(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/compare".
-
include "datatypes/bool.ma".
include "datatypes/compare.ma".
include "nat/orders.ma".
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 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 (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.
+*)
+
+definition ltb ≝λn,m. leb n m ∧ notb (eqb n m).
+
+theorem ltb_to_Prop :
+ ∀n,m.
+ match ltb n m with
+ [ true ⇒ n < m
+ | false ⇒ n ≮ m
+ ].
+intros;
+unfold ltb;
+apply leb_elim;
+apply eqb_elim;
+intros;
+simplify;
+[ rewrite < H;
+ apply le_to_not_lt;
+ constructor 1
+| apply (not_eq_to_le_to_lt ? ? H H1)
+| rewrite < H;
+ apply le_to_not_lt;
+ constructor 1
+| apply le_to_not_lt;
+ generalize in match (not_le_to_lt ? ? H1);
+ clear H1;
+ intro;
+ apply lt_to_le;
+ assumption
+].
+qed.
+
+theorem ltb_elim: ∀n,m:nat. ∀P:bool → Prop.
+(n < m → (P true)) → (n ≮ m → (P false)) →
+P (ltb n m).
+intros.
+cut
+(match (ltb n m) with
+[ true ⇒ n < m
+| false ⇒ n ≮ m] → (P (ltb n m))).
+apply Hcut.apply ltb_to_Prop.
+elim (ltb n m).
+apply ((H H2)).
+apply ((H1 H2)).
+qed.
let rec nat_compare n m: compare \def
match n with
intros.simplify.reflexivity.
qed.
-theorem S_pred: \forall n:nat.lt O n \to eq nat n (S (pred n)).
-intro.elim n.apply False_ind.exact (not_le_Sn_O O H).
-apply eq_f.apply pred_Sn.
-qed.
-
theorem nat_compare_pred_pred:
\forall n,m:nat.lt O n \to lt O m \to
eq compare (nat_compare n m) (nat_compare (pred n) (pred m)).
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.