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_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}.

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.