mathlib documentation

data.finset.nat_antidiagonal

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.nat_antidiagonal and data.multiset.nat_antidiagonal.

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

Equations
@[simp]

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

@[simp]

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

@[simp]

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

theorem finset.nat.antidiagonal_congr {n : } {p q : × } (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.