--- /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/library_autobatch/Z/times".
+
+include "auto/nat/lt_arith.ma".
+include "auto/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/library_autobatch/Z/times/Ztimes.con x y).
+
+theorem Ztimes_z_OZ: \forall z:Z. z*OZ = OZ.
+intro.
+elim z;autobatch.
+ (*simplify;reflexivity.*)
+qed.
+
+theorem Ztimes_neg_Zopp: \forall n:nat.\forall x:Z.
+neg n * x = - (pos n * x).
+intros.
+elim x;autobatch.
+ (*simplify;reflexivity.*)
+qed.
+
+theorem symmetric_Ztimes : symmetric Z Ztimes.
+change with (\forall x,y:Z. x*y = y*x).
+intros.
+elim x
+[ autobatch
+ (*rewrite > Ztimes_z_OZ.
+ reflexivity*)
+| elim y
+ [ autobatch
+ (*simplify.
+ reflexivity*)
+ | change with (pos (pred ((S n) * (S n1))) = pos (pred ((S n1) * (S n)))).
+ autobatch
+ (*rewrite < sym_times.
+ reflexivity*)
+ | change with (neg (pred ((S n) * (S n1))) = neg (pred ((S n1) * (S n)))).
+ autobatch
+ (*rewrite < sym_times.
+ reflexivity*)
+ ]
+| elim y
+ [ autobatch
+ (*simplify.
+ reflexivity*)
+ | change with (neg (pred ((S n) * (S n1))) = neg (pred ((S n1) * (S n)))).
+ autobatch
+ (*rewrite < sym_times.
+ reflexivity*)
+ | change with (pos (pred ((S n) * (S n1))) = pos (pred ((S n1) * (S n)))).
+ autobatch
+ (*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.
+unfold associative.
+intros.
+elim x
+[ autobatch
+ (*simplify.
+ reflexivity*)
+| elim y
+ [ autobatch
+ (*simplify.
+ reflexivity*)
+ | elim z
+ [ autobatch
+ (*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;autobatch
+ (*[ 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;autobatch
+ (*[ rewrite < assoc_times.
+ reflexivity
+ | apply lt_O_times_S_S
+ ]*)
+ | apply lt_O_times_S_S
+ ]
+ ]
+ | elim z
+ [ autobatch
+ (*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;autobatch
+ (*[ 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;autobatch
+ (*[ rewrite < assoc_times.
+ reflexivity
+ | apply lt_O_times_S_S
+ ]*)
+ | apply lt_O_times_S_S
+ ]
+ ]
+ ]
+| elim y
+ [ autobatch
+ (*simplify.
+ reflexivity*)
+ | elim z
+ [ autobatch
+ (*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;autobatch
+ (*[ 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;autobatch
+ (*[ rewrite < assoc_times.
+ reflexivity
+ | apply lt_O_times_S_S
+ ]*)
+ | apply lt_O_times_S_S
+ ]
+ ]
+ | elim z
+ [ autobatch
+ (*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;autobatch
+ (*[ 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;autobatch
+ (*[ 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
+ [ autobatch
+ (*rewrite < distr_times_minus.
+ reflexivity*)
+
+ (* we now close all positivity conditions *)
+ | apply lt_O_times_S_S
+ | apply lt_O_times_S_S
+ ]
+| unfold lt.autobatch
+ (*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.
+ autobatch
+ (*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.
+autobatch.
+(*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)))).
+ autobatch
+ (*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)))).
+ autobatch
+ (*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.
+autobatch.
+(*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;autobatch.
+(*[ simplify.
+ reflexivity
+| apply distr_Ztimes_Zplus_pos
+| 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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