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.

def Multiset.antidiagonal {α : Type u_1} (s : Multiset α) : Multiset (Multiset α × Multiset α)

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

Equations

• Multiset.antidiagonal s = Quot.liftOn s (fun (l : List α) => ↑(List.revzip (Multiset.powersetAux l))) ⋯

theorem Multiset.antidiagonal_coe {α : Type u_1} (l : List α) : Multiset.antidiagonal ↑l = ↑(List.revzip (Multiset.powersetAux l))

@[simp]

theorem Multiset.antidiagonal_coe' {α : Type u_1} (l : List α) : Multiset.antidiagonal ↑l = ↑(List.revzip (Multiset.powersetAux' l))

@[simp]

theorem Multiset.mem_antidiagonal {α : Type u_1} {s : Multiset α} {x : Multiset α × Multiset α} : x ∈ Multiset.antidiagonal s ↔ x.1 + x.2 = s

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

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theorem Multiset.antidiagonal_map_fst {α : Type u_1} (s : Multiset α) : Multiset.map Prod.fst (Multiset.antidiagonal s) = Multiset.powerset s

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theorem Multiset.antidiagonal_map_snd {α : Type u_1} (s : Multiset α) : Multiset.map Prod.snd (Multiset.antidiagonal s) = Multiset.powerset s

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theorem Multiset.antidiagonal_zero {α : Type u_1} : Multiset.antidiagonal 0 = {(0, 0)}

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theorem Multiset.antidiagonal_cons {α : Type u_1} (a : α) (s : Multiset α) : Multiset.antidiagonal (a ::ₘ s) = Multiset.map (Prod.map id (Multiset.cons a)) (Multiset.antidiagonal s) + Multiset.map (Prod.map (Multiset.cons a) id) (Multiset.antidiagonal s)

theorem Multiset.antidiagonal_eq_map_powerset {α : Type u_1} [DecidableEq α] (s : Multiset α) : Multiset.antidiagonal s = Multiset.map (fun (t : Multiset α) => (s - t, t)) (Multiset.powerset s)

@[simp]

theorem Multiset.card_antidiagonal {α : Type u_1} (s : Multiset α) : Multiset.card (Multiset.antidiagonal s) = 2 ^ Multiset.card s

theorem Multiset.prod_map_add {α : Type u_1} {β : Type u_2} [CommSemiring β] {s : Multiset α} {f : α → β} {g : α → β} : Multiset.prod (Multiset.map (fun (a : α) => f a + g a) s) = Multiset.sum (Multiset.map (fun (p : Multiset α × Multiset α) => Multiset.prod (Multiset.map f p.1) * Multiset.prod (Multiset.map g p.2)) (Multiset.antidiagonal s))