X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2FZ%2Fz.ma;h=d18c80b23ac323b996280a835a52912fc2ffc8ce;hb=4167cea65ca58897d1a3dbb81ff95de5074700cc;hp=a5e0fba8f97271cc3102a686631d71e26e5be71b;hpb=03fcee16d9c262aad38a47d0a409b684a965cc3f;p=helm.git

diff --git a/helm/matita/library/Z/z.ma b/helm/matita/library/Z/z.ma
index a5e0fba8f..d18c80b23 100644
--- a/helm/matita/library/Z/z.ma
+++ b/helm/matita/library/Z/z.ma
@@ -14,8 +14,8 @@
 
 set "baseuri" "cic:/matita/Z/z".
 
+include "datatypes/bool.ma".
 include "nat/nat.ma".
-include "higher_order_defs/functions.ma".
 
 inductive Z : Set \def
   OZ : Z
@@ -51,28 +51,20 @@ match z with
 theorem OZ_test_to_Prop :\forall z:Z.
 match OZ_test z with
 [true \Rightarrow z=OZ 
-|false \Rightarrow \lnot (z=OZ)].
+|false \Rightarrow z \neq OZ].
 intros.elim z.
 simplify.reflexivity.
-simplify.intros.
-cut match neg n with 
-[ OZ \Rightarrow True 
-| (pos n) \Rightarrow False
-| (neg n) \Rightarrow False].
-apply Hcut.rewrite > H.simplify.exact I.
-simplify.intros.
-cut match pos n with 
-[ OZ \Rightarrow True 
-| (pos n) \Rightarrow False
-| (neg n) \Rightarrow False].
-apply Hcut. rewrite > H.simplify.exact I.
+simplify. unfold Not. intros (H).
+discriminate H.
+simplify. unfold Not. intros (H).
+discriminate H.
 qed.
 
 (* discrimination *)
 theorem injective_pos: injective nat Z pos.
-simplify.
+unfold injective.
 intros.
-change with abs (pos x) = abs (pos y).
+change with (abs (pos x) = abs (pos y)).
 apply eq_f.assumption.
 qed.
 
@@ -80,66 +72,63 @@ variant inj_pos : \forall n,m:nat. pos n = pos m \to n = m
 \def injective_pos.
 
 theorem injective_neg: injective nat Z neg.
-simplify.
+unfold injective.
 intros.
-change with abs (neg x) = abs (neg y).
+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. \lnot (OZ = (pos n)).
-simplify.intros.
-change with
-  match pos n with
-  [ OZ \Rightarrow True
-  | (pos n) \Rightarrow False
-  | (neg n) \Rightarrow False].
-rewrite < H.
-simplify.exact I.
+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. \lnot (OZ = (neg n)).
-simplify.intros.
-change with
-  match neg n with
-  [ OZ \Rightarrow True
-  | (pos n) \Rightarrow False
-  | (neg n) \Rightarrow False].
-rewrite < H.
-simplify.exact I.
+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. \lnot ((pos n) = (neg m)).
-simplify.intros.
-change with
-  match neg m with
-  [ OZ \Rightarrow False
-  | (pos n) \Rightarrow True
-  | (neg n) \Rightarrow False].
-rewrite < H.
-simplify.exact I.
+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.simplify.
-elim x.elim y.
-left.reflexivity.
-right.apply not_eq_OZ_neg.
-right.apply not_eq_OZ_pos.
-elim y.right.intro.
-apply not_eq_OZ_neg n ?.apply sym_eq.assumption.
-elim (decidable_eq_nat n n1:(Or (n=n1) ((n=n1) \to False))).
-left.apply eq_f.assumption.
-right.intro.apply H.apply injective_neg.assumption.
-right.intro.apply not_eq_pos_neg n1 n ?.apply sym_eq.assumption.
-elim y.right.intro.
-apply not_eq_OZ_pos n ?.apply sym_eq.assumption.
-right.apply not_eq_pos_neg.
-elim (decidable_eq_nat n n1:(Or (n=n1) ((n=n1) \to False))).
-left.apply eq_f.assumption.
-right.intro.apply H.apply injective_pos.assumption.
+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 *)
@@ -163,16 +152,22 @@ definition Zpred \def
 | (neg n) \Rightarrow neg (S n)].
 
 theorem Zpred_Zsucc: \forall z:Z. Zpred (Zsucc z) = z.
-intros.elim z.reflexivity.
-elim n.reflexivity.
-reflexivity.
-reflexivity.
+intros.
+elim z.
+  reflexivity.
+  reflexivity.
+  elim n.
+    reflexivity.
+    reflexivity.
 qed.
 
 theorem Zsucc_Zpred: \forall z:Z. Zsucc (Zpred z) = z.
-intros.elim z.reflexivity.
-reflexivity.
-elim n.reflexivity.
-reflexivity.
+intros.
+elim z.
+  reflexivity.
+  elim n.
+    reflexivity.
+    reflexivity.
+  reflexivity.
 qed.