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.

Equations
Instances For
    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.

    Equations
    Instances For
      @[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.

      Equations
      • One or more equations did not get rendered due to their size.
      Instances For

        A BigOperator binder in parentheses

        Equations
        • One or more equations did not get rendered due to their size.
        Instances For

          A list of parenthesized binders

          Equations
          • One or more equations did not get rendered due to their size.
          Instances For

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

            Equations
            • One or more equations did not get rendered due to their size.
            Instances For
              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.

              Equations
              • One or more equations did not get rendered due to their size.
              Instances For
                def BigOperators.processBigOpBinders (binders : Lean.TSyntax `BigOperators.bigOpBinders) :

                Collects the binder/Finset pairs for the given bigOpBinders.

                Equations
                • One or more equations did not get rendered due to their size.
                Instances For

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

                  Equations
                  • One or more equations did not get rendered due to their size.
                  Instances For

                    Collect the terms into a product of sets.

                    Equations
                    • One or more equations did not get rendered due to their size.
                    Instances For
                      • ∑ 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

                      Equations
                      • One or more equations did not get rendered due to their size.
                      Instances For
                        • ∏ 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

                        Equations
                        • One or more equations did not get rendered due to their size.
                        Instances For

                          (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.

                          Equations
                          • One or more equations did not get rendered due to their size.
                          Instances For

                            (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.

                            Equations
                            • One or more equations did not get rendered due to their size.
                            Instances For

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

                              Equations
                              • One or more equations did not get rendered due to their size.
                              Instances For

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

                                Equations
                                • One or more equations did not get rendered due to their size.
                                Instances For
                                  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 (x x_1 : β) => x + x_1) 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 (x x_1 : β) => x * x_1) 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
                                  abbrev Finset.sum_image'.match_1 {α : Type u_2} {γ : Type u_1} {s : Finset γ} {g : γα} (_x : α) (motive : (as, g a = _x)Prop) :
                                  ∀ (x : as, g a = _x), (∀ (c : γ) (hcs : c s) (hc : g c = _x), motive )motive x
                                  Equations
                                  • =
                                  Instances For
                                    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.

                                    abbrev Finset.sum_comm'.match_1 {α : Type u_2} {γ : Type u_1} (motive : γ × αProp) :
                                    ∀ (x : γ × α), (∀ (x : γ) (y : α), motive (x, y))motive x
                                    Equations
                                    • =
                                    Instances For
                                      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.

                                      abbrev Finset.sum_congr_set.match_1 {β : Type u_1} [Fintype β] (s : Set β) [DecidablePred fun (x : β) => x s] (motive : (x : { x : β // x s }) → x Finset.univProp) :
                                      ∀ (x : { x : β // x s }) (x_1 : x Finset.univ), (∀ (x : β) (h : x s) (x_2 : x, h Finset.univ), motive x, h x_2)motive x x_1
                                      Equations
                                      • =
                                      Instances For
                                        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
                                        abbrev Finset.sum_range_succ'.match_1 (motive : Prop) :
                                        ∀ (x : ), (Unitmotive 0)(∀ (n : ), motive n.succ)motive x
                                        Equations
                                        • =
                                        Instances For
                                          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} [CanonicallyOrderedAddCommMonoid α] [Sub α] [OrderedSub α] [ContravariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] {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 =