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Mathlib.Data.Multiset.Antidiagonal

The antidiagonal on a multiset. #

The antidiagonal of a multiset s consists of all pairs (t₁, t₂) such that t₁ + t₂ = s. These pairs are counted with multiplicities.

The antidiagonal of a multiset s consists of all pairs (t₁, t₂) such that t₁ + t₂ = s. These pairs are counted with multiplicities.

Equations
Instances For
    theorem Multiset.antidiagonal_coe {α : Type u_1} (l : List α) :
    (↑l).antidiagonal = (Multiset.powersetAux l).revzip
    @[simp]
    theorem Multiset.antidiagonal_coe' {α : Type u_1} (l : List α) :
    (↑l).antidiagonal = (Multiset.powersetAux' l).revzip
    @[simp]
    theorem Multiset.mem_antidiagonal {α : Type u_1} {s : Multiset α} {x : Multiset α × Multiset α} :
    x s.antidiagonal x.1 + x.2 = s

    A pair (t₁, t₂) of multisets is contained in antidiagonal s if and only if t₁ + t₂ = s.

    @[simp]
    theorem Multiset.antidiagonal_map_fst {α : Type u_1} (s : Multiset α) :
    Multiset.map Prod.fst s.antidiagonal = s.powerset
    @[simp]
    theorem Multiset.antidiagonal_map_snd {α : Type u_1} (s : Multiset α) :
    Multiset.map Prod.snd s.antidiagonal = s.powerset
    @[simp]
    @[simp]
    theorem Multiset.antidiagonal_cons {α : Type u_1} (a : α) (s : Multiset α) :
    (a ::ₘ s).antidiagonal = Multiset.map (Prod.map id (Multiset.cons a)) s.antidiagonal + Multiset.map (Prod.map (Multiset.cons a) id) s.antidiagonal
    theorem Multiset.antidiagonal_eq_map_powerset {α : Type u_1} [DecidableEq α] (s : Multiset α) :
    s.antidiagonal = Multiset.map (fun (t : Multiset α) => (s - t, t)) s.powerset
    @[simp]
    theorem Multiset.card_antidiagonal {α : Type u_1} (s : Multiset α) :
    Multiset.card s.antidiagonal = 2 ^ Multiset.card s