Bind operation for multisets

This file defines a few basic operations on Multiset, notably the monadic bind.

Main declarations

• Multiset.join: The join, aka union or sum, of multisets.
• Multiset.bind: The bind of a multiset-indexed family of multisets.
• Multiset.product: Cartesian product of two multisets.
• Multiset.sigma: Disjoint sum of multisets in a sigma type.

Join

def Multiset.join {α : Type u_1} : Multiset (Multiset α) → Multiset α

join S, where S is a multiset of multisets, is the lift of the list join operation, that is, the union of all the sets. For example, join {{1, 2}, {1, 2}, {0, 1}} = {0, 1, 1, 1, 2, 2}.

Equations

• Multiset.join = Multiset.sum

theorem Multiset.coe_join {α : Type u_1} (L : List (List α)) : (↑(List.map ofList L)).join = ↑L.flatten

@[simp]

theorem Multiset.join_zero {α : Type u_1} : join 0 = 0

@[simp]

theorem Multiset.join_cons {α : Type u_1} (s : Multiset α) (S : Multiset (Multiset α)) : (s ::ₘ S).join = s + S.join

@[simp]

theorem Multiset.join_add {α : Type u_1} (S T : Multiset (Multiset α)) : (S + T).join = S.join + T.join

@[simp]

theorem Multiset.singleton_join {α : Type u_1} (a : Multiset α) : {a}.join = a

@[simp]

theorem Multiset.mem_join {α : Type u_1} {a : α} {S : Multiset (Multiset α)} : a ∈ S.join ↔ ∃ (s : Multiset α), s ∈ S ∧ a ∈ s

@[simp]

theorem Multiset.card_join {α : Type u_1} (S : Multiset (Multiset α)) : S.join.card = (map card S).sum

@[simp]

theorem Multiset.map_join {α : Type u_1} {β : Type v} (f : α → β) (S : Multiset (Multiset α)) : map f S.join = (map (map f) S).join

@[simp]

theorem Multiset.prod_join {α : Type u_1} [CommMonoid α] {S : Multiset (Multiset α)} : S.join.prod = (map prod S).prod

@[simp]

theorem Multiset.sum_join {α : Type u_1} [AddCommMonoid α] {S : Multiset (Multiset α)} : S.join.sum = (map sum S).sum

theorem Multiset.rel_join {α : Type u_1} {β : Type v} {r : α → β → Prop} {s : Multiset (Multiset α)} {t : Multiset (Multiset β)} (h : Rel (Rel r) s t) : Rel r s.join t.join

theorem Multiset.filter_join {α : Type u_1} (S : Multiset (Multiset α)) (p : α → Prop) [DecidablePred p] : filter p S.join = (map (filter p) S).join

theorem Multiset.filterMap_join {α : Type u_1} {β : Type v} (S : Multiset (Multiset α)) (f : α → Option β) : filterMap f S.join = (map (filterMap f) S).join

Bind

def Multiset.bind {α : Type u_1} {β : Type v} (s : Multiset α) (f : α → Multiset β) : Multiset β

s.bind f is the monad bind operation, defined as (s.map f).join. It is the union of f a as a ranges over s.

Equations

• s.bind f = (Multiset.map f s).join

@[simp]

theorem Multiset.coe_bind {α : Type u_1} {β : Type v} (l : List α) (f : α → List β) : ((↑l).bind fun (a : α) => ↑(f a)) = ↑(List.flatMap f l)

@[simp]

theorem Multiset.zero_bind {α : Type u_1} {β : Type v} (f : α → Multiset β) : bind 0 f = 0

@[simp]

theorem Multiset.cons_bind {α : Type u_1} {β : Type v} (a : α) (s : Multiset α) (f : α → Multiset β) : (a ::ₘ s).bind f = f a + s.bind f

@[simp]

theorem Multiset.singleton_bind {α : Type u_1} {β : Type v} (a : α) (f : α → Multiset β) : {a}.bind f = f a

@[simp]

theorem Multiset.add_bind {α : Type u_1} {β : Type v} (s t : Multiset α) (f : α → Multiset β) : (s + t).bind f = s.bind f + t.bind f

@[simp]

theorem Multiset.bind_zero {α : Type u_1} {β : Type v} (s : Multiset α) : (s.bind fun (x : α) => 0) = 0

@[simp]

theorem Multiset.bind_add {α : Type u_1} {β : Type v} (s : Multiset α) (f g : α → Multiset β) : (s.bind fun (a : α) => f a + g a) = s.bind f + s.bind g

@[simp]

theorem Multiset.bind_cons {α : Type u_1} {β : Type v} (s : Multiset α) (f : α → β) (g : α → Multiset β) : (s.bind fun (a : α) => f a ::ₘ g a) = map f s + s.bind g

@[simp]

theorem Multiset.bind_singleton {α : Type u_1} {β : Type v} (s : Multiset α) (f : α → β) : (s.bind fun (x : α) => {f x}) = map f s

@[simp]

theorem Multiset.mem_bind {α : Type u_1} {β : Type v} {b : β} {s : Multiset α} {f : α → Multiset β} : b ∈ s.bind f ↔ ∃ (a : α), a ∈ s ∧ b ∈ f a

@[simp]

theorem Multiset.card_bind {α : Type u_1} {β : Type v} (s : Multiset α) (f : α → Multiset β) : (s.bind f).card = (map (card ∘ f) s).sum

theorem Multiset.bind_congr {α : Type u_1} {β : Type v} {f g : α → Multiset β} {m : Multiset α} : (∀ (a : α), a ∈ m → f a = g a) → m.bind f = m.bind g

theorem Multiset.bind_hcongr {α : Type u_1} {β β' : Type v} {m : Multiset α} {f : α → Multiset β} {f' : α → Multiset β'} (h : β = β') (hf : ∀ (a : α), a ∈ m → f a ≍ f' a) : m.bind f ≍ m.bind f'

theorem Multiset.map_bind {α : Type u_1} {β : Type v} {γ : Type u_2} (m : Multiset α) (n : α → Multiset β) (f : β → γ) : map f (m.bind n) = m.bind fun (a : α) => map f (n a)

theorem Multiset.bind_map {α : Type u_1} {β : Type v} {γ : Type u_2} (m : Multiset α) (n : β → Multiset γ) (f : α → β) : (map f m).bind n = m.bind fun (a : α) => n (f a)

theorem Multiset.bind_assoc {α : Type u_1} {β : Type v} {γ : Type u_2} {s : Multiset α} {f : α → Multiset β} {g : β → Multiset γ} : (s.bind f).bind g = s.bind fun (a : α) => (f a).bind g

theorem Multiset.bind_bind {α : Type u_1} {β : Type v} {γ : Type u_2} (m : Multiset α) (n : Multiset β) {f : α → β → Multiset γ} : (m.bind fun (a : α) => n.bind fun (b : β) => f a b) = n.bind fun (b : β) => m.bind fun (a : α) => f a b

theorem Multiset.bind_map_comm {α : Type u_1} {β : Type v} {γ : Type u_2} (m : Multiset α) (n : Multiset β) {f : α → β → γ} : (m.bind fun (a : α) => map (fun (b : β) => f a b) n) = n.bind fun (b : β) => map (fun (a : α) => f a b) m

theorem Multiset.filter_eq_bind {α : Type u_1} (m : Multiset α) (p : α → Prop) [DecidablePred p] : filter p m = m.bind fun (a : α) => if p a then {a} else 0

theorem Multiset.bind_filter {α : Type u_1} {β : Type v} (m : Multiset α) (p : α → Prop) (f : α → Multiset β) [DecidablePred p] : (filter p m).bind f = m.bind fun (a : α) => if p a then f a else 0

theorem Multiset.filter_bind {α : Type u_1} {β : Type v} (m : Multiset α) (f : α → Multiset β) (p : β → Prop) [DecidablePred p] : filter p (m.bind f) = m.bind fun (a : α) => filter p (f a)

theorem Multiset.filterMap_eq_bind {α : Type u_1} {β : Type v} (m : Multiset α) (f : α → Option β) : filterMap f m = m.bind fun (a : α) => (Option.map singleton (f a)).getD 0

theorem Multiset.bind_filterMap {α : Type u_1} {β : Type v} {γ : Type u_2} (m : Multiset α) (f : α → Option β) (g : β → Multiset γ) : (filterMap f m).bind g = m.bind fun (a : α) => (Option.map g (f a)).getD 0

theorem Multiset.filterMap_bind {α : Type u_1} {β : Type v} {γ : Type u_2} (m : Multiset α) (f : α → Multiset β) (g : β → Option γ) : filterMap g (m.bind f) = m.bind fun (a : α) => filterMap g (f a)

@[simp]

theorem Multiset.prod_bind {α : Type u_1} {β : Type v} [CommMonoid β] (s : Multiset α) (t : α → Multiset β) : (s.bind t).prod = (map (fun (a : α) => (t a).prod) s).prod

@[simp]

theorem Multiset.sum_bind {α : Type u_1} {β : Type v} [AddCommMonoid β] (s : Multiset α) (t : α → Multiset β) : (s.bind t).sum = (map (fun (a : α) => (t a).sum) s).sum

theorem Multiset.rel_bind {α : Type u_1} {β : Type v} {γ : Type u_2} {δ : Type u_3} {r : α → β → Prop} {p : γ → δ → Prop} {s : Multiset α} {t : Multiset β} {f : α → Multiset γ} {g : β → Multiset δ} (h : Relator.LiftFun r (Rel p) f g) (hst : Rel r s t) : Rel p (s.bind f) (t.bind g)

theorem Multiset.count_sum {α : Type u_1} {β : Type v} [DecidableEq α] {m : Multiset β} {f : β → Multiset α} {a : α} : count a (map f m).sum = (map (fun (b : β) => count a (f b)) m).sum

theorem Multiset.count_bind {α : Type u_1} {β : Type v} [DecidableEq α] {m : Multiset β} {f : β → Multiset α} {a : α} : count a (m.bind f) = (map (fun (b : β) => count a (f b)) m).sum

theorem Multiset.le_bind {α : Type u_4} {β : Type u_5} {f : α → Multiset β} (S : Multiset α) {x : α} (hx : x ∈ S) : f x ≤ S.bind f

@[simp]

theorem Multiset.attach_bind_coe {α : Type u_1} {β : Type v} (s : Multiset α) (f : α → Multiset β) : (s.attach.bind fun (i : { x : α // x ∈ s }) => f ↑i) = s.bind f

@[simp]

theorem Multiset.nodup_bind {α : Type u_1} {β : Type v} {s : Multiset α} {f : α → Multiset β} : (s.bind f).Nodup ↔ (∀ (a : α), a ∈ s → (f a).Nodup) ∧ Pairwise (Function.onFun Disjoint f) s

@[simp]

theorem Multiset.dedup_bind_dedup {α : Type u_1} {β : Type v} [DecidableEq α] [DecidableEq β] (s : Multiset α) (f : α → Multiset β) : (s.dedup.bind f).dedup = (s.bind f).dedup

theorem Multiset.fold_bind {α : Type u_1} (op : α → α → α) [hc : Std.Commutative op] [ha : Std.Associative op] {ι : Type u_4} (s : Multiset ι) (t : ι → Multiset α) (b : ι → α) (b₀ : α) : fold op (fold op b₀ (map b s)) (s.bind t) = fold op b₀ (map (fun (i : ι) => fold op (b i) (t i)) s)

Product of two multisets

def Multiset.product {α : Type u_1} {β : Type v} (s : Multiset α) (t : Multiset β) : Multiset (α × β)

The multiplicity of (a, b) in s ×ˢ t is the product of the multiplicity of a in s and b in t.

Equations

• s.product t = s.bind fun (a : α) => Multiset.map (Prod.mk a) t

@[instance_reducible]

instance Multiset.instSProd {α : Type u_1} {β : Type v} : SProd (Multiset α) (Multiset β) (Multiset (α × β))

Equations

• Multiset.instSProd = { sprod := Multiset.product }

@[simp]

theorem Multiset.coe_product {α : Type u_1} {β : Type v} (l₁ : List α) (l₂ : List β) : ↑l₁ ×ˢ ↑l₂ = ↑(l₁ ×ˢ l₂)

@[simp]

theorem Multiset.zero_product {α : Type u_1} {β : Type v} (t : Multiset β) : 0 ×ˢ t = 0

@[simp]

theorem Multiset.cons_product {α : Type u_1} {β : Type v} (a : α) (s : Multiset α) (t : Multiset β) : (a ::ₘ s) ×ˢ t = map (Prod.mk a) t + s ×ˢ t

@[simp]

theorem Multiset.product_zero {α : Type u_1} {β : Type v} (s : Multiset α) : s ×ˢ 0 = 0

@[simp]

theorem Multiset.product_cons {α : Type u_1} {β : Type v} (b : β) (s : Multiset α) (t : Multiset β) : s ×ˢ (b ::ₘ t) = map (fun (a : α) => (a, b)) s + s ×ˢ t

@[simp]

theorem Multiset.product_singleton {α : Type u_1} {β : Type v} (a : α) (b : β) : {a} ×ˢ {b} = {(a, b)}

@[simp]

theorem Multiset.add_product {α : Type u_1} {β : Type v} (s t : Multiset α) (u : Multiset β) : (s + t) ×ˢ u = s ×ˢ u + t ×ˢ u

@[simp]

theorem Multiset.product_add {α : Type u_1} {β : Type v} (s : Multiset α) (t u : Multiset β) : s ×ˢ (t + u) = s ×ˢ t + s ×ˢ u

@[simp]

theorem Multiset.card_product {α : Type u_1} {β : Type v} (s : Multiset α) (t : Multiset β) : (s ×ˢ t).card = s.card * t.card

@[simp]

theorem Multiset.mem_product {α : Type u_1} {β : Type v} {s : Multiset α} {t : Multiset β} {p : α × β} : p ∈ s ×ˢ t ↔ p.1 ∈ s ∧ p.2 ∈ t

theorem Multiset.Nodup.product {α : Type u_1} {β : Type v} {s : Multiset α} {t : Multiset β} : s.Nodup → t.Nodup → (s ×ˢ t).Nodup

@[simp]

theorem Multiset.map_swap_product {α : Type u_1} {β : Type v} (s : Multiset α) (t : Multiset β) : map Prod.swap (s ×ˢ t) = t ×ˢ s

theorem Multiset.prod_map_product_eq_prod_prod {α : Type u_1} {β : Type v} {M : Type u_4} [CommMonoid M] (s : Multiset α) (t : Multiset β) (f : α × β → M) : (map f (s ×ˢ t)).prod = (map (fun (i : α) => (map (fun (j : β) => f (i, j)) t).prod) s).prod

Disjoint sum of multisets

def Multiset.sigma {α : Type u_1} {σ : α → Type u_4} (s : Multiset α) (t : (a : α) → Multiset (σ a)) : Multiset ((a : α) × σ a)

Multiset.sigma s t is the dependent version of Multiset.product. It is the sum of (a, b) as a ranges over s and b ranges over t a.

Equations

• s.sigma t = s.bind fun (a : α) => Multiset.map (Sigma.mk a) (t a)

@[simp]

theorem Multiset.coe_sigma {α : Type u_1} {σ : α → Type u_4} (l₁ : List α) (l₂ : (a : α) → List (σ a)) : ((↑l₁).sigma fun (a : α) => ↑(l₂ a)) = ↑(l₁.sigma l₂)

@[simp]

theorem Multiset.zero_sigma {α : Type u_1} {σ : α → Type u_4} (t : (a : α) → Multiset (σ a)) : Multiset.sigma 0 t = 0

@[simp]

theorem Multiset.cons_sigma {α : Type u_1} {σ : α → Type u_4} (a : α) (s : Multiset α) (t : (a : α) → Multiset (σ a)) : (a ::ₘ s).sigma t = map (Sigma.mk a) (t a) + s.sigma t

@[simp]

theorem Multiset.sigma_singleton {α : Type u_1} {β : Type v} (a : α) (b : α → β) : ({a}.sigma fun (a : α) => {b a}) = {⟨a, b a⟩}

@[simp]

theorem Multiset.add_sigma {α : Type u_1} {σ : α → Type u_4} (s t : Multiset α) (u : (a : α) → Multiset (σ a)) : (s + t).sigma u = s.sigma u + t.sigma u

@[simp]

theorem Multiset.sigma_add {α : Type u_1} {σ : α → Type u_4} (s : Multiset α) (t u : (a : α) → Multiset (σ a)) : (s.sigma fun (a : α) => t a + u a) = s.sigma t + s.sigma u

@[simp]

theorem Multiset.card_sigma {α : Type u_1} {σ : α → Type u_4} (s : Multiset α) (t : (a : α) → Multiset (σ a)) : (s.sigma t).card = (map (fun (a : α) => (t a).card) s).sum

@[simp]

theorem Multiset.mem_sigma {α : Type u_1} {σ : α → Type u_4} {s : Multiset α} {t : (a : α) → Multiset (σ a)} {p : (a : α) × σ a} : p ∈ s.sigma t ↔ p.fst ∈ s ∧ p.snd ∈ t p.fst

theorem Multiset.Nodup.sigma {α : Type u_1} {s : Multiset α} {σ : α → Type u_5} {t : (a : α) → Multiset (σ a)} : s.Nodup → (∀ (a : α), (t a).Nodup) → (s.sigma t).Nodup