Added Z/plus.ma e Z/compare.ma.

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

diff --git a/helm/matita/library/Z/compare.ma b/helm/matita/library/Z/compare.ma
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+++ b/helm/matita/library/Z/compare.ma
@@ -0,0 +1,143 @@
+(**************************************************************************)
+(*       ___                                                               *)
+(*      ||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".
+include "datatypes/bool.ma".
+include "datatypes/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 \lnot x=y].
+intros.elim x.
+elim y.
+simplify.reflexivity.
+simplify.apply not_eq_OZ_neg.
+simplify.apply not_eq_OZ_pos.
+elim y.
+simplify.intro.apply not_eq_OZ_neg n ?.apply sym_eq.assumption.
+simplify.apply eqb_elim.intro.simplify.apply eq_f.assumption.
+intro.simplify.intro.apply H.apply inj_neg.assumption.
+simplify.intro.apply not_eq_pos_neg n1 n ?.apply sym_eq.assumption.
+elim y.
+simplify.intro.apply not_eq_OZ_pos n ?.apply sym_eq.assumption.
+simplify.apply not_eq_pos_neg.
+simplify.apply eqb_elim.intro.simplify.apply eq_f.assumption.
+intro.simplify.intro.apply H.apply inj_pos.assumption.
+qed.
+
+theorem eqZb_elim: \forall x,y:Z.\forall P:bool \to Prop.
+(x=y \to (P true)) \to (\lnot x=y \to (P false)) \to P (eqZb x y).
+intros.
+cut 
+match (eqZb x y) with
+[ true \Rightarrow x=y
+| false \Rightarrow \lnot x=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. 
+(*CSC: qui uso le perche' altrimenti ci sono troppe scelte
+  per via delle coercions! *)
+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 (le (S n1) n)
+| EQ \Rightarrow neg n = neg n1
+| GT \Rightarrow (le (S n) n1)]. 
+apply Hcut. apply nat_compare_to_Prop. 
+elim (nat_compare n1 n).
+simplify.exact H.
+simplify.exact H.
+simplify.apply eq_f.apply sym_eq.exact H.
+simplify.exact I.
+elim y.simplify.exact I.
+simplify.exact I.
+simplify.
+(*CSC: qui uso le perche' altrimenti ci sono troppe scelte
+  per via delle coercions! *)
+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 (le (S n) n1)
+| EQ \Rightarrow pos n = pos n1
+| GT \Rightarrow (le (S n1) n)]. 
+apply Hcut. apply nat_compare_to_Prop. 
+elim (nat_compare n n1).
+simplify.exact H.
+simplify.exact H.
+simplify.apply eq_f.exact H.
+qed.