(**************************************************************************)
-(* ___ *)
+(* ___ *)
(* ||M|| *)
(* ||A|| A project by Andrea Asperti *)
(* ||T|| *)
set "baseuri" "cic:/matita/nat/compare".
-include "nat/orders.ma".
include "datatypes/bool.ma".
include "datatypes/compare.ma".
+include "nat/orders.ma".
+
+let rec eqb n m \def
+match n with
+ [ O \Rightarrow
+ match m with
+ [ O \Rightarrow true
+ | (S q) \Rightarrow false]
+ | (S p) \Rightarrow
+ match m with
+ [ O \Rightarrow false
+ | (S q) \Rightarrow eqb p q]].
+
+theorem eqb_to_Prop: \forall n,m:nat.
+match (eqb n m) with
+[ true \Rightarrow n = m
+| false \Rightarrow \lnot (n = m)].
+intros.
+apply nat_elim2
+(\lambda n,m:nat.match (eqb n m) with
+[ true \Rightarrow n = m
+| false \Rightarrow \lnot (n = m)]).
+intro.elim n1.
+simplify.reflexivity.
+simplify.apply not_eq_O_S.
+intro.
+simplify.
+intro. apply not_eq_O_S n1 ?.apply sym_eq.assumption.
+intros.simplify.
+generalize in match H.
+elim (eqb n1 m1).
+simplify.apply eq_f.apply H1.
+simplify.intro.apply H1.apply inj_S.assumption.
+qed.
+
+theorem eqb_elim : \forall n,m:nat.\forall P:bool \to Prop.
+(n=m \to (P true)) \to (\lnot n=m \to (P false)) \to (P (eqb n m)).
+intros.
+cut
+match (eqb n m) with
+[ true \Rightarrow n = m
+| false \Rightarrow \lnot (n = m)] \to (P (eqb n m)).
+apply Hcut.apply eqb_to_Prop.
+elim eqb n m.
+apply (H H2).
+apply (H1 H2).
+qed.
let rec leb n m \def
match n with
theorem leb_to_Prop: \forall n,m:nat.
match (leb n m) with
-[ true \Rightarrow (le n m)
-| false \Rightarrow (Not (le n m))].
+[ true \Rightarrow n \leq m
+| false \Rightarrow \lnot (n \leq m)].
intros.
apply nat_elim2
(\lambda n,m:nat.match (leb n m) with
-[ true \Rightarrow (le n m)
-| false \Rightarrow (Not (le n m))]).
+[ true \Rightarrow n \leq m
+| false \Rightarrow \lnot (n \leq m)]).
simplify.exact le_O_n.
simplify.exact not_le_Sn_O.
intros 2.simplify.elim (leb n1 m1).
qed.
theorem leb_elim: \forall n,m:nat. \forall P:bool \to Prop.
-((le n m) \to (P true)) \to ((Not (le n m)) \to (P false)) \to
+(n \leq m \to (P true)) \to (\not (n \leq m) \to (P false)) \to
P (leb n m).
intros.
cut
match (leb n m) with
-[ true \Rightarrow (le n m)
-| false \Rightarrow (Not (le n m))] \to (P (leb n m)).
+[ true \Rightarrow n \leq m
+| false \Rightarrow \lnot (n \leq m)] \to (P (leb n m)).
apply Hcut.apply leb_to_Prop.
elim leb n m.
apply (H H2).
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)).
+intros.
+apply lt_O_n_elim n H.
+apply lt_O_n_elim m H1.
+intros.
+simplify.reflexivity.
+qed.
+
theorem nat_compare_to_Prop: \forall n,m:nat.
match (nat_compare n m) with
- [ LT \Rightarrow (lt n m)
+ [ LT \Rightarrow n < m
| EQ \Rightarrow n=m
- | GT \Rightarrow (lt m n) ].
+ | GT \Rightarrow m < n ].
intros.
apply nat_elim2 (\lambda n,m.match (nat_compare n m) with
- [ LT \Rightarrow (lt n m)
+ [ LT \Rightarrow n < m
| EQ \Rightarrow n=m
- | GT \Rightarrow (lt m n) ]).
+ | GT \Rightarrow m < n ]).
intro.elim n1.simplify.reflexivity.
simplify.apply le_S_S.apply le_O_n.
intro.simplify.apply le_S_S. apply le_O_n.
intros 2.simplify.elim (nat_compare n1 m1).
simplify. apply le_S_S.apply H.
-simplify. apply le_S_S.apply H.
simplify. apply eq_f. apply H.
+simplify. apply le_S_S.apply H.
qed.
theorem nat_compare_n_m_m_n: \forall n,m:nat.
qed.
theorem nat_compare_elim : \forall n,m:nat. \forall P:compare \to Prop.
-((lt n m) \to (P LT)) \to (n=m \to (P EQ)) \to ((lt m n) \to (P GT)) \to
+(n < m \to P LT) \to (n=m \to P EQ) \to (m < n \to P GT) \to
(P (nat_compare n m)).
intros.
cut match (nat_compare n m) with
-[ LT \Rightarrow (lt n m)
+[ LT \Rightarrow n < m
| EQ \Rightarrow n=m
-| GT \Rightarrow (lt m n)] \to
+| GT \Rightarrow m < n] \to
(P (nat_compare n m)).
apply Hcut.apply nat_compare_to_Prop.
elim (nat_compare n m).
apply (H H3).
-apply (H2 H3).
apply (H1 H3).
+apply (H2 H3).
qed.