mathlib documentation

data.list.nat_antidiagonal

Antidiagonals in ℕ × ℕ as lists #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

This file defines the antidiagonals of ℕ × ℕ as lists: the n-th antidiagonal is the list 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 #

Files data.multiset.nat_antidiagonal and data.finset.nat_antidiagonal successively turn the list definition we have here into multiset and finset.

The antidiagonal of a natural number n is the list 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 length of the antidiagonal of n is n + 1.

@[simp]

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

The antidiagonal of n does not contain duplicate entries.