X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2FZ%2Ftimes.ma;fp=helm%2Fmatita%2Flibrary%2FZ%2Ftimes.ma;h=688957e57396cecb7fd0143c60f171640024b743;hb=03fcee16d9c262aad38a47d0a409b684a965cc3f;hp=7e4b10e0028428b64f7809329eb430f9123ac37e;hpb=4d6bf14d95252ecfd1aaca4568184b4d6dee9b24;p=helm.git diff --git a/helm/matita/library/Z/times.ma b/helm/matita/library/Z/times.ma index 7e4b10e00..688957e57 100644 --- a/helm/matita/library/Z/times.ma +++ b/helm/matita/library/Z/times.ma @@ -15,7 +15,8 @@ set "baseuri" "cic:/matita/Z/times". include "nat/lt_arith.ma". -include "Z/z.ma". +include "Z/plus.ma". +include "higher_order_defs/functions.ma". definition Ztimes :Z \to Z \to Z \def \lambda x,y. @@ -42,6 +43,13 @@ simplify.reflexivity. 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. @@ -120,108 +128,110 @@ 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))))))). +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. +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. +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 (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. +apply Ztimes_Zplus_pos_neg_pos. +elim z.reflexivity. +apply Ztimes_Zplus_pos_pos_neg. +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. +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 = neg n *) +apply distr_Ztimes_Zplus_neg. +(* case x = pos n *) +apply distr_Ztimes_Zplus_pos. +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. \ No newline at end of file