X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fq_function.ma;h=b1ff639e44b959bd6a2a482c001df38d552316f9;hb=591ffe6f23ec9d2a4d368d2c1e7b213986189e44;hp=32122371004982cad24e4deccf33ec5f71dad0e1;hpb=6b61a9e6698a7c1936adf217b599e34e65a5e4c9;p=helm.git

diff --git a/helm/software/matita/contribs/dama/dama/models/q_function.ma b/helm/software/matita/contribs/dama/dama/models/q_function.ma
index 321223710..b1ff639e4 100644
--- a/helm/software/matita/contribs/dama/dama/models/q_function.ma
+++ b/helm/software/matita/contribs/dama/dama/models/q_function.ma
@@ -12,97 +12,23 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "models/q_bars.ma".
+include "nat_ordered_set.ma".
+include "models/q_shift.ma".
 
-lemma initial_shift_same_values:
-  ∀l1:q_f.∀init.init < start l1 →
-   same_values l1 
-     (mk_q_f init (〈\fst (unpos (start l1 - init) ?),OQ〉:: bars l1)).  
-[apply hide; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption]
-intros; generalize in ⊢ (? ? (? ? (? ? (? ? ? (? ? ? (? ? %)) ?) ?))); intro;
-cases (unpos (start l1-init) H1); intro input;
-simplify in ⊢ (? ? ? (? ? ? (? ? ? (? (? ? (? ? (? ? ? % ?) ?)) ?))));
-cases (value (mk_q_f init (〈w,OQ〉::bars l1)) input);
-simplify in ⊢ (? ? ? (? ? ? %));
-cases (q_cmp input (start (mk_q_f init (〈w,OQ〉::bars l1)))) in H3;
-whd in ⊢ (% → ?); simplify in H3;
-[1: intro; cases H4; clear H4; rewrite > H3;
-    cases (value l1 init); simplify; cases (q_cmp init (start l1)) in H4;
-    [1: cases (?:False); apply (q_lt_corefl init); rewrite > H4 in ⊢ (?? %); apply H;
-    |3: cases (?:False); apply (q_lt_antisym init (start l1)); assumption;
-    |2: whd in ⊢ (% → ?); intro; rewrite > H8; clear H8 H4;
-        rewrite > H7; clear H7; rewrite > (?:\fst w1 = O); [reflexivity]
-        symmetry; apply le_n_O_to_eq;
-        rewrite > (sum_bases_O (mk_q_f init (〈w,OQ〉::bars l1)) (\fst w1)); [apply le_n]   
-        clear H6 w2; simplify in H5:(? ? (? ? %));  
-        destruct H3; rewrite > q_d_x_x in H5; assumption;]
-|2: intros; cases (value l1 input); simplify in ⊢ (? ? (? ? ? %) ?);
-    cases (q_cmp input (start l1)) in H5; whd in ⊢ (% → ?);
-    [1: cases (?:False); clear w2 H4 w1 H2 w H1; 
-        apply (q_lt_antisym init (start l1)); [assumption] rewrite < H5; assumption
-    |2: intros; rewrite > H6; clear H6; rewrite > H4; reflexivity;
-    |3: cases (?:False); apply (q_lt_antisym input (start l1)); [2: assumption]
-        apply (q_lt_trans ??? H3 H);]
-|3: intro; cases H4; clear H4;   
-    cases (value l1 input); simplify; cases (q_cmp input (start l1)) in H4; whd in ⊢ (% → ?);
-    [1: intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
-        simplify in ⊢ (? ? ? (? ? ? (? ? % ? ?)));
-        cut (\fst w1 = S (\fst w2)) as Key; [rewrite > Key; reflexivity;]
-        cut (\fst w2 = O); [2: clear H10;
-          symmetry; apply le_n_O_to_eq; rewrite > (sum_bases_O l1 (\fst w2)); [apply le_n]
-          apply (q_le_trans ??? H9); rewrite < H4; rewrite > q_d_x_x; 
-          apply q_eq_to_le; reflexivity;]
-        rewrite > Hcut; clear Hcut H10 H9; simplify in H5 H6;
-        cut (ⅆ[input,init] = Qpos w) as E; [2:
-          rewrite > H2; rewrite < H4; rewrite > q_d_sym; 
-          rewrite > q_d_noabs; [reflexivity] apply q_lt_to_le; assumption;]
-        cases (\fst w1) in H5 H6; intros;
-        [1: cases (?:False); clear H5; simplify in H6;
-            apply (q_lt_corefl ⅆ[input,init]);
-            rewrite > E in ⊢ (??%); rewrite < q_plus_OQ in ⊢ (??%);
-            rewrite > q_plus_sym; assumption;
-        |2: cases n in H5 H6; [intros; reflexivity] intros;
-            cases (?:False); clear H6; cases (bars l1) in H5; simplify; intros;
-            [apply (q_pos_OQ one);|apply (q_pos_OQ (\fst b));] 
-            apply (q_le_S ??? (sum_bases_ge_OQ ? n1));[apply []|3:apply l]
-            simplify in ⊢ (? (? (? % ?) ?) ?); rewrite < (q_plus_minus (Qpos w));
-            rewrite > q_elim_minus; apply q_le_minus_r; 
-            rewrite > q_elim_opp; rewrite < E in ⊢ (??%); assumption;]
-    |2: intros; rewrite > H8; rewrite > H7; clear H8 H7;
-        simplify in H5 H6 ⊢ %; 
-        cases (\fst w1) in H5 H6; [intros; reflexivity]
-        cases (bars l1);
-        [1: intros; simplify; elim n [reflexivity] simplify; assumption;
-        |2: simplify; intros; cases (?:False); clear H6;
-            apply (q_lt_le_incompat (input - init) (Qpos w) );
-            [1: rewrite > H2; do 2 rewrite > q_elim_minus;
-                apply q_lt_plus; rewrite > q_elim_minus;
-                rewrite < q_plus_assoc; rewrite < q_elim_minus;
-                rewrite > q_plus_minus;
-                rewrite > q_plus_OQ; assumption;
-            |2: rewrite < q_d_noabs; [2: apply q_lt_to_le; assumption]
-                rewrite > q_d_sym; apply (q_le_S ???? H5);
-                apply sum_bases_ge_OQ;]]
-    |3:
-    
-        
-STOP            
-        
 alias symbol "pi2" = "pair pi2".
 alias symbol "pi1" = "pair pi1".
 definition rebase_spec ≝ 
  ∀l1,l2:q_f.∃p:q_f × q_f.
    And4
-    (*len (bars (\fst p)) = len (bars (\snd p))*)
     (start (\fst p) = start (\snd p))
-    (same_bases (\fst p) (\snd p))
+    (same_bases (bars (\fst p)) (bars (\snd p)))
     (same_values l1 (\fst p)) 
     (same_values l2 (\snd p)).
 
 definition rebase_spec_simpl ≝ 
  λstart.λl1,l2:list bar.λp:(list bar) × (list bar).
    And3
-    (same_bases (mk_q_f start (\fst p)) (mk_q_f start (\snd p)))
+    (same_bases (\fst p) (\snd p))
     (same_values (mk_q_f start l1) (mk_q_f start (\fst p))) 
     (same_values (mk_q_f start l2) (mk_q_f start (\snd p))).
 
@@ -115,6 +41,8 @@ definition eject ≝
 coercion eject.
 definition inject ≝ λP.λp:(list bar) × (list bar).λh:P p. ex_introT ? P p h.
 coercion inject with 0 1 nocomposites.
+
+axiom devil : False.
         
 definition rebase: rebase_spec.
 intros 2 (f1 f2); cases f1 (s1 l1); cases f2 (s2 l2); clear f1 f2;
@@ -159,7 +87,7 @@ in aux : ∀l1,l2,m.∃z.∀s.spec s l1 l2 m z); unfold spec;
         [1,2: assumption;
         |3: intro; apply (H3 input);
         |4: intro; rewrite > H in H4; 
-            rewrite > (H4 input); reflexivity;]
+            rewrite > (H4 input) in ⊢ (? ? % ?); reflexivity;]
     |2: letin l2' ≝ (〈\fst (unpos (s2-s1) ?),OQ〉::l2);[
           apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
           assumption]
@@ -169,23 +97,97 @@ in aux : ∀l1,l2,m.∃z.∀s.spec s l1 l2 m z); unfold spec;
         [1: reflexivity
         |2: assumption;
         |3: assumption;
-        |4: intro; rewrite < (H4 input); clear H3 H4 H2 w;
-            cases (value (mk_q_f s1 l2') input);
-            cases (q_cmp input (start (mk_q_f s1 l2'))) in H1;
-            whd in ⊢ (% → ?);
-            [1: intros; cases H2; clear H2; whd in ⊢ (??? %);
-                cases (value (mk_q_f s2 l2) input);
-                cases (q_cmp input (start (mk_q_f s2 l2))) in H2;
-                whd in ⊢ (% → ?);
-                [1: intros; cases H6; clear H6; change with (w1 = w);
-                          
-            (* TODO *) ]]    
+        |4: intro; 
+            rewrite > (initial_shift_same_values (mk_q_f s2 l2) s1 H input) in ⊢ (? ? % ?);
+            rewrite < (H4 input)in ⊢ (? ? ? %); reflexivity;]
+    |3: letin l1' ≝ (〈\fst (unpos (s1-s2) ?),OQ〉::l1);[
+          apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+          assumption]
+        cases (aux l1' l2 (S (len l1' + len l2)));
+        cases (H1 s2 (le_n ?)); clear H1 aux;
+        exists [apply 〈mk_q_f s2 (\fst w), mk_q_f s2 (\snd w)〉] split;
+        [1: reflexivity
+        |2: assumption;
+        |4: assumption;
+        |3: intro; simplify in ⊢ (? ? ? (? ? ? (? ? ? (? % ?))));
+            rewrite > (initial_shift_same_values (mk_q_f s1 l1) s2 H input) in ⊢ (? ? % ?);
+            rewrite < (H3 input) in ⊢ (? ? ? %); reflexivity;]]
 |1,2: unfold rest; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
       assumption;        
-|3:(* TODO *)
-|4:(* TODO *)
-|5:(* TODO *)
-|6:(* TODO *)
-|7:(* TODO *)
-|8: intros; cases (?:False); apply (not_le_Sn_O ? H1);]
+|8: intros; cases (?:False); apply (not_le_Sn_O ? H1);
+|3: intros; generalize in match (unpos ??); intro X; cases X; clear X;
+    simplify in ⊢ (???? (??? (??? (??? (?? (? (?? (??? % ?) ?) ??)))) ?));
+    simplify in ⊢ (???? (???? (??? (??? (?? (? (?? (??? % ?) ?) ??)))))); 
+    clear H4; cases (aux (〈w,\snd b〉::l4) l5 n1); clear aux;
+    cut (len (〈w,\snd b〉::l4) + len l5 < n1) as K;[2:
+      simplify in H5; simplify; rewrite > sym_plus in H5; simplify in H5;
+      rewrite > sym_plus in H5; apply le_S_S_to_le; apply H5;] 
+    split;
+    [1: simplify in ⊢ (? % ?); simplify in ⊢ (? ? %); 
+        cases (H4 s K); clear K H4; intro input; cases input; [reflexivity]
+        simplify; apply H7; 
+    |2: simplify in ⊢ (? ? %); cases (H4 s K); clear H4 K H5 spec;
+        intro;
+        (* input < s + b1 || input >= s + b1 *)
+    |3: simplify in ⊢ (? ? %);]   
+|4: intros; generalize in match (unpos ??); intro X; cases X; clear X;
+    (* duale del 3 *)
+|5: intros; (* triviale, caso in cui non fa nulla *)
+|6,7: (* casi base in cui allunga la lista più corta *) 
+]
+elim devil;
 qed.
+
+include "Q/q/qtimes.ma".
+
+let rec area (l:list bar) on l ≝
+  match l with 
+  [ nil ⇒ OQ
+  | cons he tl ⇒ area tl + Qpos (\fst he) * ⅆ[OQ,\snd he]].
+
+alias symbol "pi1" = "exT \fst".
+alias symbol "minus" = "Q minus".
+alias symbol "exists" = "CProp exists".
+definition minus_spec_bar ≝
+ λf,g,h:list bar.
+   same_bases f g → len f = len g →
+     ∀s,i:ℚ. \snd (\fst (value (mk_q_f s h) i)) = 
+       \snd (\fst (value (mk_q_f s f) i)) - \snd (\fst (value (mk_q_f s g) i)). 
+
+definition minus_spec ≝
+ λf,g:q_f.
+   ∃h:q_f. 
+     ∀i:ℚ. \snd (\fst (value h i)) = 
+       \snd (\fst (value f i)) - \snd (\fst (value g i)). 
+
+definition eject_bar : ∀P:list bar → CProp.(∃l:list bar.P l) → list bar ≝
+ λP.λp.match p with [ex_introT x _ ⇒ x].
+definition inject_bar ≝ ex_introT (list bar).
+
+coercion inject_bar with 0 1 nocomposites.
+coercion eject_bar with 0 0 nocomposites.
+
+lemma minus_q_f : ∀f,g. minus_spec f g.
+intros;
+letin aux ≝ (
+  let rec aux (l1, l2 : list bar) on l1 ≝
+    match l1 with
+    [ nil ⇒ []
+    | cons he1 tl1 ⇒
+        match l2 with
+        [ nil ⇒ []
+        | cons he2 tl2 ⇒ 〈\fst he1, \snd he1 - \snd he2〉 :: aux tl1 tl2]]
+  in aux : ∀l1,l2 : list bar.∃h.minus_spec_bar l1 l2 h);
+[2: intros 4; simplify in H3; destruct H3;
+|3: intros 4; simplify in H3; cases l1 in H2; [2: intro X; simplify in X; destruct X]    
+    intros; rewrite > (value_OQ_e (mk_q_f s []) i); [2: reflexivity]
+    rewrite > q_elim_minus; rewrite > q_plus_OQ; reflexivity;
+|1: cases (aux l2 l3); unfold in H2; intros 4;
+    simplify in ⊢ (? ? (? ? ? (? ? ? (? % ?))) ?);
+    cases (q_cmp i (s + Qpos (\fst b)));
+    
+
+
+definition excess ≝ 
+  λf,g.∃i.\snd (\fst (value f i)) < \snd (\fst (value g i)).
+