New version of the library.

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

diff --git a/helm/matita/library/Z/z.ma b/helm/matita/library/Z/z.ma
index 6ba305d98c05d2b8b50468f47dee83f5b8b41023..f59d7b369f8607454b24ebc4c3f74c617f2e65ed 100644 (file)
--- a/helm/matita/library/Z/z.ma
+++ b/helm/matita/library/Z/z.ma
@@ -14,8 +14,9 @@
 
 set "baseuri" "cic:/matita/Z/".
 
-include "nat/nat.ma".
-include "datatypes/bool.ma".
+include "nat/compare.ma".
+include "nat/minus.ma".
+include "higher_order_defs/functions.ma".
 
 inductive Z : Set \def
   OZ : Z
@@ -34,23 +35,26 @@ 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.
+theorem OZ_test_to_Prop :\forall z:Z.
+match OZ_test z with
+[true \Rightarrow eq Z z OZ 
+|false \Rightarrow Not (eq Z z OZ)].
+intros.elim z.
+simplify.reflexivity.
 simplify.intros.
 cut match neg e1 with 
 [ OZ \Rightarrow True 
@@ -83,21 +87,22 @@ 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.
+theorem Zpred_Zsucc: \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.
+theorem Zsucc_Zpred: \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
+definition Zplus :Z \to Z \to Z \def
+\lambda x,y.
   match x with
     [ OZ \Rightarrow y
     | (pos m) \Rightarrow
@@ -119,32 +124,34 @@ let rec Zplus x y : Z \def
                 | 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.
+theorem Zplus_z_OZ:  \forall z:Z. eq Z (Zplus 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. eq Z (Zplus x y) (Zplus y x).
-intros.elim x.simplify.rewrite > Zplus_z_O.reflexivity.
+intros.elim x.rewrite > Zplus_z_OZ.reflexivity.
 elim y.simplify.reflexivity.
 simplify.
 rewrite < sym_plus.reflexivity.
 simplify.
-rewrite > nat_compare_invert.
+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_invert.
+simplify.rewrite > nat_compare_n_m_m_n.
 simplify.elim nat_compare ? ?.simplify.reflexivity.
 simplify. reflexivity.
 simplify. reflexivity.
-simplify.elim (sym_plus ? ?).reflexivity.
+simplify.rewrite < sym_plus.reflexivity.
 qed.
 
-theorem Zpred_neg : \forall z:Z. eq Z (Zpred z) (Zplus (neg O) z).
+theorem Zpred_Zplus_neg_O : \forall z:Z. eq Z (Zpred z) (Zplus (neg O) z).
 intros.elim z.
 simplify.reflexivity.
 simplify.reflexivity.
@@ -152,7 +159,7 @@ elim e2.simplify.reflexivity.
 simplify.reflexivity.
 qed.
 
-theorem Zsucc_pos : \forall z:Z. eq Z (Zsucc z) (Zplus (pos O) z).
+theorem Zsucc_Zplus_pos_O : \forall z:Z. eq Z (Zsucc z) (Zplus (pos O) z).
 intros.elim z.
 simplify.reflexivity.
 elim e1.simplify.reflexivity.
@@ -160,7 +167,7 @@ simplify.reflexivity.
 simplify.reflexivity.
 qed.
 
-theorem Zplus_succ_pred_pp :
+theorem Zplus_pos_pos:
 \forall n,m. eq Z (Zplus (pos n) (pos m)) (Zplus (Zsucc (pos n)) (Zpred (pos m))).
 intros.
 elim n.elim m.
@@ -173,12 +180,12 @@ simplify.
 rewrite < plus_n_Sm.reflexivity.
 qed.
 
-theorem Zplus_succ_pred_pn :
+theorem Zplus_pos_neg:
 \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 :
+theorem Zplus_neg_pos :
 \forall n,m. eq Z (Zplus (neg n) (pos m)) (Zplus (Zsucc (neg n)) (Zpred (pos m))).
 intros.
 elim n.elim m.
@@ -189,7 +196,7 @@ simplify.reflexivity.
 simplify.reflexivity.
 qed.
 
-theorem Zplus_succ_pred_nn:
+theorem Zplus_neg_neg:
 \forall n,m. eq Z (Zplus (neg n) (neg m)) (Zplus (Zsucc (neg n)) (Zpred (neg m))).
 intros.
 elim n.elim m.
@@ -200,32 +207,34 @@ simplify.rewrite < plus_n_Sm.reflexivity.
 simplify.rewrite > plus_n_Sm.reflexivity.
 qed.
 
-theorem Zplus_succ_pred:
+theorem Zplus_Zsucc_Zpred:
 \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.
+rewrite < Zsucc_Zplus_pos_O.
+rewrite > Zsucc_Zpred.reflexivity.
 elim y.rewrite < sym_Zplus.rewrite < sym_Zplus (Zpred OZ).
-rewrite < Zpred_neg.rewrite > Zpred_succ.
+rewrite < Zpred_Zplus_neg_O.
+rewrite > Zpred_Zsucc.
 simplify.reflexivity.
-rewrite < Zplus_succ_pred_nn.reflexivity.
-apply Zplus_succ_pred_np.
+rewrite < Zplus_neg_neg.reflexivity.
+apply Zplus_neg_pos.
 elim y.simplify.reflexivity.
-apply Zplus_succ_pred_pn.
-apply Zplus_succ_pred_pp.
+apply Zplus_pos_neg.
+apply Zplus_pos_pos.
 qed.
 
-theorem Zsucc_plus_pp : 
+theorem Zplus_Zsucc_pos_pos : 
 \forall n,m. eq Z (Zplus (Zsucc (pos n)) (pos m)) (Zsucc (Zplus (pos n) (pos m))).
 intros.reflexivity.
 qed.
 
-theorem Zsucc_plus_pn : 
+theorem Zplus_Zsucc_pos_neg: 
 \forall n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m))).
 intros.
-apply nat_double_ind
+apply nat_elim2
 (\lambda n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m)))).intro.
 intros.elim n1.
 simplify. reflexivity.
@@ -235,14 +244,14 @@ intros. elim n1.
 simplify. reflexivity.
 simplify.reflexivity.
 intros.
-rewrite < (Zplus_succ_pred_pn ? m1).
+rewrite < (Zplus_pos_neg ? m1).
 elim H.reflexivity.
 qed.
 
-theorem Zsucc_plus_nn : 
+theorem Zplus_Zsucc_neg_neg : 
 \forall n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m))).
 intros.
-apply nat_double_ind
+apply nat_elim2
 (\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m)))).intro.
 intros.elim n1.
 simplify. reflexivity.
@@ -252,14 +261,14 @@ intros. elim n1.
 simplify. reflexivity.
 simplify.reflexivity.
 intros.
-rewrite < (Zplus_succ_pred_nn ? m1).
+rewrite < (Zplus_neg_neg ? m1).
 reflexivity.
 qed.
 
-theorem Zsucc_plus_np : 
+theorem Zplus_Zsucc_neg_pos: 
 \forall n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m))).
 intros.
-apply nat_double_ind
+apply nat_elim2
 (\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m)))).
 intros.elim n1.
 simplify. reflexivity.
@@ -270,53 +279,76 @@ simplify. reflexivity.
 simplify.reflexivity.
 intros.
 rewrite < H.
-rewrite < (Zplus_succ_pred_np ? (S m1)).
+rewrite < (Zplus_neg_pos ? (S m1)).
 reflexivity.
 qed.
 
-
-theorem Zsucc_plus : \forall x,y:Z. eq Z (Zplus (Zsucc x) y) (Zsucc (Zplus x y)).
+theorem Zplus_Zsucc : \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.
+rewrite < Zsucc_Zplus_pos_O.reflexivity.
 simplify.reflexivity.
 elim y.rewrite < sym_Zplus.rewrite < sym_Zplus OZ.simplify.reflexivity.
-apply Zsucc_plus_nn.
-apply Zsucc_plus_np.
+apply Zplus_Zsucc_neg_neg.
+apply Zplus_Zsucc_neg_pos.
 elim y.
 rewrite < sym_Zplus OZ.reflexivity.
-apply Zsucc_plus_pn.
-apply Zsucc_plus_pp.
+apply Zplus_Zsucc_pos_neg.
+apply Zplus_Zsucc_pos_pos.
 qed.
 
-theorem Zpred_plus : \forall x,y:Z. eq Z (Zplus (Zpred x) y) (Zpred (Zplus x y)).
+theorem Zplus_Zpred: \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.
+rewrite > Zplus_Zsucc.
+rewrite > Zpred_Zsucc.
+reflexivity.
+rewrite > Zsucc_Zpred.
+reflexivity.
+qed.
+
+
+theorem associative_Zplus: associative Z Zplus.
+(* change with (\forall x,y,z. eq ?(Zplus (Zplus x y) z) (Zplus x (Zplus y z))).*)
+simplify.
+intros.elim x.simplify.reflexivity.
+elim e1.rewrite < (Zpred_Zplus_neg_O (Zplus y z)).
+drop.
+rewrite < (Zpred_Zplus_neg_O y).
+rewrite < Zplus_Zpred.
 reflexivity.
-rewrite > Zsucc_pred.
+rewrite > Zplus_Zpred (neg e).
+rewrite > Zplus_Zpred (neg e).
+rewrite > Zplus_Zpred (Zplus (neg e) y).
+apply eq_f.assumption.
+elim e2.rewrite < Zsucc_Zplus_pos_O.
+rewrite < Zsucc_Zplus_pos_O.
+rewrite > Zplus_Zsucc.
 reflexivity.
+rewrite > Zplus_Zsucc (pos e1).
+rewrite > Zplus_Zsucc (pos e1).
+rewrite > Zplus_Zsucc (Zplus (pos e1) y).
+apply eq_f.assumption.
 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.
+elim e1.rewrite < (Zpred_Zplus_neg_O (Zplus y z)).
+rewrite < (Zpred_Zplus_neg_O y).
+rewrite < Zplus_Zpred.
 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.
+rewrite > Zplus_Zpred (neg e).
+rewrite > Zplus_Zpred (neg e).
+rewrite > Zplus_Zpred (Zplus (neg e) y).
+apply eq_f.assumption.
+elim e2.rewrite < Zsucc_Zplus_pos_O.
+rewrite < Zsucc_Zplus_pos_O.
+rewrite > Zplus_Zsucc.
 reflexivity.
-rewrite > Zsucc_plus (pos e1).
-rewrite > Zsucc_plus (pos e1).
-rewrite > Zsucc_plus (Zplus (pos e1) y).
-apply f_equal.assumption.
+rewrite > Zplus_Zsucc (pos e1).
+rewrite > Zplus_Zsucc (pos e1).
+rewrite > Zplus_Zsucc (Zplus (pos e1) y).
+apply eq_f.assumption.
 qed.