The Coq Proof Assistant

# Library Coq.Arith.Compare

Equality is decidable on nat
Local Open Scope nat_scope.
Notation not_eq_sym := not_eq_sym (only parsing).
Implicit Types m n p q : nat.
Require Import Arith_base.
Require Import Peano_dec.
Require Import Compare_dec.
Definition le_or_le_S := le_le_S_dec.
Definition Pcompare := gt_eq_gt_dec.
Lemma le_dec : forall n m, {n <= m} + {m <= n}.
Definition lt_or_eq n m := {m > n} + {n = m}.
Lemma le_decide : forall n m, n <= m -> lt_or_eq n m.
Lemma le_le_S_eq : forall n m, n <= m -> S n <= m \/ n = m.
Lemma discrete_nat :
forall n m, n < m -> S n = m \/ (exists r : nat, m = S (S (n + r))).
Require Export Wf_nat.