# Library Coq.Arith.Compare

``` ```
 Equality is decidable on `nat`
``` Open Local Scope nat_scope. Notation not_eq_sym := sym_not_eq. Implicit Types m n p q : nat. Require Import Arith. 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}. Proof le_ge_dec. Definition lt_or_eq n m := {m > n} + {n = m}. Lemma le_decide : forall n m, n <= m -> lt_or_eq n m. Proof le_lt_eq_dec. Lemma le_le_S_eq : forall n m, n <= m -> S n <= m \/ n = m. Proof le_lt_or_eq. Lemma discrete_nat :  forall n m, n < m -> S n = m \/ (exists r : nat, m = S (S (n + r))). Proof. intros m n H. lapply (lt_le_S m n); auto with arith. intro H'; lapply (le_lt_or_eq (S m) n); auto with arith. induction 1; auto with arith. right; exists (n - S (S m)); simpl in |- *. rewrite (plus_comm m (n - S (S m))). rewrite (plus_n_Sm (n - S (S m)) m). rewrite (plus_n_Sm (n - S (S m)) (S m)). rewrite (plus_comm (n - S (S m)) (S (S m))); auto with arith. Qed. Require Export Wf_nat. Require Export Min. ```
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