The cartesian product of multisets

def Multiset.Pi.empty {α : Type u_1} (δ : α → Sort u_2) (a : α) : a ∈ 0 → δ a

Given δ : α → Type*, Pi.empty δ is the trivial dependent function out of the empty multiset.

Equations

• Multiset.Pi.empty δ a✝ a = nomatch a✝, a

def Multiset.Pi.cons {α : Type u_1} [DecidableEq α] {δ : α → Sort v} (m : Multiset α) (a : α) (b : δ a) (f : (a : α) → a ∈ m → δ a) (a' : α) : a' ∈ a ::ₘ m → δ a'

Given δ : α → Type*, a multiset m and a term a, as well as a term b : δ a and a function f such that f a' : δ a' for all a' in m, Pi.cons m a b f is a function g such that g a'' : δ a'' for all a'' in a ::ₘ m.

Equations

• Multiset.Pi.cons m a b f a' ha' = if h : a' = a then ⋯ ▸ b else f a' ⋯

theorem Multiset.Pi.cons_same {α : Type u_1} [DecidableEq α] {δ : α → Sort v} {m : Multiset α} {a : α} {b : δ a} {f : (a : α) → a ∈ m → δ a} (h : a ∈ a ::ₘ m) : Multiset.Pi.cons m a b f a h = b

theorem Multiset.Pi.cons_ne {α : Type u_1} [DecidableEq α] {δ : α → Sort v} {m : Multiset α} {a : α} {a' : α} {b : δ a} {f : (a : α) → a ∈ m → δ a} (h' : a' ∈ a ::ₘ m) (h : a' ≠ a) : Multiset.Pi.cons m a b f a' h' = f a' ⋯

theorem Multiset.Pi.cons_swap {α : Type u_1} [DecidableEq α] {δ : α → Sort v} {a : α} {a' : α} {b : δ a} {b' : δ a'} {m : Multiset α} {f : (a : α) → a ∈ m → δ a} (h : a ≠ a') : HEq (Multiset.Pi.cons (a' ::ₘ m) a b (Multiset.Pi.cons m a' b' f)) (Multiset.Pi.cons (a ::ₘ m) a' b' (Multiset.Pi.cons m a b f))

@[simp]

theorem Multiset.pi.cons_eta {α : Type u_1} [DecidableEq α] {δ : α → Sort v} {m : Multiset α} {a : α} (f : (a' : α) → a' ∈ a ::ₘ m → δ a') : (Multiset.Pi.cons m a (f a ⋯) fun (a' : α) (ha' : a' ∈ m) => f a' ⋯) = f

theorem Multiset.Pi.cons_injective {α : Type u_1} [DecidableEq α] {δ : α → Sort v} {a : α} {b : δ a} {s : Multiset α} (hs : a ∉ s) : Function.Injective (Multiset.Pi.cons s a b)

def Multiset.pi {α : Type u_1} [DecidableEq α] {β : α → Type u} (m : Multiset α) (t : (a : α) → Multiset (β a)) : Multiset ((a : α) → a ∈ m → β a)

pi m t constructs the Cartesian product over t indexed by m.

Equations

• m.pi t = m.recOn {Multiset.Pi.empty β} (fun (a : α) (m : Multiset α) (p : Multiset ((a : α) → a ∈ m → β a)) => (t a).bind fun (b : β a) => Multiset.map (Multiset.Pi.cons m a b) p) ⋯

@[simp]

theorem Multiset.pi_zero {α : Type u_1} [DecidableEq α] {β : α → Type u} (t : (a : α) → Multiset (β a)) : Multiset.pi 0 t = {Multiset.Pi.empty β}

@[simp]

theorem Multiset.pi_cons {α : Type u_1} [DecidableEq α] {β : α → Type u} (m : Multiset α) (t : (a : α) → Multiset (β a)) (a : α) : (a ::ₘ m).pi t = (t a).bind fun (b : β a) => Multiset.map (Multiset.Pi.cons m a b) (m.pi t)

theorem Multiset.card_pi {α : Type u_1} [DecidableEq α] {β : α → Type u} (m : Multiset α) (t : (a : α) → Multiset (β a)) : Multiset.card (m.pi t) = (Multiset.map (fun (a : α) => Multiset.card (t a)) m).prod

theorem Multiset.Nodup.pi {α : Type u_1} [DecidableEq α] {β : α → Type u} {s : Multiset α} {t : (a : α) → Multiset (β a)} : s.Nodup → (∀ a ∈ s, (t a).Nodup) → (s.pi t).Nodup

theorem Multiset.mem_pi {α : Type u_1} [DecidableEq α] {β : α → Type u} (m : Multiset α) (t : (a : α) → Multiset (β a)) (f : (a : α) → a ∈ m → β a) : f ∈ m.pi t ↔ ∀ (a : α) (h : a ∈ m), f a h ∈ t a