Antidiagonals in ℕ × ℕ as finsets #

This file defines the antidiagonals of ℕ × ℕ as finsets: the n-th antidiagonal is the finset of pairs (i, j) such that i + j = n. This is useful for polynomial multiplication and more generally for sums going from 0 to n.

Notes #

This refines files Data.List.NatAntidiagonal and Data.Multiset.NatAntidiagonal.

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def Finset.Nat.antidiagonal (n : ℕ) :

Finset (ℕ × ℕ)

The antidiagonal of a natural number n is the finset of pairs (i, j) such that i + j = n.

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Finset.Nat.antidiagonal n = { val := Multiset.Nat.antidiagonal n, nodup := (_ : Multiset.Nodup (Multiset.Nat.antidiagonal n)) }

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@[simp]

theorem Finset.Nat.mem_antidiagonal {n : ℕ} {x : ℕ × ℕ} :

x ∈ Finset.Nat.antidiagonal n ↔ x.fst + x.snd = n

A pair (i, j) is contained in the antidiagonal of n if and only if i + j = n.

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@[simp]

theorem Finset.Nat.card_antidiagonal (n : ℕ) :

Finset.card (Finset.Nat.antidiagonal n) = n + 1

The cardinality of the antidiagonal of n is n + 1.

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@[simp]

theorem Finset.Nat.antidiagonal_zero :

Finset.Nat.antidiagonal 0 = {(0, 0)}

The antidiagonal of 0 is the list [(0, 0)]

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theorem Finset.Nat.antidiagonal_succ (n : ℕ) :

Finset.Nat.antidiagonal (n + 1) = Finset.cons (0, n + 1) (Finset.map (Function.Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Function.Embedding.refl ℕ)) (Finset.Nat.antidiagonal n)) (_ : ¬(0, n + 1) ∈ Finset.map (Function.Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Function.Embedding.refl ℕ)) (Finset.Nat.antidiagonal n))

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theorem Finset.Nat.antidiagonal_succ' (n : ℕ) :

Finset.Nat.antidiagonal (n + 1) = Finset.cons (n + 1, 0) (Finset.map (Function.Embedding.prodMap (Function.Embedding.refl ℕ) { toFun := Nat.succ, inj' := Nat.succ_injective }) (Finset.Nat.antidiagonal n)) (_ : ¬(n + 1, 0) ∈ Finset.map (Function.Embedding.prodMap (Function.Embedding.refl ℕ) { toFun := Nat.succ, inj' := Nat.succ_injective }) (Finset.Nat.antidiagonal n))

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theorem Finset.Nat.antidiagonal_succ_succ' {n : ℕ} :

Finset.Nat.antidiagonal (n + 2) = Finset.cons (0, n + 2) (Finset.cons (n + 2, 0) (Finset.map (Function.Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } { toFun := Nat.succ, inj' := Nat.succ_injective }) (Finset.Nat.antidiagonal n)) (_ : ¬(n + 2, 0) ∈ Finset.map (Function.Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } { toFun := Nat.succ, inj' := Nat.succ_injective }) (Finset.Nat.antidiagonal n))) (_ : ¬(0, n + 2) ∈ Finset.cons (n + 2, 0) (Finset.map (Function.Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } { toFun := Nat.succ, inj' := Nat.succ_injective }) (Finset.Nat.antidiagonal n)) (_ : ¬(n + 2, 0) ∈ Finset.map (Function.Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } { toFun := Nat.succ, inj' := Nat.succ_injective }) (Finset.Nat.antidiagonal n)))

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theorem Finset.Nat.map_swap_antidiagonal {n : ℕ} :

Finset.map { toFun := Prod.swap, inj' := (_ : Function.Injective Prod.swap) } (Finset.Nat.antidiagonal n) = Finset.Nat.antidiagonal n

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theorem Finset.Nat.antidiagonal_congr {n : ℕ} {p : ℕ × ℕ} {q : ℕ × ℕ} (hp : p ∈ Finset.Nat.antidiagonal n) (hq : q ∈ Finset.Nat.antidiagonal n) :

p = q ↔ p.fst = q.fst

A point in the antidiagonal is determined by its first co-ordinate.

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theorem Finset.Nat.antidiagonal.fst_le {n : ℕ} {kl : ℕ × ℕ} (hlk : kl ∈ Finset.Nat.antidiagonal n) :

kl.fst ≤ n

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theorem Finset.Nat.antidiagonal.snd_le {n : ℕ} {kl : ℕ × ℕ} (hlk : kl ∈ Finset.Nat.antidiagonal n) :

kl.snd ≤ n

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theorem Finset.Nat.filter_fst_eq_antidiagonal (n : ℕ) (m : ℕ) :

Finset.filter (fun x => x.fst = m) (Finset.Nat.antidiagonal n) = if m ≤ n then {(m, n - m)} else ∅

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theorem Finset.Nat.filter_snd_eq_antidiagonal (n : ℕ) (m : ℕ) :

Finset.filter (fun x => x.snd = m) (Finset.Nat.antidiagonal n) = if m ≤ n then {(n - m, m)} else ∅

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@[simp]

theorem Finset.Nat.sigmaAntidiagonalEquivProd_symm_apply_snd_coe (x : ℕ × ℕ) :

↑(↑(Equiv.symm Finset.Nat.sigmaAntidiagonalEquivProd) x).snd = x

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theorem Finset.Nat.sigmaAntidiagonalEquivProd_symm_apply_fst (x : ℕ × ℕ) :

(↑(Equiv.symm Finset.Nat.sigmaAntidiagonalEquivProd) x).fst = x.fst + x.snd

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theorem Finset.Nat.sigmaAntidiagonalEquivProd_apply (x : (n : ℕ) × { x // x ∈ Finset.Nat.antidiagonal n }) :

↑Finset.Nat.sigmaAntidiagonalEquivProd x = ↑x.snd

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def Finset.Nat.sigmaAntidiagonalEquivProd :

(n : ℕ) × { x // x ∈ Finset.Nat.antidiagonal n } ≃ ℕ × ℕ

The disjoint union of antidiagonals Σ (n : ℕ), antidiagonal n is equivalent to the product ℕ × ℕ. This is such an equivalence, obtained by mapping (n, (k, l)) to (k, l).

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