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+(**************************************************************************)
+(*       __                                                               *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
+(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/Z/times".
+
+include "nat/lt_arith.ma".
+include "Z/plus.ma".
+
+definition Ztimes :Z \to Z \to Z \def
+\lambda x,y.
+  match x with
+    [ OZ \Rightarrow OZ
+    | (pos m) \Rightarrow
+        match y with
+         [ OZ \Rightarrow OZ
+         | (pos n) \Rightarrow (pos (pred ((S m) * (S n))))
+         | (neg n) \Rightarrow (neg (pred ((S m) * (S n))))]
+    | (neg m) \Rightarrow
+        match y with
+         [ OZ \Rightarrow OZ
+         | (pos n) \Rightarrow (neg (pred ((S m) * (S n))))
+         | (neg n) \Rightarrow (pos (pred ((S m) * (S n))))]].
+         
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "integer times" 'times x y = (cic:/matita/Z/times/Ztimes.con x y).
+
+theorem Ztimes_z_OZ:  \forall z:Z. z*OZ = OZ.
+intro.elim z.
+simplify.reflexivity.
+simplify.reflexivity.
+simplify.reflexivity.
+qed.
+
+theorem Ztimes_neg_Zopp: \forall n:nat.\forall x:Z.
+neg n * x = - (pos n * x).
+intros.elim x.
+simplify.reflexivity.
+simplify.reflexivity.
+simplify.reflexivity.
+qed.
+theorem symmetric_Ztimes : symmetric Z Ztimes.
+change with (\forall x,y:Z. x*y = y*x).
+intros.elim x.rewrite > Ztimes_z_OZ.reflexivity.
+elim y.simplify.reflexivity. 
+change with (pos (pred ((S n) * (S n1))) = pos (pred ((S n1) * (S n)))).
+rewrite < sym_times.reflexivity.
+change with (neg (pred ((S n) * (S n1))) = neg (pred ((S n1) * (S n)))).
+rewrite < sym_times.reflexivity.
+elim y.simplify.reflexivity.
+change with (neg (pred ((S n) * (S n1))) = neg (pred ((S n1) * (S n)))).
+rewrite < sym_times.reflexivity.
+change with (pos (pred ((S n) * (S n1))) = pos (pred ((S n1) * (S n)))).
+rewrite < sym_times.reflexivity.
+qed.
+
+variant sym_Ztimes : \forall x,y:Z. x*y = y*x
+\def symmetric_Ztimes.
+
+theorem associative_Ztimes: associative Z Ztimes.
+change with (\forall x,y,z:Z. (x*y)*z = x*(y*z)).
+intros.elim x.
+  simplify.reflexivity. 
+  elim y.
+    simplify.reflexivity.
+    elim z.
+      simplify.reflexivity.
+      change with 
+       (pos (pred ((S (pred ((S n) * (S n1)))) * (S n2))) =
+       pos (pred ((S n) * (S (pred ((S n1) * (S n2))))))).
+        rewrite < S_pred.rewrite < S_pred.rewrite < assoc_times.reflexivity.
+         apply lt_O_times_S_S.apply lt_O_times_S_S.
+      change with 
+       (neg (pred ((S (pred ((S n) * (S n1)))) * (S n2))) =
+       neg (pred ((S n) * (S (pred ((S n1) * (S n2))))))).
+        rewrite < S_pred.rewrite < S_pred.rewrite < assoc_times.reflexivity.
+         apply lt_O_times_S_S.apply lt_O_times_S_S.
+    elim z.
+      simplify.reflexivity.
+      change with 
+       (neg (pred ((S (pred ((S n) * (S n1)))) * (S n2))) =
+       neg (pred ((S n) * (S (pred ((S n1) * (S n2))))))).
+        rewrite < S_pred.rewrite < S_pred.rewrite < assoc_times.reflexivity.
+         apply lt_O_times_S_S.apply lt_O_times_S_S.
+      change with 
+       (pos (pred ((S (pred ((S n) * (S n1)))) * (S n2))) =
+       pos(pred ((S n) * (S (pred ((S n1) * (S n2))))))).
+        rewrite < S_pred.rewrite < S_pred.rewrite < assoc_times.reflexivity.
+        apply lt_O_times_S_S.apply lt_O_times_S_S.
+  elim y.
+    simplify.reflexivity.
+    elim z.
+      simplify.reflexivity.
+      change with 
+       (neg (pred ((S (pred ((S n) * (S n1)))) * (S n2))) =
+       neg (pred ((S n) * (S (pred ((S n1) * (S n2))))))).
+        rewrite < S_pred.rewrite < S_pred.rewrite < assoc_times.reflexivity.
+         apply lt_O_times_S_S.apply lt_O_times_S_S.
+      change with 
+       (pos (pred ((S (pred ((S n) * (S n1)))) * (S n2))) =
+       pos (pred ((S n) * (S (pred ((S n1) * (S n2))))))).
+        rewrite < S_pred.rewrite < S_pred.rewrite < assoc_times.reflexivity.
+         apply lt_O_times_S_S.apply lt_O_times_S_S.
+    elim z.
+      simplify.reflexivity.
+      change with 
+       (pos (pred ((S (pred ((S n) * (S n1)))) * (S n2))) =
+       pos (pred ((S n) * (S (pred ((S n1) * (S n2))))))).
+        rewrite < S_pred.rewrite < S_pred.rewrite < assoc_times.reflexivity.
+         apply lt_O_times_S_S.apply lt_O_times_S_S.
+      change with 
+       (neg (pred ((S (pred ((S n) * (S n1)))) * (S n2))) =
+       neg(pred ((S n) * (S (pred ((S n1) * (S n2))))))).
+        rewrite < S_pred.rewrite < S_pred.rewrite < assoc_times.reflexivity.
+         apply lt_O_times_S_S.apply lt_O_times_S_S.
+qed.
+
+variant assoc_Ztimes : \forall x,y,z:Z. 
+(x * y) * z = x * (y * z) \def 
+associative_Ztimes.
+
+lemma times_minus1: \forall n,p,q:nat. lt q p \to
+(S n) * (S (pred ((S p) - (S q)))) =
+pred ((S n) * (S p)) - pred ((S n) * (S q)).
+intros.
+rewrite < S_pred.  
+rewrite > minus_pred_pred.
+rewrite < distr_times_minus. 
+reflexivity.
+(* we now close all positivity conditions *)
+apply lt_O_times_S_S.                    
+apply lt_O_times_S_S.
+simplify.unfold lt.
+apply le_SO_minus.  exact H.
+qed.
+
+lemma Ztimes_Zplus_pos_neg_pos: \forall n,p,q:nat.
+(pos n)*((neg p)+(pos q)) = (pos n)*(neg p)+ (pos n)*(pos q). 
+intros.
+simplify. 
+change in match (p + n * (S p)) with (pred ((S n) * (S p))).
+change in match (q + n * (S q)) with (pred ((S n) * (S q))).
+rewrite < nat_compare_pred_pred.
+rewrite < nat_compare_times_l.
+rewrite < nat_compare_S_S.
+apply (nat_compare_elim p q).
+intro.
+(* uff *)
+change with (pos (pred ((S n) * (S (pred ((S q) - (S p)))))) =
+            pos (pred ((pred ((S n) * (S q))) - (pred ((S n) * (S p)))))).
+rewrite < (times_minus1 n q p H).reflexivity.
+intro.rewrite < H.simplify.reflexivity.
+intro.
+change with (neg (pred ((S n) * (S (pred ((S p) - (S q)))))) =
+            neg (pred ((pred ((S n) * (S p))) - (pred ((S n) * (S q)))))). 
+rewrite < (times_minus1 n p q H).reflexivity.                                 
+(* two more positivity conditions from nat_compare_pred_pred *)   
+apply lt_O_times_S_S.  
+apply lt_O_times_S_S. 
+qed. 
+
+lemma Ztimes_Zplus_pos_pos_neg: \forall n,p,q:nat.
+(pos n)*((pos p)+(neg q)) = (pos n)*(pos p)+ (pos n)*(neg q).
+intros.
+rewrite < sym_Zplus.
+rewrite > Ztimes_Zplus_pos_neg_pos.
+apply sym_Zplus.
+qed.
+
+lemma distributive2_Ztimes_pos_Zplus: 
+distributive2 nat Z (\lambda n,z. (pos n) * z) Zplus.
+change with (\forall n,y,z.
+(pos n) * (y + z) = (pos n) * y + (pos n) * z).  
+intros.elim y.
+  reflexivity.
+  elim z.
+    reflexivity.
+    change with
+     (pos (pred ((S n) * ((S n1) + (S n2)))) =
+     pos (pred ((S n) * (S n1) + (S n) * (S n2)))).
+      rewrite < distr_times_plus.reflexivity.
+    apply Ztimes_Zplus_pos_pos_neg.
+  elim z.
+    reflexivity.
+    apply Ztimes_Zplus_pos_neg_pos.
+    change with
+     (neg (pred ((S n) * ((S n1) + (S n2)))) =
+     neg (pred ((S n) * (S n1) + (S n) * (S n2)))).
+    rewrite < distr_times_plus.reflexivity.
+qed.
+
+variant distr_Ztimes_Zplus_pos: \forall n,y,z.
+(pos n) * (y + z) = ((pos n) * y + (pos n) * z) \def
+distributive2_Ztimes_pos_Zplus.
+
+lemma distributive2_Ztimes_neg_Zplus : 
+distributive2 nat Z (\lambda n,z. (neg n) * z) Zplus.
+change with (\forall n,y,z.
+(neg n) * (y + z) = (neg n) * y + (neg n) * z).  
+intros.
+rewrite > Ztimes_neg_Zopp. 
+rewrite > distr_Ztimes_Zplus_pos.
+rewrite > Zopp_Zplus.
+rewrite < Ztimes_neg_Zopp. rewrite < Ztimes_neg_Zopp.
+reflexivity.
+qed.
+
+variant distr_Ztimes_Zplus_neg: \forall n,y,z.
+(neg n) * (y + z) = (neg n) * y + (neg n) * z \def
+distributive2_Ztimes_neg_Zplus.
+
+theorem distributive_Ztimes_Zplus: distributive Z Ztimes Zplus.
+change with (\forall x,y,z:Z. x * (y + z) = x*y + x*z).
+intros.elim x.
+(* case x = OZ *)
+simplify.reflexivity.
+(* case x = pos n *)
+apply distr_Ztimes_Zplus_pos.
+(* case x = neg n *)
+apply distr_Ztimes_Zplus_neg.
+qed.
+
+variant distr_Ztimes_Zplus: \forall x,y,z.
+x * (y + z) = x*y + x*z \def
+distributive_Ztimes_Zplus.