOpposite.op (f x)

                                  Moving to the opposite additive commutative monoid commutes with summing.

                                  @[simp]
                                  theorem Finset.unop_sum {α : Type u_3} {β : Type u_4} [AddCommMonoid β] {s : Finset α} (f : αβᵐᵒᵖ) :
                                  MulOpposite.unop (∑ xs, f x) = xs, MulOpposite.unop (f x)
                                  @[simp]
                                  theorem Finset.sum_neg_distrib {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} [SubtractionCommMonoid β] :
                                  xs, -f x = -xs, f x
                                  @[simp]
                                  theorem Finset.prod_inv_distrib {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} [DivisionCommMonoid β] :
                                  xs, (f x)⁻¹ = (∏ xs, f x)⁻¹
                                  @[simp]
                                  theorem Finset.sum_sub_distrib {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} {g : αβ} [SubtractionCommMonoid β] :
                                  xs, (f x - g x) = xs, f x - xs, g x
                                  @[simp]
                                  theorem Finset.prod_div_distrib {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} {g : αβ} [DivisionCommMonoid β] :
                                  xs, f x / g x = (∏ xs, f x) / xs, g x
                                  theorem Finset.sum_zsmul {α : Type u_3} {β : Type u_4} [SubtractionCommMonoid β] (f : αβ) (s : Finset α) (n : ) :
                                  as, n f a = n as, f a
                                  theorem Finset.prod_zpow {α : Type u_3} {β : Type u_4} [DivisionCommMonoid β] (f : αβ) (s : Finset α) (n : ) :
                                  as, f a ^ n = (∏ as, f a) ^ n
                                  @[simp]
                                  theorem Finset.sum_sdiff_eq_sub {α : Type u_3} {β : Type u_4} {s₁ : Finset α} {s₂ : Finset α} {f : αβ} [AddCommGroup β] [DecidableEq α] (h : s₁ s₂) :
                                  xs₂ \ s₁, f x = xs₂, f x - xs₁, f x
                                  @[simp]
                                  theorem Finset.prod_sdiff_eq_div {α : Type u_3} {β : Type u_4} {s₁ : Finset α} {s₂ : Finset α} {f : αβ} [CommGroup β] [DecidableEq α] (h : s₁ s₂) :
                                  xs₂ \ s₁, f x = (∏ xs₂, f x) / xs₁, f x
                                  theorem Finset.sum_sdiff_sub_sum_sdiff {α : Type u_3} {β : Type u_4} {s₁ : Finset α} {s₂ : Finset α} {f : αβ} [AddCommGroup β] [DecidableEq α] :
                                  xs₂ \ s₁, f x - xs₁ \ s₂, f x = xs₂, f x - xs₁, f x
                                  theorem Finset.prod_sdiff_div_prod_sdiff {α : Type u_3} {β : Type u_4} {s₁ : Finset α} {s₂ : Finset α} {f : αβ} [CommGroup β] [DecidableEq α] :
                                  (∏ xs₂ \ s₁, f x) / xs₁ \ s₂, f x = (∏ xs₂, f x) / xs₁, f x
                                  @[simp]
                                  theorem Finset.sum_erase_eq_sub {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} [AddCommGroup β] [DecidableEq α] {a : α} (h : a s) :
                                  xs.erase a, f x = xs, f x - f a
                                  @[simp]
                                  theorem Finset.prod_erase_eq_div {α : Type u_3} {β : Type u_4} {s : Finset α} {f : αβ} [CommGroup β] [DecidableEq α] {a : α} (h : a s) :
                                  xs.erase a, f x = (∏ xs, f x) / f a
                                  @[simp]
                                  theorem Finset.card_sigma {α : Type u_3} {σ : αType u_6} (s : Finset α) (t : (a : α) → Finset (σ a)) :
                                  (s.sigma t).card = as, (t a).card
                                  @[simp]
                                  theorem Finset.card_disjiUnion {α : Type u_3} {β : Type u_4} (s : Finset α) (t : αFinset β) (h : (↑s).PairwiseDisjoint t) :
                                  (s.disjiUnion t h).card = is, (t i).card
                                  theorem Finset.card_biUnion {α : Type u_3} {β : Type u_4} [DecidableEq β] {s : Finset α} {t : αFinset β} (h : xs, ys, x yDisjoint (t x) (t y)) :
                                  (s.biUnion t).card = us, (t u).card
                                  theorem Finset.card_biUnion_le {α : Type u_3} {β : Type u_4} [DecidableEq β] {s : Finset α} {t : αFinset β} :
                                  (s.biUnion t).card as, (t a).card
                                  theorem Finset.card_eq_sum_card_fiberwise {α : Type u_3} {β : Type u_4} [DecidableEq β] {f : αβ} {s : Finset α} {t : Finset β} (H : xs, f x t) :
                                  s.card = at, (Finset.filter (fun (x : α) => f x = a) s).card
                                  theorem Finset.card_eq_sum_card_image {α : Type u_3} {β : Type u_4} [DecidableEq β] (f : αβ) (s : Finset α) :
                                  s.card = aFinset.image f s, (Finset.filter (fun (x : α) => f x = a) s).card
                                  theorem Finset.mem_sum {α : Type u_3} {β : Type u_4} {f : αMultiset β} (s : Finset α) (b : β) :
                                  b xs, f x as, b f a
                                  theorem Finset.sum_unique_nonempty {α : Type u_6} {β : Type u_7} [AddCommMonoid β] [Unique α] (s : Finset α) (f : αβ) (h : s.Nonempty) :
                                  xs, f x = f default
                                  theorem Finset.prod_unique_nonempty {α : Type u_6} {β : Type u_7} [CommMonoid β] [Unique α] (s : Finset α) (f : αβ) (h : s.Nonempty) :
                                  xs, f x = f default
                                  theorem Finset.sum_nat_mod {α : Type u_3} (s : Finset α) (n : ) (f : α) :
                                  (∑ is, f i) % n = (∑ is, f i % n) % n
                                  theorem Finset.prod_nat_mod {α : Type u_3} (s : Finset α) (n : ) (f : α) :
                                  (∏ is, f i) % n = (∏ is, f i % n) % n
                                  theorem Finset.sum_int_mod {α : Type u_3} (s : Finset α) (n : ) (f : α) :
                                  (∑ is, f i) % n = (∑ is, f i % n) % n
                                  theorem Finset.prod_int_mod {α : Type u_3} (s : Finset α) (n : ) (f : α) :
                                  (∏ is, f i) % n = (∏ is, f i % n) % n
                                  theorem Fintype.sum_bijective {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [AddCommMonoid α] (e : ικ) (he : Function.Bijective e) (f : ια) (g : κα) (h : ∀ (x : ι), f x = g (e x)) :
                                  x : ι, f x = x : κ, g x

                                  Fintype.sum_bijective is a variant of Finset.sum_bij that accepts Function.Bijective.

                                  See Function.Bijective.sum_comp for a version without h.

                                  theorem Fintype.prod_bijective {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [CommMonoid α] (e : ικ) (he : Function.Bijective e) (f : ια) (g : κα) (h : ∀ (x : ι), f x = g (e x)) :
                                  x : ι, f x = x : κ, g x

                                  Fintype.prod_bijective is a variant of Finset.prod_bij that accepts Function.Bijective.

                                  See Function.Bijective.prod_comp for a version without h.

                                  theorem Function.Bijective.finset_prod {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [CommMonoid α] (e : ικ) (he : Function.Bijective e) (f : ια) (g : κα) (h : ∀ (x : ι), f x = g (e x)) :
                                  x : ι, f x = x : κ, g x

                                  Alias of Fintype.prod_bijective.


                                  Fintype.prod_bijective is a variant of Finset.prod_bij that accepts Function.Bijective.

                                  See Function.Bijective.prod_comp for a version without h.

                                  theorem Function.Bijective.finset_sum {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [AddCommMonoid α] (e : ικ) (he : Function.Bijective e) (f : ια) (g : κα) (h : ∀ (x : ι), f x = g (e x)) :
                                  x : ι, f x = x : κ, g x
                                  theorem Fintype.sum_equiv {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [AddCommMonoid α] (e : ι κ) (f : ια) (g : κα) (h : ∀ (x : ι), f x = g (e x)) :
                                  x : ι, f x = x : κ, g x

                                  Fintype.sum_equiv is a specialization of Finset.sum_bij that automatically fills in most arguments.

                                  See Equiv.sum_comp for a version without h.

                                  theorem Fintype.prod_equiv {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [CommMonoid α] (e : ι κ) (f : ια) (g : κα) (h : ∀ (x : ι), f x = g (e x)) :
                                  x : ι, f x = x : κ, g x

                                  Fintype.prod_equiv is a specialization of Finset.prod_bij that automatically fills in most arguments.

                                  See Equiv.prod_comp for a version without h.

                                  theorem Function.Bijective.sum_comp {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [AddCommMonoid α] {e : ικ} (he : Function.Bijective e) (g : κα) :
                                  i : ι, g (e i) = i : κ, g i
                                  theorem Function.Bijective.prod_comp {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [CommMonoid α] {e : ικ} (he : Function.Bijective e) (g : κα) :
                                  i : ι, g (e i) = i : κ, g i
                                  theorem Equiv.sum_comp {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [AddCommMonoid α] (e : ι κ) (g : κα) :
                                  i : ι, g (e i) = i : κ, g i
                                  theorem Equiv.prod_comp {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [CommMonoid α] (e : ι κ) (g : κα) :
                                  i : ι, g (e i) = i : κ, g i
                                  theorem Fintype.sum_of_injective {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [AddCommMonoid α] (e : ικ) (he : Function.Injective e) (f : ια) (g : κα) (h' : iSet.range e, g i = 0) (h : ∀ (i : ι), f i = g (e i)) :
                                  i : ι, f i = j : κ, g j
                                  theorem Fintype.prod_of_injective {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [CommMonoid α] (e : ικ) (he : Function.Injective e) (f : ια) (g : κα) (h' : iSet.range e, g i = 1) (h : ∀ (i : ι), f i = g (e i)) :
                                  i : ι, f i = j : κ, g j
                                  theorem Fintype.sum_fiberwise {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [AddCommMonoid α] [DecidableEq κ] (g : ικ) (f : ια) :
                                  j : κ, i : { i : ι // g i = j }, f i = i : ι, f i
                                  theorem Fintype.prod_fiberwise {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [CommMonoid α] [DecidableEq κ] (g : ικ) (f : ια) :
                                  j : κ, i : { i : ι // g i = j }, f i = i : ι, f i
                                  theorem Fintype.sum_fiberwise' {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [AddCommMonoid α] [DecidableEq κ] (g : ικ) (f : κα) :
                                  j : κ, _i : { i : ι // g i = j }, f j = i : ι, f (g i)
                                  theorem Fintype.prod_fiberwise' {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [CommMonoid α] [DecidableEq κ] (g : ικ) (f : κα) :
                                  j : κ, _i : { i : ι // g i = j }, f j = i : ι, f (g i)
                                  theorem Fintype.sum_unique {α : Type u_9} {β : Type u_10} [AddCommMonoid β] [Unique α] [Fintype α] (f : αβ) :
                                  x : α, f x = f default
                                  theorem Fintype.prod_unique {α : Type u_9} {β : Type u_10} [CommMonoid β] [Unique α] [Fintype α] (f : αβ) :
                                  x : α, f x = f default
                                  theorem Fintype.sum_empty {α : Type u_9} {β : Type u_10} [AddCommMonoid β] [IsEmpty α] [Fintype α] (f : αβ) :
                                  x : α, f x = 0
                                  theorem Fintype.prod_empty {α : Type u_9} {β : Type u_10} [CommMonoid β] [IsEmpty α] [Fintype α] (f : αβ) :
                                  x : α, f x = 1
                                  theorem Fintype.sum_subsingleton {α : Type u_9} {β : Type u_10} [AddCommMonoid β] [Subsingleton α] [Fintype α] (f : αβ) (a : α) :
                                  x : α, f x = f a
                                  theorem Fintype.prod_subsingleton {α : Type u_9} {β : Type u_10} [CommMonoid β] [Subsingleton α] [Fintype α] (f : αβ) (a : α) :
                                  x : α, f x = f a
                                  theorem Fintype.sum_subtype_add_sum_subtype {α : Type u_9} {β : Type u_10} [Fintype α] [AddCommMonoid β] (p : αProp) (f : αβ) [DecidablePred p] :
                                  i : { x : α // p x }, f i + i : { x : α // ¬p x }, f i = i : α, f i
                                  theorem Fintype.prod_subtype_mul_prod_subtype {α : Type u_9} {β : Type u_10} [Fintype α] [CommMonoid β] (p : αProp) (f : αβ) [DecidablePred p] :
                                  (∏ i : { x : α // p x }, f i) * i : { x : α // ¬p x }, f i = i : α, f i
                                  theorem Fintype.sum_subset {ι : Type u_6} {α : Type u_8} [Fintype ι] [AddCommMonoid α] {s : Finset ι} {f : ια} (h : ∀ (i : ι), f i 0i s) :
                                  is, f i = i : ι, f i
                                  theorem Fintype.prod_subset {ι : Type u_6} {α : Type u_8} [Fintype ι] [CommMonoid α] {s : Finset ι} {f : ια} (h : ∀ (i : ι), f i 1i s) :
                                  is, f i = i : ι, f i
                                  theorem Fintype.sum_ite_eq_ite_exists {ι : Type u_6} {α : Type u_8} [Fintype ι] [AddCommMonoid α] (p : ιProp) [DecidablePred p] (h : ∀ (i j : ι), p ip ji = j) (a : α) :
                                  (∑ i : ι, if p i then a else 0) = if ∃ (i : ι), p i then a else 0
                                  theorem Fintype.prod_ite_eq_ite_exists {ι : Type u_6} {α : Type u_8} [Fintype ι] [CommMonoid α] (p : ιProp) [DecidablePred p] (h : ∀ (i j : ι), p ip ji = j) (a : α) :
                                  (∏ i : ι, if p i then a else 1) = if ∃ (i : ι), p i then a else 1
                                  theorem Fintype.sum_dite_eq {ι : Type u_6} {α : Type u_8} [Fintype ι] [AddCommMonoid α] [DecidableEq ι] (i : ι) (f : (j : ι) → i = jα) :
                                  (∑ j : ι, if h : i = j then f j h else 0) = f i

                                  See also Finset.sum_dite_eq.

                                  theorem Fintype.prod_dite_eq {ι : Type u_6} {α : Type u_8} [Fintype ι] [CommMonoid α] [DecidableEq ι] (i : ι) (f : (j : ι) → i = jα) :
                                  (∏ j : ι, if h : i = j then f j h else 1) = f i

                                  See also Finset.prod_dite_eq.

                                  theorem Fintype.sum_dite_eq' {ι : Type u_6} {α : Type u_8} [Fintype ι] [AddCommMonoid α] [DecidableEq ι] (i : ι) (f : (j : ι) → j = iα) :
                                  (∑ j : ι, if h : j = i then f j h else 0) = f i

                                  See also Finset.sum_dite_eq'.

                                  theorem Fintype.prod_dite_eq' {ι : Type u_6} {α : Type u_8} [Fintype ι] [CommMonoid α] [DecidableEq ι] (i : ι) (f : (j : ι) → j = iα) :
                                  (∏ j : ι, if h : j = i then f j h else 1) = f i

                                  See also Finset.prod_dite_eq'.

                                  theorem Fintype.sum_ite_eq {ι : Type u_6} {α : Type u_8} [Fintype ι] [AddCommMonoid α] [DecidableEq ι] (i : ι) (f : ια) :
                                  (∑ j : ι, if i = j then f j else 0) = f i

                                  See also Finset.sum_ite_eq.

                                  theorem Fintype.prod_ite_eq {ι : Type u_6} {α : Type u_8} [Fintype ι] [CommMonoid α] [DecidableEq ι] (i : ι) (f : ια) :
                                  (∏ j : ι, if i = j then f j else 1) = f i

                                  See also Finset.prod_ite_eq.

                                  theorem Fintype.sum_ite_eq' {ι : Type u_6} {α : Type u_8} [Fintype ι] [AddCommMonoid α] [DecidableEq ι] (i : ι) (f : ια) :
                                  (∑ j : ι, if j = i then f j else 0) = f i

                                  See also Finset.sum_ite_eq'.

                                  theorem Fintype.prod_ite_eq' {ι : Type u_6} {α : Type u_8} [Fintype ι] [CommMonoid α] [DecidableEq ι] (i : ι) (f : ια) :
                                  (∏ j : ι, if j = i then f j else 1) = f i

                                  See also Finset.prod_ite_eq'.

                                  theorem Fintype.sum_pi_single {ι : Type u_6} [Fintype ι] [DecidableEq ι] {α : ιType u_9} [(i : ι) → AddCommMonoid (α i)] (i : ι) (f : (i : ι) → α i) :
                                  j : ι, Pi.single j (f j) i = f i

                                  See also Finset.sum_pi_single.

                                  theorem Fintype.prod_pi_mulSingle {ι : Type u_6} [Fintype ι] [DecidableEq ι] {α : ιType u_9} [(i : ι) → CommMonoid (α i)] (i : ι) (f : (i : ι) → α i) :
                                  j : ι, Pi.mulSingle j (f j) i = f i

                                  See also Finset.prod_pi_mulSingle.

                                  theorem Fintype.sum_pi_single' {ι : Type u_6} {α : Type u_8} [Fintype ι] [AddCommMonoid α] [DecidableEq ι] (i : ι) (a : α) :
                                  j : ι, Pi.single i a j = a

                                  See also Finset.sum_pi_single'.

                                  theorem Fintype.prod_pi_mulSingle' {ι : Type u_6} {α : Type u_8} [Fintype ι] [CommMonoid α] [DecidableEq ι] (i : ι) (a : α) :
                                  j : ι, Pi.mulSingle i a j = a

                                  See also Finset.prod_pi_mulSingle'.

                                  @[simp]
                                  theorem Finset.sum_attach_univ {ι : Type u_1} {α : Type u_3} [AddCommMonoid α] [Fintype ι] (f : { i : ι // i Finset.univ }α) :
                                  iFinset.univ.attach, f i = i : ι, f i,
                                  @[simp]
                                  theorem Finset.prod_attach_univ {ι : Type u_1} {α : Type u_3} [CommMonoid α] [Fintype ι] (f : { i : ι // i Finset.univ }α) :
                                  iFinset.univ.attach, f i = i : ι, f i,
                                  theorem Finset.sum_erase_attach {ι : Type u_1} {α : Type u_3} [AddCommMonoid α] [DecidableEq ι] {s : Finset ι} (f : ια) (i : { x : ι // x s }) :
                                  js.attach.erase i, f j = js.erase i, f j
                                  theorem Finset.prod_erase_attach {ι : Type u_1} {α : Type u_3} [CommMonoid α] [DecidableEq ι] {s : Finset ι} (f : ια) (i : { x : ι // x s }) :
                                  js.attach.erase i, f j = js.erase i, f j
                                  theorem List.sum_toFinset {α : Type u_3} {M : Type u_6} [DecidableEq α] [AddCommMonoid M] (f : αM) {l : List α} (_hl : l.Nodup) :
                                  l.toFinset.sum f = (List.map f l).sum
                                  theorem List.prod_toFinset {α : Type u_3} {M : Type u_6} [DecidableEq α] [CommMonoid M] (f : αM) {l : List α} (_hl : l.Nodup) :
                                  l.toFinset.prod f = (List.map f l).prod
                                  @[simp]
                                  theorem List.sum_toFinset_count_eq_length {α : Type u_3} [DecidableEq α] (l : List α) :
                                  al.toFinset, List.count a l = l.length
                                  theorem Multiset.disjoint_list_sum_left {α : Type u_3} {a : Multiset α} {l : List (Multiset α)} :
                                  l.sum.Disjoint a bl, b.Disjoint a
                                  theorem Multiset.disjoint_list_sum_right {α : Type u_3} {a : Multiset α} {l : List (Multiset α)} :
                                  a.Disjoint l.sum bl, a.Disjoint b
                                  theorem Multiset.disjoint_sum_left {α : Type u_3} {a : Multiset α} {i : Multiset (Multiset α)} :
                                  i.sum.Disjoint a bi, b.Disjoint a
                                  theorem Multiset.disjoint_sum_right {α : Type u_3} {a : Multiset α} {i : Multiset (Multiset α)} :
                                  a.Disjoint i.sum bi, a.Disjoint b
                                  theorem Multiset.disjoint_finset_sum_left {α : Type u_3} {β : Type u_6} {i : Finset β} {f : βMultiset α} {a : Multiset α} :
                                  (i.sum f).Disjoint a bi, (f b).Disjoint a
                                  theorem Multiset.disjoint_finset_sum_right {α : Type u_3} {β : Type u_6} {i : Finset β} {f : βMultiset α} {a : Multiset α} :
                                  a.Disjoint (i.sum f) bi, a.Disjoint (f b)
                                  @[simp]
                                  theorem Multiset.mem_sum {ι : Type u_1} {α : Type u_3} {a : α} {s : Finset ι} {m : ιMultiset α} :
                                  a is, m i is, a m i
                                  theorem Multiset.toFinset_sum_count_eq {α : Type u_3} [DecidableEq α] (s : Multiset α) :
                                  as.toFinset, Multiset.count a s = Multiset.card s
                                  @[simp]
                                  theorem Multiset.sum_count_eq_card {α : Type u_3} [DecidableEq α] {s : Finset α} {m : Multiset α} (hms : am, a s) :
                                  as, Multiset.count a m = Multiset.card m
                                  @[deprecated Multiset.sum_count_eq_card]
                                  theorem Multiset.sum_count_eq {α : Type u_3} [DecidableEq α] [Fintype α] (s : Multiset α) :
                                  a : α, Multiset.count a s = Multiset.card s
                                  theorem Multiset.count_sum' {α : Type u_3} {β : Type u_4} [DecidableEq α] {s : Finset β} {a : α} {f : βMultiset α} :
                                  Multiset.count a (∑ xs, f x) = xs, Multiset.count a (f x)
                                  @[simp]
                                  theorem Multiset.toFinset_sum_count_nsmul_eq {α : Type u_3} [DecidableEq α] (s : Multiset α) :
                                  as.toFinset, Multiset.count a s {a} = s
                                  theorem Multiset.exists_smul_of_dvd_count {α : Type u_3} [DecidableEq α] (s : Multiset α) {k : } (h : as, k Multiset.count a s) :
                                  ∃ (u : Multiset α), s = k u
                                  theorem Multiset.toFinset_prod_dvd_prod {α : Type u_3} [DecidableEq α] [CommMonoid α] (S : Multiset α) :
                                  S.toFinset.prod id S.prod
                                  theorem Multiset.sum_sum {α : Type u_6} {ι : Type u_7} [AddCommMonoid α] (f : ιMultiset α) (s : Finset ι) :
                                  (∑ xs, f x).sum = xs, (f x).sum
                                  theorem Multiset.prod_sum {α : Type u_6} {ι : Type u_7} [CommMonoid α] (f : ιMultiset α) (s : Finset ι) :
                                  (∑ xs, f x).prod = xs, (f x).prod
                                  @[simp]
                                  theorem Units.coe_prod {α : Type u_3} {M : Type u_6} [CommMonoid M] (f : αMˣ) (s : Finset α) :
                                  (∏ is, f i) = is, (f i)
                                  theorem nat_abs_sum_le {ι : Type u_6} (s : Finset ι) (f : ι) :
                                  (∑ is, f i).natAbs is, (f i).natAbs

                                  Additive, Multiplicative #

                                  @[simp]
                                  theorem ofMul_list_prod {α : Type u_3} [Monoid α] (s : List α) :
                                  Additive.ofMul s.prod = (List.map (⇑Additive.ofMul) s).sum
                                  @[simp]
                                  theorem toMul_list_sum {α : Type u_3} [Monoid α] (s : List (Additive α)) :
                                  Additive.toMul s.sum = (List.map (⇑Additive.toMul) s).prod
                                  @[simp]
                                  theorem ofAdd_list_prod {α : Type u_3} [AddMonoid α] (s : List α) :
                                  Multiplicative.ofAdd s.sum = (List.map (⇑Multiplicative.ofAdd) s).prod
                                  @[simp]
                                  theorem toAdd_list_sum {α : Type u_3} [AddMonoid α] (s : List (Multiplicative α)) :
                                  Multiplicative.toAdd s.prod = (List.map (⇑Multiplicative.toAdd) s).sum
                                  @[simp]
                                  theorem ofMul_multiset_prod {α : Type u_3} [CommMonoid α] (s : Multiset α) :
                                  Additive.ofMul s.prod = (Multiset.map (⇑Additive.ofMul) s).sum
                                  @[simp]
                                  theorem toMul_multiset_sum {α : Type u_3} [CommMonoid α] (s : Multiset (Additive α)) :
                                  Additive.toMul s.sum = (Multiset.map (⇑Additive.toMul) s).prod
                                  @[simp]
                                  theorem ofMul_prod {ι : Type u_1} {α : Type u_3} [CommMonoid α] (s : Finset ι) (f : ια) :
                                  Additive.ofMul (∏ is, f i) = is, Additive.ofMul (f i)
                                  @[simp]
                                  theorem toMul_sum {ι : Type u_1} {α : Type u_3} [CommMonoid α] (s : Finset ι) (f : ιAdditive α) :
                                  Additive.toMul (∑ is, f i) = is, Additive.toMul (f i)
                                  @[simp]
                                  theorem ofAdd_multiset_prod {α : Type u_3} [AddCommMonoid α] (s : Multiset α) :
                                  Multiplicative.ofAdd s.sum = (Multiset.map (⇑Multiplicative.ofAdd) s).prod
                                  @[simp]
                                  theorem toAdd_multiset_sum {α : Type u_3} [AddCommMonoid α] (s : Multiset (Multiplicative α)) :
                                  Multiplicative.toAdd s.prod = (Multiset.map (⇑Multiplicative.toAdd) s).sum
                                  @[simp]
                                  theorem ofAdd_sum {ι : Type u_1} {α : Type u_3} [AddCommMonoid α] (s : Finset ι) (f : ια) :
                                  Multiplicative.ofAdd (∑ is, f i) = is, Multiplicative.ofAdd (f i)
                                  @[simp]
                                  theorem toAdd_prod {ι : Type u_1} {α : Type u_3} [AddCommMonoid α] (s : Finset ι) (f : ιMultiplicative α) :
                                  Multiplicative.toAdd (∏ is, f i) = is, Multiplicative.toAdd (f i)
                                  @[deprecated Fintype.prod_equiv]
                                  theorem Equiv.prod_comp' {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [CommMonoid α] (e : ι κ) (f : ια) (g : κα) (h : ∀ (x : ι), f x = g (e x)) :
                                  x : ι, f x = x : κ, g x

                                  Alias of Fintype.prod_equiv.


                                  Fintype.prod_equiv is a specialization of Finset.prod_bij that automatically fills in most arguments.

                                  See Equiv.prod_comp for a version without h.

                                  @[deprecated Fintype.sum_equiv]
                                  theorem Equiv.sum_comp' {ι : Type u_6} {κ : Type u_7} {α : Type u_8} [Fintype ι] [Fintype κ] [AddCommMonoid α] (e : ι κ) (f : ια) (g : κα) (h : ∀ (x : ι), f x = g (e x)) :
                                  x : ι, f x = x : κ, g x

                                  Alias of Fintype.sum_equiv.


                                  Fintype.sum_equiv is a specialization of Finset.sum_bij that automatically fills in most arguments.

                                  See Equiv.sum_comp for a version without h.

                                  theorem Finset.sum_sym2_filter_not_isDiag {ι : Type u_6} {α : Type u_7} [LinearOrder ι] [AddCommMonoid α] (s : Finset ι) (p : Sym2 ια) :
                                  iFinset.filter (fun (x : Sym2 ι) => ¬x.IsDiag) s.sym2, p i = iFinset.filter (fun (i : ι × ι) => i.1 < i.2) s.offDiag, p s(i.1, i.2)