Coq.Arith.Compare

# Module Coq.Arith.Compare

``

Equality is decidable on `nat`

`

`Lemma` not_eq_sym : (A:Set)(p,q:A)(~p=q) -> ~(q=p).
`Proof` sym_not_eq.
`Hints` `Immediate` not_eq_sym : arith.

`Require` Arith.
`Require` Peano_dec.
`Require` Compare_dec.

`Definition` le_or_le_S := le_le_S_dec.

`Definition` compare := gt_eq_gt_dec.

`Lemma` le_dec : (n,m:nat) {le n m} + {le m n}.
`Proof` le_ge_dec.

`Definition` lt_or_eq := [n,m:nat]{(gt m n)}+{n=m}.

`Lemma` le_decide : (n,m:nat)(le n m)->(lt_or_eq n m).
`Proof` le_lt_eq_dec.

`Lemma` le_le_S_eq : (p,q:nat)(le p q)->((le (S p) q)\/(p=q)).
`Proof` le_lt_or_eq.

`Lemma` discrete_nat : (m, n: nat) (lt m n) ->
(S m) = n \/ (EX r: nat | n = (S (S (plus m 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.
NewInduction 1; Auto with arith.
Right; Exists (minus n (S (S m))); Simpl.
Rewrite (plus_sym m (minus n (S (S m)))).
Rewrite (plus_n_Sm (minus n (S (S m))) m).
Rewrite (plus_n_Sm (minus n (S (S m))) (S m)).
Rewrite (plus_sym (minus n (S (S m))) (S (S m))); Auto with arith.
`Qed`.

`Require` `Export` Wf_nat.

`Require` `Export` Min.

`

Index