X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2Flibrary_auto%2Fauto%2FZ%2Ftimes.ma;fp=matita%2Fcontribs%2Flibrary_auto%2Fauto%2FZ%2Ftimes.ma;h=1fa633d9fb04b3f4da63bbaff719d19541224a49;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1

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+(**************************************************************************)
+(*       __                                                               *)
+(*      ||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.