ocaml 3.09 transition

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

diff --git a/helm/matita/library/Z/plus.ma b/helm/matita/library/Z/plus.ma
index b1942d209f45077d6060ba5f993ccbea2041ca89..976f6cfb3ce01d379244fc66ec964a99563917d3 100644 (file)
--- a/helm/matita/library/Z/plus.ma
+++ b/helm/matita/library/Z/plus.ma
@@ -15,7 +15,6 @@
 set "baseuri" "cic:/matita/Z/plus".
 
 include "Z/z.ma".
-include "nat/compare.ma".
 include "nat/minus.ma".
 
 definition Zplus :Z \to Z \to Z \def
@@ -30,7 +29,7 @@ definition Zplus :Z \to Z \to Z \def
               match nat_compare m n with
                 [ LT \Rightarrow (neg (pred (n-m)))
                 | EQ \Rightarrow OZ
-                | GT \Rightarrow (pos (pred (m-n)))]]
+                | GT \Rightarrow (pos (pred (m-n)))] ]
     | (neg m) \Rightarrow
         match y with
          [ OZ \Rightarrow x
@@ -39,7 +38,7 @@ definition Zplus :Z \to Z \to Z \def
                 [ LT \Rightarrow (pos (pred (n-m)))
                 | EQ \Rightarrow OZ
                 | GT \Rightarrow (neg (pred (m-n)))]     
-         | (neg n) \Rightarrow (neg (pred ((S m)+(S 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/plus/Zplus.con x y).
@@ -60,12 +59,12 @@ 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.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.elim nat_compare.simplify.reflexivity.
 simplify. reflexivity.
 simplify. reflexivity.
 simplify.rewrite < plus_n_Sm. rewrite < plus_n_Sm.rewrite < sym_plus.reflexivity.
@@ -73,18 +72,20 @@ 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.
+  simplify.reflexivity.
+  elim n.
+    simplify.reflexivity.
+    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.
+  simplify.reflexivity.
+  simplify.reflexivity.
+  elim n.
+    simplify.reflexivity.
+    simplify.reflexivity.
 qed.
 
 theorem Zplus_pos_pos:
@@ -129,21 +130,20 @@ 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.
+intros.elim x. 
+  elim y.
+    simplify.reflexivity.
+    rewrite < Zsucc_Zplus_pos_O.rewrite > Zsucc_Zpred.reflexivity.
+    simplify.reflexivity.
+  elim y.
+    simplify.reflexivity.
+    apply Zplus_pos_pos.
+    apply Zplus_pos_neg.
+  elim y.
+    rewrite < sym_Zplus.rewrite < (sym_Zplus (Zpred OZ)).
+     rewrite < Zpred_Zplus_neg_O.rewrite > Zpred_Zsucc.simplify.reflexivity.
+    apply Zplus_neg_pos.
+    rewrite < Zplus_neg_neg.reflexivity.
 qed.
 
 theorem Zplus_Zsucc_pos_pos : 
@@ -154,8 +154,8 @@ 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.
+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.
@@ -169,10 +169,10 @@ elim H.reflexivity.
 qed.
 
 theorem Zplus_Zsucc_neg_neg : 
-\forall n,m. (Zsucc (neg n))+(neg m) = Zsucc ((neg n)+(neg m)).
+\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.
+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.
@@ -188,8 +188,8 @@ 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))).
+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.
@@ -204,22 +204,24 @@ 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.
+intros.elim x.
+  elim y.
+    simplify. reflexivity.
+    simplify.reflexivity.
+    rewrite < Zsucc_Zplus_pos_O.reflexivity.
+  elim y.
+    rewrite < (sym_Zplus OZ).reflexivity.
+    apply Zplus_Zsucc_pos_pos.
+    apply Zplus_Zsucc_pos_neg.
+  elim y.
+    rewrite < sym_Zplus.rewrite < (sym_Zplus OZ).simplify.reflexivity.
+    apply Zplus_Zsucc_neg_pos.
+    apply Zplus_Zsucc_neg_neg.
 qed.
 
 theorem Zplus_Zpred: \forall x,y:Z. (Zpred x)+y = Zpred (x+y).
 intros.
-cut Zpred (x+y) = Zpred ((Zsucc (Zpred x))+y).
+cut (Zpred (x+y) = Zpred ((Zsucc (Zpred x))+y)).
 rewrite > Hcut.
 rewrite > Zplus_Zsucc.
 rewrite > Zpred_Zsucc.
@@ -230,25 +232,20 @@ qed.
 
 
 theorem associative_Zplus: associative Z Zplus.
-change with \forall x,y,z:Z. (x + y) + z = x + (y + z). 
+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.
+intros.elim x.
+  simplify.reflexivity.
+  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.
+  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.
 qed.
 
 variant assoc_Zplus : \forall x,y,z:Z.  (x+y)+z = x+(y+z)
@@ -286,8 +283,7 @@ intro.simplify.reflexivity.
 simplify.reflexivity.
 qed.
 
-(* --x non gli piace, ma lo stampa *)
-theorem Zopp_Zopp: \forall x:Z. -(-x) = x.
+theorem Zopp_Zopp: \forall x:Z. --x = x.
 intro. elim x.
 reflexivity.reflexivity.reflexivity.
 qed.