X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2FZ%2Fz.ma;h=d18c80b23ac323b996280a835a52912fc2ffc8ce;hb=4167cea65ca58897d1a3dbb81ff95de5074700cc;hp=6ba305d98c05d2b8b50468f47dee83f5b8b41023;hpb=f263e4ec717d5ec2e7f9c057855f8223f81baae8;p=helm.git

diff --git a/helm/matita/library/Z/z.ma b/helm/matita/library/Z/z.ma
index 6ba305d98..d18c80b23 100644
--- a/helm/matita/library/Z/z.ma
+++ b/helm/matita/library/Z/z.ma
@@ -12,10 +12,10 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/Z/".
+set "baseuri" "cic:/matita/Z/z".
 
-include "nat/nat.ma".
 include "datatypes/bool.ma".
+include "nat/nat.ma".
 
 inductive Z : Set \def
   OZ : Z
@@ -34,37 +34,105 @@ definition neg_Z_of_nat \def
 [ O \Rightarrow  OZ 
 | (S n)\Rightarrow  neg n].
 
-definition absZ \def
+definition abs \def
 \lambda z.
  match z with 
 [ OZ \Rightarrow O
 | (pos n) \Rightarrow n
 | (neg n) \Rightarrow n].
 
-definition OZ_testb \def
+definition OZ_test \def
 \lambda z.
 match z with 
 [ OZ \Rightarrow true
 | (pos n) \Rightarrow false
 | (neg n) \Rightarrow false].
 
-theorem OZ_discr :
-\forall z. if_then_else (OZ_testb z) (eq Z z OZ) (Not (eq Z z OZ)). 
-intros.elim z.simplify.reflexivity.
-simplify.intros.
-cut match neg e1 with 
-[ OZ \Rightarrow True 
-| (pos n) \Rightarrow False
-| (neg n) \Rightarrow False].
-apply Hcut.rewrite > H.simplify.exact I.
-simplify.intros.
-cut match pos e2 with 
-[ OZ \Rightarrow True 
-| (pos n) \Rightarrow False
-| (neg n) \Rightarrow False].
-apply Hcut. rewrite > H.simplify.exact I.
+theorem OZ_test_to_Prop :\forall z:Z.
+match OZ_test z with
+[true \Rightarrow z=OZ 
+|false \Rightarrow z \neq OZ].
+intros.elim z.
+simplify.reflexivity.
+simplify. unfold Not. intros (H).
+discriminate H.
+simplify. unfold Not. intros (H).
+discriminate H.
+qed.
+
+(* discrimination *)
+theorem injective_pos: injective nat Z pos.
+unfold injective.
+intros.
+change with (abs (pos x) = abs (pos y)).
+apply eq_f.assumption.
 qed.
 
+variant inj_pos : \forall n,m:nat. pos n = pos m \to n = m
+\def injective_pos.
+
+theorem injective_neg: injective nat Z neg.
+unfold injective.
+intros.
+change with (abs (neg x) = abs (neg y)).
+apply eq_f.assumption.
+qed.
+
+variant inj_neg : \forall n,m:nat. neg n = neg m \to n = m
+\def injective_neg.
+
+theorem not_eq_OZ_pos: \forall n:nat. OZ \neq pos n.
+unfold Not.intros (n H).
+discriminate H.
+qed.
+
+theorem not_eq_OZ_neg :\forall n:nat. OZ \neq neg n.
+unfold Not.intros (n H).
+discriminate H.
+qed.
+
+theorem not_eq_pos_neg :\forall n,m:nat. pos n \neq neg m.
+unfold Not.intros (n m H).
+discriminate H.
+qed.
+
+theorem decidable_eq_Z : \forall x,y:Z. decidable (x=y).
+intros.unfold decidable.
+elim x.
+(* goal: x=OZ *)
+  elim y.
+  (* goal: x=OZ y=OZ *)
+    left.reflexivity.
+  (* goal: x=OZ 2=2 *)
+    right.apply not_eq_OZ_pos.
+  (* goal: x=OZ 2=3 *)
+    right.apply not_eq_OZ_neg.
+(* goal: x=pos *)
+  elim y.
+  (* goal: x=pos y=OZ *)
+    right.unfold Not.intro.
+    apply (not_eq_OZ_pos n). symmetry. assumption.
+  (* goal: x=pos y=pos *)
+    elim (decidable_eq_nat n n1:((n=n1) \lor ((n=n1) \to False))).
+    left.apply eq_f.assumption.
+    right.unfold Not.intros (H_inj).apply H. injection H_inj. assumption.
+  (* goal: x=pos y=neg *)
+    right.unfold Not.intro.apply (not_eq_pos_neg n n1). assumption.
+(* goal: x=neg *)
+  elim y.
+  (* goal: x=neg y=OZ *)
+    right.unfold Not.intro.
+    apply (not_eq_OZ_neg n). symmetry. assumption.
+  (* goal: x=neg y=pos *)
+    right. unfold Not.intro. apply (not_eq_pos_neg n1 n). symmetry. assumption.
+  (* goal: x=neg y=neg *)
+    elim (decidable_eq_nat n n1:((n=n1) \lor ((n=n1) \to False))).
+    left.apply eq_f.assumption.
+    right.unfold Not.intro.apply H.apply injective_neg.assumption.
+qed.
+
+(* end discrimination *)
+
 definition Zsucc \def
 \lambda z. match z with
 [ OZ \Rightarrow pos O
@@ -83,240 +151,23 @@ definition Zpred \def
 	  | (S p) \Rightarrow pos p]
 | (neg n) \Rightarrow neg (S n)].
 
-theorem Zpred_succ: \forall z:Z. eq Z (Zpred (Zsucc z)) z.
-intros.elim z.reflexivity.
-elim e1.reflexivity.
-reflexivity.
-reflexivity.
-qed.
-
-theorem Zsucc_pred: \forall z:Z. eq Z (Zsucc (Zpred z)) z.
-intros.elim z.reflexivity.
-reflexivity.
-elim e2.reflexivity.
-reflexivity.
-qed.
-
-let rec Zplus x y : Z \def
-  match x with
-    [ OZ \Rightarrow y
-    | (pos m) \Rightarrow
-        match y with
-         [ OZ \Rightarrow x
-         | (pos n) \Rightarrow (pos (S (plus m n)))
-         | (neg n) \Rightarrow 
-              match nat_compare m n with
-                [ LT \Rightarrow (neg (pred (minus n m)))
-                | EQ \Rightarrow OZ
-                | GT \Rightarrow (pos (pred (minus m n)))]]
-    | (neg m) \Rightarrow
-        match y with
-         [ OZ \Rightarrow x
-         | (pos n) \Rightarrow 
-              match nat_compare m n with
-                [ LT \Rightarrow (pos (pred (minus n m)))
-                | EQ \Rightarrow OZ
-                | GT \Rightarrow (neg (pred (minus m n)))]     
-         | (neg n) \Rightarrow (neg (S (plus m n)))]].
-         
-theorem Zplus_z_O:  \forall z:Z. eq Z (Zplus z OZ) z.
-intro.elim z.
-simplify.reflexivity.
-simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-theorem sym_Zplus : \forall x,y:Z. eq Z (Zplus x y) (Zplus y x).
-intros.elim x.simplify.rewrite > Zplus_z_O.reflexivity.
-elim y.simplify.reflexivity.
-simplify.
-rewrite < sym_plus.reflexivity.
-simplify.
-rewrite > nat_compare_invert.
-simplify.elim nat_compare ? ?.simplify.reflexivity.
-simplify. reflexivity.
-simplify. reflexivity.
-elim y.simplify.reflexivity.
-simplify.rewrite > nat_compare_invert.
-simplify.elim nat_compare ? ?.simplify.reflexivity.
-simplify. reflexivity.
-simplify. reflexivity.
-simplify.elim (sym_plus ? ?).reflexivity.
-qed.
-
-theorem Zpred_neg : \forall z:Z. eq Z (Zpred z) (Zplus (neg O) z).
-intros.elim z.
-simplify.reflexivity.
-simplify.reflexivity.
-elim e2.simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-theorem Zsucc_pos : \forall z:Z. eq Z (Zsucc z) (Zplus (pos O) z).
-intros.elim z.
-simplify.reflexivity.
-elim e1.simplify.reflexivity.
-simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-theorem Zplus_succ_pred_pp :
-\forall n,m. eq Z (Zplus (pos n) (pos m)) (Zplus (Zsucc (pos n)) (Zpred (pos m))).
+theorem Zpred_Zsucc: \forall z:Z. Zpred (Zsucc z) = z.
 intros.
-elim n.elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-elim m.
-simplify.
-rewrite < plus_n_O.reflexivity.
-simplify.
-rewrite < plus_n_Sm.reflexivity.
+elim z.
+  reflexivity.
+  reflexivity.
+  elim n.
+    reflexivity.
+    reflexivity.
 qed.
 
-theorem Zplus_succ_pred_pn :
-\forall n,m. eq Z (Zplus (pos n) (neg m)) (Zplus (Zsucc (pos n)) (Zpred (neg m))).
-intros.reflexivity.
-qed.
-
-theorem Zplus_succ_pred_np :
-\forall n,m. eq Z (Zplus (neg n) (pos m)) (Zplus (Zsucc (neg n)) (Zpred (pos m))).
+theorem Zsucc_Zpred: \forall z:Z. Zsucc (Zpred z) = z.
 intros.
-elim n.elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-theorem Zplus_succ_pred_nn:
-\forall n,m. eq Z (Zplus (neg n) (neg m)) (Zplus (Zsucc (neg n)) (Zpred (neg m))).
-intros.
-elim n.elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-elim m.
-simplify.rewrite < plus_n_Sm.reflexivity.
-simplify.rewrite > plus_n_Sm.reflexivity.
-qed.
-
-theorem Zplus_succ_pred:
-\forall x,y. eq Z (Zplus x y) (Zplus (Zsucc x) (Zpred y)).
-intros.
-elim x. elim y.
-simplify.reflexivity.
-simplify.reflexivity.
-rewrite < Zsucc_pos.rewrite > Zsucc_pred.reflexivity.
-elim y.rewrite < sym_Zplus.rewrite < sym_Zplus (Zpred OZ).
-rewrite < Zpred_neg.rewrite > Zpred_succ.
-simplify.reflexivity.
-rewrite < Zplus_succ_pred_nn.reflexivity.
-apply Zplus_succ_pred_np.
-elim y.simplify.reflexivity.
-apply Zplus_succ_pred_pn.
-apply Zplus_succ_pred_pp.
-qed.
-
-theorem Zsucc_plus_pp : 
-\forall n,m. eq Z (Zplus (Zsucc (pos n)) (pos m)) (Zsucc (Zplus (pos n) (pos m))).
-intros.reflexivity.
-qed.
-
-theorem Zsucc_plus_pn : 
-\forall n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m))).
-intros.
-apply nat_double_ind
-(\lambda n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m)))).intro.
-intros.elim n1.
-simplify. reflexivity.
-elim e1.simplify. reflexivity.
-simplify. reflexivity.
-intros. elim n1.
-simplify. reflexivity.
-simplify.reflexivity.
-intros.
-rewrite < (Zplus_succ_pred_pn ? m1).
-elim H.reflexivity.
-qed.
-
-theorem Zsucc_plus_nn : 
-\forall n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m))).
-intros.
-apply nat_double_ind
-(\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m)))).intro.
-intros.elim n1.
-simplify. reflexivity.
-elim e1.simplify. reflexivity.
-simplify. reflexivity.
-intros. elim n1.
-simplify. reflexivity.
-simplify.reflexivity.
-intros.
-rewrite < (Zplus_succ_pred_nn ? m1).
-reflexivity.
+elim z.
+  reflexivity.
+  elim n.
+    reflexivity.
+    reflexivity.
+  reflexivity.
 qed.
 
-theorem Zsucc_plus_np : 
-\forall n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m))).
-intros.
-apply nat_double_ind
-(\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m)))).
-intros.elim n1.
-simplify. reflexivity.
-elim e1.simplify. reflexivity.
-simplify. reflexivity.
-intros. elim n1.
-simplify. reflexivity.
-simplify.reflexivity.
-intros.
-rewrite < H.
-rewrite < (Zplus_succ_pred_np ? (S m1)).
-reflexivity.
-qed.
-
-
-theorem Zsucc_plus : \forall x,y:Z. eq Z (Zplus (Zsucc x) y) (Zsucc (Zplus x y)).
-intros.elim x.elim y.
-simplify. reflexivity.
-rewrite < Zsucc_pos.reflexivity.
-simplify.reflexivity.
-elim y.rewrite < sym_Zplus.rewrite < sym_Zplus OZ.simplify.reflexivity.
-apply Zsucc_plus_nn.
-apply Zsucc_plus_np.
-elim y.
-rewrite < sym_Zplus OZ.reflexivity.
-apply Zsucc_plus_pn.
-apply Zsucc_plus_pp.
-qed.
-
-theorem Zpred_plus : \forall x,y:Z. eq Z (Zplus (Zpred x) y) (Zpred (Zplus x y)).
-intros.
-cut eq Z (Zpred (Zplus x y)) (Zpred (Zplus (Zsucc (Zpred x)) y)).
-rewrite > Hcut.
-rewrite > Zsucc_plus.
-rewrite > Zpred_succ.
-reflexivity.
-rewrite > Zsucc_pred.
-reflexivity.
-qed.
-
-theorem assoc_Zplus : 
-\forall x,y,z:Z. eq Z (Zplus x (Zplus y z)) (Zplus (Zplus x y) z).
-intros.elim x.simplify.reflexivity.
-elim e1.rewrite < (Zpred_neg (Zplus y z)).
-rewrite < (Zpred_neg y).
-rewrite < Zpred_plus.
-reflexivity.
-rewrite > Zpred_plus (neg e).
-rewrite > Zpred_plus (neg e).
-rewrite > Zpred_plus (Zplus (neg e) y).
-apply f_equal.assumption.
-elim e2.rewrite < Zsucc_pos.
-rewrite < Zsucc_pos.
-rewrite > Zsucc_plus.
-reflexivity.
-rewrite > Zsucc_plus (pos e1).
-rewrite > Zsucc_plus (pos e1).
-rewrite > Zsucc_plus (Zplus (pos e1) y).
-apply f_equal.assumption.
-qed.