X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2FZ%2Fz.ma;h=d18c80b23ac323b996280a835a52912fc2ffc8ce;hb=4167cea65ca58897d1a3dbb81ff95de5074700cc;hp=cdc0e44d7c28f29fb01681a6d2b52a139a2955d0;hpb=7bbce6bc163892cfd99cfcda65db42001b86789f;p=helm.git

diff --git a/helm/matita/library/Z/z.ma b/helm/matita/library/Z/z.ma
index cdc0e44d7..d18c80b23 100644
--- a/helm/matita/library/Z/z.ma
+++ b/helm/matita/library/Z/z.ma
@@ -14,9 +14,8 @@
 
 set "baseuri" "cic:/matita/Z/z".
 
-include "nat/compare.ma".
-include "nat/minus.ma".
-include "higher_order_defs/functions.ma".
+include "datatypes/bool.ma".
+include "nat/nat.ma".
 
 inductive Z : Set \def
   OZ : Z
@@ -52,23 +51,88 @@ 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.
+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
@@ -88,273 +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.
-qed.
-
-theorem Zsucc_Zpred: \forall z:Z. Zsucc (Zpred z) = z.
-intros.elim z.reflexivity.
-reflexivity.
-elim n.reflexivity.
-reflexivity.
-qed.
-
-definition Zplus :Z \to Z \to Z \def
-\lambda x,y.
-  match x with
-    [ OZ \Rightarrow y
-    | (pos m) \Rightarrow
-        match y with
-         [ OZ \Rightarrow x
-         | (pos n) \Rightarrow (pos (pred ((S m)+(S n))))
-         | (neg n) \Rightarrow 
-              match nat_compare m n with
-                [ LT \Rightarrow (neg (pred (n-m)))
-                | EQ \Rightarrow OZ
-                | GT \Rightarrow (pos (pred (m-n)))]]
-    | (neg m) \Rightarrow
-        match y with
-         [ OZ \Rightarrow x
-         | (pos n) \Rightarrow 
-              match nat_compare m n with
-                [ LT \Rightarrow (pos (pred (n-m)))
-                | EQ \Rightarrow OZ
-                | GT \Rightarrow (neg (pred (m-n)))]     
-         | (neg n) \Rightarrow (neg (pred ((S m)+(S n))))]].
-
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "integer plus" 'plus x y = (cic:/matita/Z/z/Zplus.con x y).
-         
-theorem Zplus_z_OZ:  \forall z:Z. z+OZ = z.
-intro.elim z.
-simplify.reflexivity.
-simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-(* theorem symmetric_Zplus: symmetric Z Zplus. *)
-
-theorem sym_Zplus : \forall x,y:Z. x+y = y+x.
-intros.elim x.rewrite > Zplus_z_OZ.reflexivity.
-elim y.simplify.reflexivity.
-simplify.
-rewrite < plus_n_Sm. rewrite < plus_n_Sm.rewrite < sym_plus.reflexivity.
-simplify.
-rewrite > nat_compare_n_m_m_n.
-simplify.elim nat_compare ? ?.simplify.reflexivity.
-simplify. reflexivity.
-simplify. reflexivity.
-elim y.simplify.reflexivity.
-simplify.rewrite > nat_compare_n_m_m_n.
-simplify.elim nat_compare ? ?.simplify.reflexivity.
-simplify. reflexivity.
-simplify. reflexivity.
-simplify.rewrite < plus_n_Sm. rewrite < plus_n_Sm.rewrite < sym_plus.reflexivity.
-qed.
-
-theorem Zpred_Zplus_neg_O : \forall z:Z. Zpred z = (neg O)+z.
-intros.elim z.
-simplify.reflexivity.
-simplify.reflexivity.
-elim n.simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-theorem Zsucc_Zplus_pos_O : \forall z:Z. Zsucc z = (pos O)+z.
-intros.elim z.
-simplify.reflexivity.
-elim n.simplify.reflexivity.
-simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-theorem Zplus_pos_pos:
-\forall n,m. (pos n)+(pos m) = (Zsucc (pos n))+(Zpred (pos m)).
 intros.
-elim n.elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-elim m.
-simplify.rewrite < plus_n_Sm.
-rewrite < plus_n_O.reflexivity.
-simplify.rewrite < plus_n_Sm.
-rewrite < plus_n_Sm.reflexivity.
-qed.
-
-theorem Zplus_pos_neg:
-\forall n,m. (pos n)+(neg m) = (Zsucc (pos n))+(Zpred (neg m)).
-intros.reflexivity.
+elim z.
+  reflexivity.
+  reflexivity.
+  elim n.
+    reflexivity.
+    reflexivity.
 qed.
 
-theorem Zplus_neg_pos :
-\forall n,m. (neg n)+(pos m) = (Zsucc (neg n))+(Zpred (pos m)).
-intros.
-elim n.elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-theorem Zplus_neg_neg:
-\forall n,m. (neg n)+(neg m) = (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_Zsucc_Zpred:
-\forall x,y. x+y = (Zsucc x)+(Zpred y).
-intros.
-elim x. elim y.
-simplify.reflexivity.
-simplify.reflexivity.
-rewrite < Zsucc_Zplus_pos_O.
-rewrite > Zsucc_Zpred.reflexivity.
-elim y.rewrite < sym_Zplus.rewrite < sym_Zplus (Zpred OZ).
-rewrite < Zpred_Zplus_neg_O.
-rewrite > Zpred_Zsucc.
-simplify.reflexivity.
-rewrite < Zplus_neg_neg.reflexivity.
-apply Zplus_neg_pos.
-elim y.simplify.reflexivity.
-apply Zplus_pos_neg.
-apply Zplus_pos_pos.
-qed.
-
-theorem Zplus_Zsucc_pos_pos : 
-\forall n,m. (Zsucc (pos n))+(pos m) = Zsucc ((pos n)+(pos m)).
-intros.reflexivity.
-qed.
-
-theorem Zplus_Zsucc_pos_neg: 
-\forall n,m. (Zsucc (pos n))+(neg m) = (Zsucc ((pos n)+(neg m))).
-intros.
-apply nat_elim2
-(\lambda n,m. (Zsucc (pos n))+(neg m) = (Zsucc ((pos n)+(neg m)))).intro.
-intros.elim n1.
-simplify. reflexivity.
-elim n2.simplify. reflexivity.
-simplify. reflexivity.
-intros. elim n1.
-simplify. reflexivity.
-simplify.reflexivity.
-intros.
-rewrite < (Zplus_pos_neg ? m1).
-elim H.reflexivity.
-qed.
-
-theorem Zplus_Zsucc_neg_neg : 
-\forall n,m. (Zsucc (neg n))+(neg m) = Zsucc ((neg n)+(neg m)).
-intros.
-apply nat_elim2
-(\lambda n,m. ((Zsucc (neg n))+(neg m)) = Zsucc ((neg n)+(neg m))).intro.
-intros.elim n1.
-simplify. reflexivity.
-elim n2.simplify. reflexivity.
-simplify. reflexivity.
-intros. elim n1.
-simplify. reflexivity.
-simplify.reflexivity.
-intros.
-rewrite < (Zplus_neg_neg ? m1).
-reflexivity.
-qed.
-
-theorem Zplus_Zsucc_neg_pos: 
-\forall n,m. Zsucc (neg n)+(pos m) = Zsucc ((neg n)+(pos m)).
-intros.
-apply nat_elim2
-(\lambda n,m. (Zsucc (neg n))+(pos m) = Zsucc ((neg n)+(pos m))).
-intros.elim n1.
-simplify. reflexivity.
-elim n2.simplify. reflexivity.
-simplify. reflexivity.
-intros. elim n1.
-simplify. reflexivity.
-simplify.reflexivity.
-intros.
-rewrite < H.
-rewrite < (Zplus_neg_pos ? (S m1)).
-reflexivity.
-qed.
-
-theorem Zplus_Zsucc : \forall x,y:Z. (Zsucc x)+y = Zsucc (x+y).
-intros.elim x.elim y.
-simplify. reflexivity.
-rewrite < Zsucc_Zplus_pos_O.reflexivity.
-simplify.reflexivity.
-elim y.rewrite < sym_Zplus.rewrite < sym_Zplus OZ.simplify.reflexivity.
-apply Zplus_Zsucc_neg_neg.
-apply Zplus_Zsucc_neg_pos.
-elim y.
-rewrite < sym_Zplus OZ.reflexivity.
-apply Zplus_Zsucc_pos_neg.
-apply Zplus_Zsucc_pos_pos.
-qed.
-
-theorem Zplus_Zpred: \forall x,y:Z. (Zpred x)+y = Zpred (x+y).
+theorem Zsucc_Zpred: \forall z:Z. Zsucc (Zpred z) = z.
 intros.
-cut Zpred (x+y) = Zpred ((Zsucc (Zpred x))+y).
-rewrite > Hcut.
-rewrite > Zplus_Zsucc.
-rewrite > Zpred_Zsucc.
-reflexivity.
-rewrite > Zsucc_Zpred.
-reflexivity.
-qed.
-
-
-theorem associative_Zplus: associative Z Zplus.
-change with \forall x,y,z:Z. (x + y) + z = x + (y + z). 
-(* simplify. *)
-intros.elim x.simplify.reflexivity.
-elim n.rewrite < (Zpred_Zplus_neg_O (y+z)).
-rewrite < (Zpred_Zplus_neg_O y).
-rewrite < Zplus_Zpred.
-reflexivity.
-rewrite > Zplus_Zpred (neg n1).
-rewrite > Zplus_Zpred (neg n1).
-rewrite > Zplus_Zpred ((neg n1)+y).
-apply eq_f.assumption.
-elim n.rewrite < Zsucc_Zplus_pos_O.
-rewrite < Zsucc_Zplus_pos_O.
-rewrite > Zplus_Zsucc.
-reflexivity.
-rewrite > Zplus_Zsucc (pos n1).
-rewrite > Zplus_Zsucc (pos n1).
-rewrite > Zplus_Zsucc ((pos n1)+y).
-apply eq_f.assumption.
-qed.
-
-variant assoc_Zplus : \forall x,y,z:Z.  (x+y)+z = x+(y+z)
-\def associative_Zplus.
-
-(* Zopp *)
-definition Zopp : Z \to Z \def
-\lambda x:Z. match x with
-[ OZ \Rightarrow OZ
-| (pos n) \Rightarrow (neg n)
-| (neg n) \Rightarrow (pos n) ].
-
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "integer unary minus" 'uminus x = (cic:/matita/Z/z/Zopp.con x).
-
-theorem Zplus_Zopp: \forall x:Z. x+ -x = OZ.
-intro.elim x.
-apply refl_eq.
-simplify.
-rewrite > nat_compare_n_n.
-simplify.apply refl_eq.
-simplify.
-rewrite > nat_compare_n_n.
-simplify.apply refl_eq.
+elim z.
+  reflexivity.
+  elim n.
+    reflexivity.
+    reflexivity.
+  reflexivity.
 qed.