Compare.v

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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}.forall n m : nat, {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.forall n m : nat, 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.forall n m : nat, n <= m -> S n <= m \/ n = m
Proof le_lt_or_eq.

(* By special request of G. Kahn - Used in Group Theory *)
Lemma discrete_nat :
  forall n m, n < m -> S n = m \/ (exists r : nat, m = S (S (n + r))).forall n m : nat,
n < m ->
S n = m \/ (exists r : nat, m = S (S (n + r)))
Proof.forall n m : nat,
n < m ->
S n = m \/ (exists r : nat, m = S (S (n + r)))
  intros m n H.m, n:nat
H:m < n
S m = n \/ (exists r : nat, n = S (S (m + r)))
  lapply (lt_le_S m n); auto with arith.m, n:nat
H:m < n
S m <= n ->
S m = n \/ (exists r : nat, n = S (S (m + r)))
  intro H'; lapply (le_lt_or_eq (S m) n); auto with arith.m, n:nat
H:m < n
H':S m <= n
S m < n \/ S m = n ->
S m = n \/ (exists r : nat, n = S (S (m + r)))
  induction 1; auto with arith.m, n:nat
H:m < n
H':S m <= n
H0:S m < n
S m = n \/ (exists r : nat, n = S (S (m + r)))
  right; exists (n - S (S m)); simpl.m, n:nat
H:m < n
H':S m <= n
H0:S m < n
n = S (S (m + (n - S (S m))))
  rewrite (plus_comm m (n - S (S m))).m, n:nat
H:m < n
H':S m <= n
H0:S m < n
n = S (S (n - S (S m) + m))
  rewrite (plus_n_Sm (n - S (S m)) m).m, n:nat
H:m < n
H':S m <= n
H0:S m < n
n = S (n - S (S m) + S m)
  rewrite (plus_n_Sm (n - S (S m)) (S m)).m, n:nat
H:m < n
H':S m <= n
H0:S m < n
n = n - S (S m) + S (S m)
  rewrite (plus_comm (n - S (S m)) (S (S m))); auto with arith.
Qed.

Require Export Wf_nat.

Require Export Min Max.