mathlib documentation

data.list.nat_antidiagonal

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def list.nat.antidiagonal (n : ℕ) :
list (ℕ × ℕ)

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

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@[simp]
theorem list.nat.mem_antidiagonal {n : ℕ} {x : ℕ × ℕ} :
x ∈ list.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 list.nat.length_antidiagonal (n : ℕ) :
(list.nat.antidiagonal n).length = n + 1

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

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@[simp]
theorem list.nat.antidiagonal_zero  :
list.nat.antidiagonal 0 = [(0, 0)]

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

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theorem list.nat.nodup_antidiagonal (n : ℕ) :
(list.nat.antidiagonal n).nodup

The antidiagonal of n does not contain duplicate entries.

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@[simp]
theorem list.nat.antidiagonal_succ {n : ℕ} :
list.nat.antidiagonal (n + 1) = (0, n + 1) :: list.map (prod.map nat.succ id) (list.nat.antidiagonal n)