(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/compare".
-
include "datatypes/bool.ma".
include "datatypes/compare.ma".
include "nat/orders.ma".
]
qed.
+theorem leb_true_to_le:\forall n,m.
+leb n m = true \to (n \le m).
+intros 2.
+apply leb_elim
+ [intros.assumption
+ |intros.destruct H1.
+ ]
+qed.
+
+theorem leb_false_to_not_le:\forall n,m.
+leb n m = false \to \lnot (n \le m).
+intros 2.
+apply leb_elim
+ [intros.destruct H1
+ |intros.assumption
+ ]
+qed.
(*
theorem decidable_le: \forall n,m. n \leq m \lor n \nleq m.
intros.
apply ((H1 H3)).
apply ((H2 H3)).
qed.
+
+inductive cmp_cases (n,m:nat) : CProp ≝
+ | cmp_le : n ≤ m → cmp_cases n m
+ | cmp_gt : m < n → cmp_cases n m.
+
+lemma cmp_nat: ∀n,m.cmp_cases n m.
+intros; generalize in match (nat_compare_to_Prop n m);
+cases (nat_compare n m); intros;
+[constructor 1;apply lt_to_le|constructor 1;rewrite > H|constructor 2]
+try assumption; apply le_n;
+qed.