X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2FZ%2Fplus.ma;fp=helm%2Fmatita%2Flibrary%2FZ%2Fplus.ma;h=976f6cfb3ce01d379244fc66ec964a99563917d3;hp=0000000000000000000000000000000000000000;hb=792b5d29ebae8f917043d9dd226692919b5d6ca1;hpb=a14a8c7637fd0b95e9d4deccb20c6abc98e8f953 diff --git a/helm/matita/library/Z/plus.ma b/helm/matita/library/Z/plus.ma new file mode 100644 index 000000000..976f6cfb3 --- /dev/null +++ b/helm/matita/library/Z/plus.ma @@ -0,0 +1,301 @@ +(**************************************************************************) +(* ___ *) +(* ||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/plus". + +include "Z/z.ma". +include "nat/minus.ma". + +definition Zplus :Z \to Z \to Z \def +\lambda x,y. + match x with + [ OZ \Rightarrow y + | (pos m) \Rightarrow + match y with + [ OZ \Rightarrow x + | (pos n) \Rightarrow (pos (pred ((S m)+(S n)))) + | (neg n) \Rightarrow + match nat_compare m n with + [ LT \Rightarrow (neg (pred (n-m))) + | EQ \Rightarrow OZ + | GT \Rightarrow (pos (pred (m-n)))] ] + | (neg m) \Rightarrow + match y with + [ OZ \Rightarrow x + | (pos n) \Rightarrow + match nat_compare m n with + [ LT \Rightarrow (pos (pred (n-m))) + | EQ \Rightarrow OZ + | GT \Rightarrow (neg (pred (m-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). + +theorem Zplus_z_OZ: \forall z:Z. z+OZ = z. +intro.elim z. +simplify.reflexivity. +simplify.reflexivity. +simplify.reflexivity. +qed. + +(* theorem symmetric_Zplus: symmetric Z Zplus. *) + +theorem sym_Zplus : \forall x,y:Z. x+y = y+x. +intros.elim x.rewrite > Zplus_z_OZ.reflexivity. +elim y.simplify.reflexivity. +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. reflexivity. +simplify. reflexivity. +elim y.simplify.reflexivity. +simplify.rewrite > nat_compare_n_m_m_n. +simplify.elim nat_compare.simplify.reflexivity. +simplify. reflexivity. +simplify. reflexivity. +simplify.rewrite < plus_n_Sm. rewrite < plus_n_Sm.rewrite < sym_plus.reflexivity. +qed. + +theorem Zpred_Zplus_neg_O : \forall z:Z. Zpred z = (neg O)+z. +intros.elim z. + 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. + simplify.reflexivity. + elim n. + simplify.reflexivity. + simplify.reflexivity. +qed. + +theorem Zplus_pos_pos: +\forall n,m. (pos n)+(pos m) = (Zsucc (pos n))+(Zpred (pos m)). +intros. +elim n.elim m. +simplify.reflexivity. +simplify.reflexivity. +elim m. +simplify.rewrite < plus_n_Sm. +rewrite < plus_n_O.reflexivity. +simplify.rewrite < plus_n_Sm. +rewrite < plus_n_Sm.reflexivity. +qed. + +theorem Zplus_pos_neg: +\forall n,m. (pos n)+(neg m) = (Zsucc (pos n))+(Zpred (neg m)). +intros.reflexivity. +qed. + +theorem Zplus_neg_pos : +\forall n,m. (neg n)+(pos m) = (Zsucc (neg n))+(Zpred (pos m)). +intros. +elim n.elim m. +simplify.reflexivity. +simplify.reflexivity. +elim m. +simplify.reflexivity. +simplify.reflexivity. +qed. + +theorem Zplus_neg_neg: +\forall n,m. (neg n)+(neg m) = (Zsucc (neg n))+(Zpred (neg m)). +intros. +elim n.elim m. +simplify.reflexivity. +simplify.reflexivity. +elim m. +simplify.rewrite > plus_n_Sm.reflexivity. +simplify.rewrite > plus_n_Sm.reflexivity. +qed. + +theorem Zplus_Zsucc_Zpred: +\forall x,y. x+y = (Zsucc x)+(Zpred y). +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 : +\forall n,m. (Zsucc (pos n))+(pos m) = Zsucc ((pos n)+(pos m)). +intros.reflexivity. +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. +intros.elim n1. +simplify. reflexivity. +elim n2.simplify. reflexivity. +simplify. reflexivity. +intros. elim n1. +simplify. reflexivity. +simplify.reflexivity. +intros. +rewrite < (Zplus_pos_neg ? m1). +elim H.reflexivity. +qed. + +theorem Zplus_Zsucc_neg_neg : +\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. +intros.elim n1. +simplify. reflexivity. +elim n2.simplify. reflexivity. +simplify. reflexivity. +intros. elim n1. +simplify. reflexivity. +simplify.reflexivity. +intros. +rewrite < (Zplus_neg_neg ? m1). +reflexivity. +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))). +intros.elim n1. +simplify. reflexivity. +elim n2.simplify. reflexivity. +simplify. reflexivity. +intros. elim n1. +simplify. reflexivity. +simplify.reflexivity. +intros. +rewrite < H. +rewrite < (Zplus_neg_pos ? (S m1)). +reflexivity. +qed. + +theorem Zplus_Zsucc : \forall x,y:Z. (Zsucc x)+y = Zsucc (x+y). +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)). +rewrite > Hcut. +rewrite > Zplus_Zsucc. +rewrite > Zpred_Zsucc. +reflexivity. +rewrite > Zsucc_Zpred. +reflexivity. +qed. + + +theorem associative_Zplus: associative Z Zplus. +change with (\forall x,y,z:Z. (x + y) + z = x + (y + z)). +(* simplify. *) +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) +\def associative_Zplus. + +(* Zopp *) +definition Zopp : Z \to Z \def +\lambda x:Z. match x with +[ OZ \Rightarrow OZ +| (pos n) \Rightarrow (neg n) +| (neg n) \Rightarrow (pos n) ]. + +(*CSC: the URI must disappear: there is a bug now *) +interpretation "integer unary minus" 'uminus x = (cic:/matita/Z/plus/Zopp.con x). + +theorem Zopp_Zplus: \forall x,y:Z. -(x+y) = -x + -y. +intros. +elim x.elim y. +simplify. reflexivity. +simplify. reflexivity. +simplify. reflexivity. +elim y. +simplify. reflexivity. +simplify. reflexivity. +simplify. apply nat_compare_elim. +intro.simplify.reflexivity. +intro.simplify.reflexivity. +intro.simplify.reflexivity. +elim y. +simplify. reflexivity. +simplify. apply nat_compare_elim. +intro.simplify.reflexivity. +intro.simplify.reflexivity. +intro.simplify.reflexivity. +simplify.reflexivity. +qed. + +theorem Zopp_Zopp: \forall x:Z. --x = x. +intro. elim x. +reflexivity.reflexivity.reflexivity. +qed. + +theorem Zplus_Zopp: \forall x:Z. x+ -x = OZ. +intro.elim x. +apply refl_eq. +simplify. +rewrite > nat_compare_n_n. +simplify.apply refl_eq. +simplify. +rewrite > nat_compare_n_n. +simplify.apply refl_eq. +qed. +