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

diff --git a/matita/library/Z/compare.ma b/matita/library/Z/compare.ma
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+(**************************************************************************)
+(*       ___                                                               *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
+(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/Z/compare".
+
+include "Z/orders.ma".
+include "nat/compare.ma".
+
+(* boolean equality *)
+definition eqZb : Z \to Z \to bool \def
+\lambda x,y:Z.
+  match x with
+  [ OZ \Rightarrow 
+      match y with
+        [ OZ \Rightarrow true
+        | (pos q) \Rightarrow false
+        | (neg q) \Rightarrow false]
+  | (pos p) \Rightarrow 
+      match y with
+        [ OZ \Rightarrow false
+        | (pos q) \Rightarrow eqb p q
+        | (neg q) \Rightarrow false]     
+  | (neg p) \Rightarrow 
+      match y with
+        [ OZ \Rightarrow false
+        | (pos q) \Rightarrow false
+        | (neg q) \Rightarrow eqb p q]].
+
+theorem eqZb_to_Prop: 
+\forall x,y:Z. 
+match eqZb x y with
+[ true \Rightarrow x=y
+| false \Rightarrow x \neq y].
+intros.
+elim x.
+  elim y.
+    simplify.reflexivity.
+    simplify.apply not_eq_OZ_pos.
+    simplify.apply not_eq_OZ_neg.
+  elim y.
+    simplify.unfold Not.intro.apply (not_eq_OZ_pos n).apply sym_eq.assumption.
+    simplify.apply eqb_elim.
+      intro.simplify.apply eq_f.assumption.
+      intro.simplify.unfold Not.intro.apply H.apply inj_pos.assumption.
+    simplify.apply not_eq_pos_neg.
+  elim y.
+    simplify.unfold Not.intro.apply (not_eq_OZ_neg n).apply sym_eq.assumption.
+    simplify.unfold Not.intro.apply (not_eq_pos_neg n1 n).apply sym_eq.assumption.
+    simplify.apply eqb_elim.
+      intro.simplify.apply eq_f.assumption.
+      intro.simplify.unfold Not.intro.apply H.apply inj_neg.assumption.
+qed.
+
+theorem eqZb_elim: \forall x,y:Z.\forall P:bool \to Prop.
+(x=y \to (P true)) \to (x \neq y \to (P false)) \to P (eqZb x y).
+intros.
+cut 
+(match (eqZb x y) with
+[ true \Rightarrow x=y
+| false \Rightarrow x \neq y] \to P (eqZb x y)).
+apply Hcut.
+apply eqZb_to_Prop.
+elim (eqZb).
+apply (H H2).
+apply (H1 H2).
+qed.
+
+definition Z_compare : Z \to Z \to compare \def
+\lambda x,y:Z.
+  match x with
+  [ OZ \Rightarrow 
+    match y with 
+    [ OZ \Rightarrow EQ
+    | (pos m) \Rightarrow LT
+    | (neg m) \Rightarrow GT ]
+  | (pos n) \Rightarrow 
+    match y with 
+    [ OZ \Rightarrow GT
+    | (pos m) \Rightarrow (nat_compare n m)
+    | (neg m) \Rightarrow GT]
+  | (neg n) \Rightarrow 
+    match y with 
+    [ OZ \Rightarrow LT
+    | (pos m) \Rightarrow LT
+    | (neg m) \Rightarrow nat_compare m n ]].
+
+theorem Z_compare_to_Prop : 
+\forall x,y:Z. match (Z_compare x y) with
+[ LT \Rightarrow x < y
+| EQ \Rightarrow x=y
+| GT \Rightarrow y < x]. 
+intros.
+elim x. 
+  elim y.
+    simplify.apply refl_eq.
+    simplify.exact I.
+    simplify.exact I.
+  elim y.
+    simplify.exact I.
+    simplify.
+      cut (match (nat_compare n n1) with
+      [ LT \Rightarrow n<n1
+      | EQ \Rightarrow n=n1
+      | GT \Rightarrow n1<n] \to 
+      match (nat_compare n n1) with
+      [ LT \Rightarrow (S n) \leq n1
+      | EQ \Rightarrow pos n = pos n1
+      | GT \Rightarrow (S n1) \leq n]). 
+        apply Hcut.apply nat_compare_to_Prop. 
+        elim (nat_compare n n1).
+          simplify.exact H.
+          simplify.apply eq_f.exact H.
+          simplify.exact H.
+    simplify.exact I.    
+  elim y. 
+    simplify.exact I.
+    simplify.exact I.
+    simplify. 
+      cut (match (nat_compare n1 n) with
+      [ LT \Rightarrow n1<n
+      | EQ \Rightarrow n1=n
+      | GT \Rightarrow n<n1] \to 
+      match (nat_compare n1 n) with
+      [ LT \Rightarrow (S n1) \leq n
+      | EQ \Rightarrow neg n = neg n1
+      | GT \Rightarrow (S n) \leq n1]). 
+        apply Hcut. apply nat_compare_to_Prop. 
+        elim (nat_compare n1 n).
+          simplify.exact H.
+          simplify.apply eq_f.apply sym_eq.exact H.
+          simplify.exact H.
+qed.