Documentation

Mathlib.Data.Multiset.Pi

The cartesian product of multisets #

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

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

Equations
Instances For
    def Multiset.Pi.cons {α : Type u_1} [DecidableEq α] {δ : αSort v} (m : Multiset α) (a : α) (b : δ a) (f : (a : α) → a mδ a) (a' : α) :
    a' a ::ₘ mδ 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
    Instances For
      theorem Multiset.Pi.cons_same {α : Type u_1} [DecidableEq α] {δ : αSort v} {m : Multiset α} {a : α} {b : δ a} {f : (a : α) → a mδ 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 : αSort v} {m : Multiset α} {a : α} {a' : α} {b : δ a} {f : (a : α) → a mδ 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 : αSort v} {a : α} {a' : α} {b : δ a} {b' : δ a'} {m : Multiset α} {f : (a : α) → a mδ 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 : αSort v} {m : Multiset α} {a : α} (f : (a' : α) → a' a ::ₘ mδ a') :
      (Multiset.Pi.cons m a (f a (_ : a a ::ₘ m)) fun (a' : α) (ha' : a' m) => f a' (_ : a' a ::ₘ m)) = f
      theorem Multiset.Pi.cons_injective {α : Type u_1} [DecidableEq α] {δ : αSort v} {a : α} {b : δ a} {s : Multiset α} (hs : as) :
      def Multiset.pi {α : Type u_1} [DecidableEq α] {β : αType u} (m : Multiset α) (t : (a : α) → Multiset (β a)) :
      Multiset ((a : α) → a mβ 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.
      Instances For
        @[simp]
        theorem Multiset.pi_zero {α : Type u_1} [DecidableEq α] {β : αType u} (t : (a : α) → Multiset (β a)) :
        @[simp]
        theorem Multiset.pi_cons {α : Type u_1} [DecidableEq α] {β : αType u} (m : Multiset α) (t : (a : α) → Multiset (β a)) (a : α) :
        Multiset.pi (a ::ₘ m) t = Multiset.bind (t a) fun (b : β a) => Multiset.map (Multiset.Pi.cons m a b) (Multiset.pi m t)
        theorem Multiset.card_pi {α : Type u_1} [DecidableEq α] {β : αType u} (m : Multiset α) (t : (a : α) → Multiset (β a)) :
        Multiset.card (Multiset.pi m t) = Multiset.prod (Multiset.map (fun (a : α) => Multiset.card (t a)) m)
        theorem Multiset.Nodup.pi {α : Type u_1} [DecidableEq α] {β : αType u} {s : Multiset α} {t : (a : α) → Multiset (β a)} :
        Multiset.Nodup s(as, Multiset.Nodup (t a))Multiset.Nodup (Multiset.pi s t)
        theorem Multiset.mem_pi {α : Type u_1} [DecidableEq α] {β : αType u} (m : Multiset α) (t : (a : α) → Multiset (β a)) (f : (a : α) → a mβ a) :
        f Multiset.pi m t ∀ (a : α) (h : a m), f a h t a