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 = Multiset.Pi.empty.match_1 (fun a a_1 => δ a) 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 (_ : a = a') ▸ b else f a' (_ : a' ∈ m)

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' (_ : a' ∈ m)

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 (_ : a ∈ a ::ₘ m)) fun a' ha' => f a' (_ : a' ∈ a ::ₘ m)) = 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

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

@[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) : Multiset.pi (a ::ₘ m) t = Multiset.bind (t a) fun b => Multiset.map (Multiset.Pi.cons m a b) (Multiset.pi m 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 (Multiset.pi m t) = Multiset.prod (Multiset.map (fun a => ↑Multiset.card (t a)) m)

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

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 ∈ Multiset.pi m t ↔ ∀ (a : α) (h : a ∈ m), f a h ∈ t a