set "baseuri" "cic:/matita/Z/times".
include "nat/lt_arith.ma".
-include "Z/z.ma".
+include "Z/plus.ma".
definition Ztimes :Z \to Z \to Z \def
\lambda x,y.
simplify.reflexivity.
qed.
+theorem Ztimes_neg_Zopp: \forall n:nat.\forall x:Z.
+eq Z (Ztimes (neg n) x) (Zopp (Ztimes (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. eq Z (Ztimes x y) (Ztimes y x).
intros.elim x.rewrite > Ztimes_z_OZ.reflexivity.
theorem associative_Ztimes: associative Z Ztimes.
change with \forall x,y,z:Z.eq Z (Ztimes (Ztimes x y) z) (Ztimes x (Ztimes y z)).
-intros.
-elim x.simplify.reflexivity.
-elim y.simplify.reflexivity.
-elim z.simplify.reflexivity.
-change with
-eq Z (neg (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
- (neg (pred (times (S n) (S (pred (times (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
-eq Z (pos (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
- (pos (pred (times (S n) (S (pred (times (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
-eq Z (pos (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
- (pos(pred (times (S n) (S (pred (times (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
-eq Z (neg (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
- (neg (pred (times (S n) (S (pred (times (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
-eq Z (pos (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
- (pos (pred (times (S n) (S (pred (times (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
-eq Z (neg (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
- (neg (pred (times (S n) (S (pred (times (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
-eq Z (neg (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
- (neg(pred (times (S n) (S (pred (times (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
-eq Z (pos (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
- (pos (pred (times (S n) (S (pred (times (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.
+intros.elim x.
+ simplify.reflexivity.
+ elim y.
+ simplify.reflexivity.
+ elim z.
+ simplify.reflexivity.
+ change with
+ eq Z (pos (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
+ (pos (pred (times (S n) (S (pred (times (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
+ eq Z (neg (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
+ (neg (pred (times (S n) (S (pred (times (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
+ eq Z (neg (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
+ (neg (pred (times (S n) (S (pred (times (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
+ eq Z (pos (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
+ (pos(pred (times (S n) (S (pred (times (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
+ eq Z (neg (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
+ (neg (pred (times (S n) (S (pred (times (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
+ eq Z (pos (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
+ (pos (pred (times (S n) (S (pred (times (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
+ eq Z (pos (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
+ (pos (pred (times (S n) (S (pred (times (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
+ eq Z (neg (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
+ (neg(pred (times (S n) (S (pred (times (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.
eq Z (Ztimes (Ztimes x y) z) (Ztimes x (Ztimes y z)) \def
associative_Ztimes.
-
-theorem distributive_Ztimes: distributive Z Ztimes Zplus.
-change with \forall x,y,z:Z.
-eq Z (Ztimes x (Zplus y z)) (Zplus (Ztimes x y) (Ztimes x z)).
-intros.elim x.
-simplify.reflexivity.
-(* case x = neg n *)
-elim y.reflexivity.
-elim z.reflexivity.
-change with
-eq Z (pos (pred (times (S n) (plus (S n1) (S n2)))))
- (pos (pred (plus (times (S n) (S n1))(times (S n) (S n2))))).
-rewrite < times_plus_distr.reflexivity.
-simplify. (* problemi di naming su n *)
-(* sintassi della change ??? *)
-cut \forall n,m:nat.eq nat (plus m (times n (S m))) (pred (times (S n) (S m))).
-rewrite > Hcut.
-rewrite > Hcut.
-rewrite < nat_compare_pred_pred ? ? ? ?.
-rewrite < nat_compare_times_l.
-rewrite < nat_compare_S_S.
-apply nat_compare_elim n1 n2.
-intro.
-(* per ricavare questo change ci ho messo un'ora;
-LA GESTIONE DELLA RIDUZIONE NON VA *)
-change with (eq Z (neg (pred (times (S n) (S (pred (minus (S n2) (S n1)))))))
- (neg (pred (minus (pred (times (S n) (S n2)))
- (pred (times (S n) (S n1))))))).
-rewrite < S_pred ? ?.
-rewrite > minus_pred_pred ? ? ? ?.
-rewrite < distr_times_minus.
-reflexivity.
-(* we start closing stupid positivity conditions *)
-apply lt_O_times_S_S.
-apply lt_O_times_S_S.
-simplify.
-apply le_SO_minus. exact H.
-intro.rewrite < H.simplify.reflexivity.
-intro.
-change with (eq Z (pos (pred (times (S n) (S (pred (minus (S n1) (S n2)))))))
- (pos (pred (minus (pred (times (S n) (S n1)))
- (pred (times (S n) (S n2))))))).
-rewrite < S_pred ? ?.
-rewrite > minus_pred_pred ? ? ? ?.
+lemma times_minus1: \forall n,p,q:nat. lt q p \to
+eq nat (times (S n) (S (pred (minus (S p) (S q)))))
+ (minus (pred (times (S n) (S p)))
+ (pred (times (S n) (S q)))).
+intros.
+rewrite < S_pred.
+rewrite > minus_pred_pred.
rewrite < distr_times_minus.
-reflexivity.
-(* we start closing stupid positivity conditions *)
+reflexivity.
+(* we now close all positivity conditions *)
apply lt_O_times_S_S.
apply lt_O_times_S_S.
simplify.
apply le_SO_minus. exact H.
-(* questi non so neppure da dove vengano *)
-apply lt_O_times_S_S.
-apply lt_O_times_S_S.
-(* adesso chiudo il cut stupido *)
-intros.reflexivity.
-elim z.reflexivity.
-simplify.
-cut \forall n,m:nat.eq nat (plus m (times n (S m))) (pred (times (S n) (S m))).
-rewrite > Hcut.
-rewrite > Hcut.
-rewrite < nat_compare_pred_pred ? ? ? ?.
+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 (plus p (times n (S p))) with (pred (times (S n) (S p))).
+change in match (plus q (times n (S q))) with (pred (times (S n) (S q))).
+rewrite < nat_compare_pred_pred.
rewrite < nat_compare_times_l.
rewrite < nat_compare_S_S.
-apply nat_compare_elim n1 n2.
+apply nat_compare_elim p q.
intro.
-(* per ricavare questo change ci ho messo un'ora;
-LA GESTIONE DELLA RIDUZIONE NON VA *)
-change with (eq Z (pos (pred (times (S n) (S (pred (minus (S n2) (S n1)))))))
- (pos (pred (minus (pred (times (S n) (S n2)))
- (pred (times (S n) (S n1))))))).
-rewrite < S_pred ? ?.
-rewrite > minus_pred_pred ? ? ? ?.
-rewrite < distr_times_minus.
-reflexivity.
-(* we start closing stupid positivity conditions *)
-apply lt_O_times_S_S.
-apply lt_O_times_S_S.
-simplify.
-apply le_SO_minus. exact H.
+(* uff *)
+change with (eq Z (pos (pred (times (S n) (S (pred (minus (S q) (S p)))))))
+ (pos (pred (minus (pred (times (S n) (S q)))
+ (pred (times (S n) (S p))))))).
+rewrite < times_minus1 n q p H.reflexivity.
intro.rewrite < H.simplify.reflexivity.
intro.
-change with (eq Z (neg (pred (times (S n) (S (pred (minus (S n1) (S n2)))))))
- (neg (pred (minus (pred (times (S n) (S n1)))
- (pred (times (S n) (S n2))))))).
-rewrite < S_pred ? ?.
-rewrite > minus_pred_pred ? ? ? ?.
-rewrite < distr_times_minus.
-reflexivity.
-(* we start closing stupid positivity conditions *)
-apply lt_O_times_S_S.
-apply lt_O_times_S_S.
-simplify.
-apply le_SO_minus. exact H.
-(* questi non so neppure da dove vengano *)
-apply lt_O_times_S_S.
+change with (eq Z (neg (pred (times (S n) (S (pred (minus (S p) (S q)))))))
+ (neg (pred (minus (pred (times (S n) (S p)))
+ (pred (times (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.
-(* adesso chiudo il cut stupido *)
-intros.reflexivity.
-change with
-eq Z (neg (pred (times (S n) (plus (S n1) (S n2)))))
+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. Ztimes (pos n) z) Zplus.
+change with \forall n,y,z.
+eq Z (Ztimes (pos n) (Zplus y z)) (Zplus (Ztimes (pos n) y) (Ztimes (pos n) z)).
+intros.elim y.
+ reflexivity.
+ elim z.
+ reflexivity.
+ change with
+ eq Z (pos (pred (times (S n) (plus (S n1) (S n2)))))
+ (pos (pred (plus (times (S n) (S n1))(times (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
+ eq Z (neg (pred (times (S n) (plus (S n1) (S n2)))))
(neg (pred (plus (times (S n) (S n1))(times (S n) (S n2))))).
-rewrite < times_plus_distr.
+ rewrite < distr_times_plus.reflexivity.
+qed.
+
+variant distr_Ztimes_Zplus_pos: \forall n,y,z.
+eq Z (Ztimes (pos n) (Zplus y z)) (Zplus (Ztimes (pos n) y) (Ztimes (pos n) z)) \def
+distributive2_Ztimes_pos_Zplus.
+
+lemma distributive2_Ztimes_neg_Zplus :
+distributive2 nat Z (\lambda n,z. Ztimes (neg n) z) Zplus.
+change with \forall n,y,z.
+eq Z (Ztimes (neg n) (Zplus y z)) (Zplus (Ztimes (neg n) y) (Ztimes (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.
-(* and now the case x = pos n *)
+qed.
+
+variant distr_Ztimes_Zplus_neg: \forall n,y,z.
+eq Z (Ztimes (neg n) (Zplus y z)) (Zplus (Ztimes (neg n) y) (Ztimes (neg n) z)) \def
+distributive2_Ztimes_neg_Zplus.
+
+theorem distributive_Ztimes_Zplus: distributive Z Ztimes Zplus.
+change with \forall x,y,z:Z.
+eq Z (Ztimes x (Zplus y z)) (Zplus (Ztimes x y) (Ztimes 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.
+eq Z (Ztimes x (Zplus y z)) (Zplus (Ztimes x y) (Ztimes x z)) \def
+distributive_Ztimes_Zplus.
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