X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fq_function.ma;h=e2187b51014799aed94ac412bef1ac57f24056d9;hb=d03c932e859d59c0ae381f941b4003d744b6b106;hp=d3d63233c37520c7d982a414ed767a51f6eaa64f;hpb=88b32d4e8fe371d59e41cd272064c9d486ae7ec5;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 d3d63233c..e2187b510 100644
--- a/helm/software/matita/contribs/dama/dama/models/q_function.ma
+++ b/helm/software/matita/contribs/dama/dama/models/q_function.ma
@@ -13,201 +13,8 @@
 (**************************************************************************)
 
 include "nat_ordered_set.ma".
-include "models/q_bars.ma".
+include "models/q_shift.ma".
 
-axiom le_le_eq: ∀x,y:Q. x ≤ y → y ≤ x → x = y.
-
-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: intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
-        simplify in H5 H6 ⊢ (? ? ? (? ? ? (? ? % ? ?)));
-
-axiom nth_nil: ∀T,n.∀d:T. nth [] d n = d.
-
-lemma key:
-  ∀init,input,l1,w1,w2,w.
-  Qpos w = start l1 - init →   
-  init < start l1 → 
-  start l1 < input →
-  sum_bases (〈w,OQ〉::bars l1) w1 ≤ ⅆ[input,init] →
-    ⅆ[input,init] < sum_bases (bars l1) w1 + (start l1-init) →
-  sum_bases (bars l1) w2 ≤ ⅆ[input,start l1] →
-    ⅆ[input,start l1] < sum_bases (bars l1) (S w2) →
-    \snd (nth (bars l1) ▭ w2) = \snd (nth (〈w,OQ〉::bars l1) ▭ w1).
-intros 4 (init input l); cases l (st l);
-change in match (start (mk_q_f st l)) with st;
-change in match (bars (mk_q_f st l)) with l;
-elim l;
-[1: rewrite > nth_nil; cases w1 in H4;
-    [1: rewrite > q_d_sym; rewrite > q_d_noabs; [2:
-          apply (q_le_trans ? st); apply q_lt_to_le; assumption]
-        do 2 rewrite > q_elim_minus; rewrite > q_plus_assoc;
-        intro X; lapply (q_lt_canc_plus_r ??? X) as Y;
-        simplify in Y; cases (?:False); 
-        apply (q_lt_corefl st); apply (q_lt_trans ??? H2);
-        apply (q_lt_le_trans ??? Y); rewrite > q_plus_sym; rewrite > q_plus_OQ;
-        apply q_eq_to_le; reflexivity;
-    |2: intros; simplify; rewrite > nth_nil; reflexivity;]       
-|2: FACTORIZE w1>0    
-    
-     (* interesting case: init < start < input *)
-        intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
-        simplify in H5 H6 ⊢ (? ? ? (? ? ? (? ? % ? ?)));
-        elim (\fst w2) in H9 H10;
-        [1: elim (\fst w1) in H5 H6;
-            [1: cases (?:False); clear H5 H8 H7; 
-                apply (q_lt_antisym input (start l1)); [2: assumption]
-                rewrite > q_d_sym in H6; rewrite > q_d_noabs in H6; 
-                  [2: apply q_lt_to_le; assumption]
-                rewrite > q_plus_sym in H6; rewrite > q_plus_OQ in H6; 
-                rewrite > H2 in H6; apply (q_lt_canc_plus_r ?? (Qopp init)); 
-                do 2 rewrite < q_elim_minus; assumption;
-            |2: 
-                
-        cut (\fst w1 = S (\fst w2)) as Key; [rewrite > Key; reflexivity;]
-        cases (\fst w1) in H5 H6; intros; [1: 
-          cases (?:False); clear H5 H9 H10; 
-          apply (q_lt_antisym input (start l1)); [2: assumption]
-          rewrite > q_d_sym in H6; rewrite > q_d_noabs in H6; 
-            [2: apply q_lt_to_le; assumption]
-          rewrite > q_plus_sym in H6; rewrite > q_plus_OQ in H6; 
-          rewrite > H2 in H6; apply (q_lt_canc_plus_r ?? (Qopp init)); 
-          do 2 rewrite < q_elim_minus; assumption;]
-       apply eq_f;
-            cut (sum_bases (bars l1) (\fst w2) < sum_bases (bars l1) (S n));[2:
-              apply (q_le_lt_trans ??? H9);
-              apply (q_lt_trans ??? ? H6);
-              rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le;assumption]
-              rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le;assumption]
-              do 2 rewrite > q_elim_minus; rewrite > (q_plus_sym ? (Qopp init));
-              apply q_lt_plus; rewrite > q_plus_sym;
-              rewrite > q_elim_minus; rewrite < q_plus_assoc;
-              rewrite < q_elim_minus; rewrite > q_plus_minus;
-              rewrite > q_plus_OQ; apply q_lt_opp_opp; assumption]
-            clear H9 H6;
-            cut (ⅆ[input,init] - Qpos w = ⅆ[input,start l1]);[2:
-              rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le;assumption]
-              rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le;assumption]
-              rewrite > H2; rewrite > (q_elim_minus (start ?));
-              rewrite > q_minus_distrib; rewrite > q_elim_opp;
-              do 2 rewrite > q_elim_minus;
-              do 2 rewrite < q_plus_assoc;
-              rewrite > (q_plus_sym ? init);
-              rewrite > (q_plus_assoc ? init);
-              rewrite > (q_plus_sym ? init);
-              rewrite < (q_elim_minus init); rewrite > q_plus_minus;
-              rewrite > (q_plus_sym OQ); rewrite > q_plus_OQ;
-              rewrite < q_elim_minus; reflexivity;]
-            cut (sum_bases (bars l1) n < sum_bases (bars l1) (S (\fst w2)));[2:
-              apply (q_le_lt_trans ???? H10); rewrite < Hcut1;
-              rewrite > q_elim_minus; apply q_le_minus_r; rewrite > q_elim_opp;
-              assumption;] clear Hcut1 H5 H10;
-            generalize in match Hcut;generalize in match Hcut2;clear Hcut Hcut2; 
-            apply (nat_elim2 ???? n (\fst w2)); 
-            [3: intros (x y); apply eq_f; apply H5; clear H5;
-                [1: clear H7; apply sum_bases_lt_canc; assumption;
-                |2: clear H6; ]
-            |2: intros; cases (?:False); clear H6;
-                cases n1 in H5; intro;
-                [1: apply (q_lt_corefl ? H5);
-                |2: cases (bars l1) in H5; intro;
-                    [1: simplify in H5; 
-                        apply (q_lt_le_incompat ?? (q_lt_canc_plus_r ??? H5));
-                        apply q_le_plus_trans; [apply sum_bases_ge_OQ]
-                        apply q_le_OQ_Qpos;
-                    |2: simplify in H5:(??%);
-                        lapply (q_lt_canc_plus_r (sum_bases l (S n2)) ?? H5) as X;
-                        apply (q_lt_le_incompat ?? X); apply sum_bases_ge_OQ]]
-            |1: intro; cases n1 [intros; reflexivity] intros; cases (?:False);
-                elim n2 in H5 H6;
-            
-            
-             elim (bars l1) 0; 
-                [1: intro; elim n1; [reflexivity] cases (?:False);
-                 
-            
-                intros; clear H5;
-                elim n1 in H6; [reflexivity] cases (?:False);
-                [1: apply (q_lt_corefl ? H5);
-                |2: cases (bars l1) in H5; intro;
-                    [1: simplify in H5; 
-                        apply (q_lt_le_incompat ?? (q_lt_canc_plus_r ??? H5));
-                        apply q_le_plus_trans; [apply sum_bases_ge_OQ]
-                        apply q_le_OQ_Qpos;
-                    |2: simplify in H5:(??%);
-                        lapply (q_lt_canc_plus_r (sum_bases l (S n2)) ?? H5) as X;
-                        apply (q_lt_le_incompat ?? X); apply sum_bases_ge_OQ]]
-qed.
-
-            
-        
 alias symbol "pi2" = "pair pi2".
 alias symbol "pi1" = "pair pi1".
 definition rebase_spec ≝ 
@@ -289,23 +96,40 @@ 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);
+            rewrite < (H4 input); 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; rewrite > (initial_shift_same_values (mk_q_f s1 l1) s2 H input);
+            rewrite < (H3 input); 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 *) 
+]
 qed.