+++ /dev/null
-(**************************************************************************)
-(* __ *)
-(* ||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.
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