New version of the library.

[helm.git] / helm / matita / library / nat / compare.ma

diff --git a/helm/matita/library/nat/compare.ma b/helm/matita/library/nat/compare.ma
index d148dfd310b2b8be46fdb8a5e4b854e71d6ac34f..fec46ae2fe1f65f8106020bc4a038553beb1667b 100644 (file)
--- a/helm/matita/library/nat/compare.ma
+++ b/helm/matita/library/nat/compare.ma
@@ -16,6 +16,7 @@ set "baseuri" "cic:/matita/nat/compare".
 
 include "nat/orders.ma".
 include "datatypes/bool.ma".
+include "datatypes/compare.ma".
 
 let rec leb n m \def 
 match n with 
@@ -41,7 +42,7 @@ simplify.apply le_S_S.apply H.
 simplify.intros.apply H.apply le_S_S_to_le.assumption.
 qed.
 
-theorem le_elim: \forall n,m:nat. \forall P:bool \to Prop. 
+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
 P (leb n m).
 intros.
@@ -55,5 +56,70 @@ apply (H H2).
 apply (H1 H2).
 qed.
 
+let rec nat_compare n m: compare \def
+match n with
+[ O \Rightarrow 
+    match m with 
+      [ O \Rightarrow EQ
+      | (S q) \Rightarrow LT ]
+| (S p) \Rightarrow 
+    match m with 
+      [ O \Rightarrow GT
+      | (S q) \Rightarrow nat_compare p q]].
 
+theorem nat_compare_n_n: \forall n:nat.(eq compare (nat_compare n n) EQ).
+intro.elim n.
+simplify.reflexivity.
+simplify.assumption.
+qed.
+
+theorem nat_compare_S_S: \forall n,m:nat. 
+eq compare (nat_compare n m) (nat_compare (S n) (S m)).
+intros.simplify.reflexivity.
+qed.
 
+theorem nat_compare_to_Prop: \forall n,m:nat. 
+match (nat_compare n m) with
+  [ LT \Rightarrow (lt n m)
+  | EQ \Rightarrow (eq nat n m)
+  | GT \Rightarrow (lt m n) ].
+intros.
+apply nat_elim2 (\lambda n,m.match (nat_compare n m) with
+  [ LT \Rightarrow (lt n m)
+  | EQ \Rightarrow (eq nat n m)
+  | GT \Rightarrow (lt 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.
+qed.
+
+theorem nat_compare_n_m_m_n: \forall n,m:nat. 
+eq compare (nat_compare n m) (compare_invert (nat_compare m n)).
+intros. 
+apply nat_elim2 (\lambda n,m.eq compare (nat_compare n m) (compare_invert (nat_compare m n))).
+intros.elim n1.simplify.reflexivity.
+simplify.reflexivity.
+intro.elim n1.simplify.reflexivity.
+simplify.reflexivity.
+intros.simplify.elim H.reflexivity.
+qed.
+     
+theorem nat_compare_elim : \forall n,m:nat. \forall P:compare \to Prop.
+((lt n m) \to (P LT)) \to ((eq nat n m) \to (P EQ)) \to ((lt m n) \to (P GT)) \to 
+(P (nat_compare n m)).
+intros.
+cut match (nat_compare n m) with
+[ LT \Rightarrow (lt n m)
+| EQ \Rightarrow (eq nat n m)
+| GT \Rightarrow (lt 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).
+qed.