X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fq_function.ma;h=d3d63233c37520c7d982a414ed767a51f6eaa64f;hb=88b32d4e8fe371d59e41cd272064c9d486ae7ec5;hp=bb3ba55fc8c49ff33bacbef1b8f5c2c6b8fd81f1;hpb=d97f69b313893900ca2d57544fcd200eb06ee286;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 bb3ba55fc..d3d63233c 100644
--- a/helm/software/matita/contribs/dama/dama/models/q_function.ma
+++ b/helm/software/matita/contribs/dama/dama/models/q_function.ma
@@ -12,8 +12,11 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "nat_ordered_set.ma".
 include "models/q_bars.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 
@@ -64,18 +67,146 @@ whd in ⊢ (% → ?); simplify in H3;
         |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 (mk_q_f init ?) n1));[apply [];|3:apply l]
-            simplify in ⊢ (? (? (? % ?) ?) ?); rewrite < (q_plus_minus w);
-            apply q_le_minus_r; rewrite < q_minus_r; 
-            rewrite < E in ⊢ (??%); assumption;]
+            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);
-        
-STOP            
+        |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".