match m with
[ O \Rightarrow false
| (S q) \Rightarrow leb p q]].
-
+
+theorem leb_elim: \forall n,m:nat. \forall P:bool \to Prop.
+(n \leq m \to (P true)) \to (n \nleq m \to (P false)) \to
+P (leb n m).
+apply nat_elim2; intros; simplify
+ [apply H.apply le_O_n
+ |apply H1.apply not_le_Sn_O.
+ |apply H;intros
+ [apply H1.apply le_S_S.assumption.
+ |apply H2.unfold Not.intros.apply H3.apply le_S_S_to_le.assumption
+ ]
+ ]
+qed.
+
+(*
+theorem decidable_le: \forall n,m. n \leq m \lor n \nleq m.
+intros.
+apply (leb_elim n m)
+ [intro.left.assumption
+ |intro.right.assumption
+ ]
+qed.
+*)
+
+theorem le_to_leb_true: \forall n,m. n \leq m \to leb n m = true.
+intros.apply leb_elim;intros
+ [reflexivity
+ |apply False_ind.apply H1.apply H.
+ ]
+qed.
+
+theorem lt_to_leb_false: \forall n,m. m < n \to leb n m = false.
+intros.apply leb_elim;intros
+ [apply False_ind.apply (le_to_not_lt ? ? H1). assumption
+ |reflexivity
+ ]
+qed.
+
theorem leb_to_Prop: \forall n,m:nat.
match (leb n m) with
[ true \Rightarrow n \leq m
| false \Rightarrow n \nleq m].
-intros.
-apply (nat_elim2
-(\lambda n,m:nat.match (leb n m) with
-[ true \Rightarrow n \leq m
-| false \Rightarrow n \nleq m])).
-simplify.exact le_O_n.
-simplify.exact not_le_Sn_O.
-intros 2.simplify.elim ((leb n1 m1)).
-simplify.apply le_S_S.apply H.
-simplify.unfold Not.intros.apply H.apply le_S_S_to_le.assumption.
+apply nat_elim2;simplify
+ [exact le_O_n
+ |exact not_le_Sn_O
+ |intros 2.simplify.
+ elim ((leb n m));simplify
+ [apply le_S_S.apply H
+ |unfold Not.intros.apply H.apply le_S_S_to_le.assumption
+ ]
+ ]
qed.
+(*
theorem leb_elim: \forall n,m:nat. \forall P:bool \to Prop.
(n \leq m \to (P true)) \to (n \nleq m \to (P false)) \to
P (leb n m).
apply ((H H2)).
apply ((H1 H2)).
qed.
+*)
let rec nat_compare n m: compare \def
match n with