X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Fmatch.ma;h=8f525c9f4f7188df65acfeb91114d4d1b072b18c;hb=afd1e4522f61a72711ed822267c5ca86a3eb6d63;hp=d986ac4e3779147c532e095a144814b13ccd1d4a;hpb=26aa4427b7075c6587faf3ead1dc44910ec86a5c;p=helm.git

diff --git a/matita/matita/lib/turing/multi_universal/match.ma b/matita/matita/lib/turing/multi_universal/match.ma
index d986ac4e3..8f525c9f4 100644
--- a/matita/matita/lib/turing/multi_universal/match.ma
+++ b/matita/matita/lib/turing/multi_universal/match.ma
@@ -215,10 +215,15 @@ definition R_compare ≝
    (current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
     current ? (nth i ? int (niltape ?)) = None ? ∨
     current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
-  (∀ls,x,xs,ci,rs,ls0,cj,rs0. 
+  (∀ls,x,xs,ci,rs,ls0,rs0. 
     nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
-    nth j ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) →
+    nth j ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
     (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → 
+    (rs0 = [ ] → 
+     outt = change_vec ?? 
+           (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
+           (mk_tape sig (reverse ? xs@x::ls0) (None ?) []) j) ∨
+    ∀cj,rs1.rs0 = cj::rs1 →
     (is_endc ci = true ∨ ci ≠ cj) → 
     outt = change_vec ?? 
            (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
@@ -336,21 +341,37 @@ definition Rtc_multi_true ≝
 definition Rtc_multi_false ≝ 
   λalpha,test,n,i.λt1,t2:Vector ? (S n).
     (∀c. current alpha (nth i ? t1 (niltape ?)) = Some ? c → test c = false) ∧ t2 = t1.
-   
+
+definition R_match_step_false ≝ 
+  λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
+  ∀ls,x,xs,end,rs.
+  nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
+  (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
+   ((current sig (nth dst (tape sig) int (niltape sig)) = None ? ) ∧ outt = int) ∨
+   (∃ls0,rs0. 
+    nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
+    ∀rsj,end,c. 
+    rs0 = c::rsj →
+    outt = change_vec ??
+           (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
+           (midtape sig (reverse ? xs@x::ls0) c rsj) dst).
+(*  
 definition R_match_step_false ≝  
   λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
    (((∃x.current ? (nth src ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
-     (* current ? (nth src ? int (niltape ?)) ≠ current ? (nth dst ? int (niltape ?)) ∨ *)   
      current sig (nth src (tape sig) int (niltape sig)) = None ? ∨
      current sig (nth dst (tape sig) int (niltape sig)) = None ? ) ∧ outt = int) ∨
-   ∃ls,ls0,rs,rs0,x,xs. ∀rsi,rsj,end,c. 
+   (∃ls,ls0,rs,rs0,x,xs. 
+    nth src ? int (niltape ?) = midtape sig ls x (xs@rs) ∧ is_endc x = false ∧
+    nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
+    ∀rsi,rsj,end,c. 
     rs = end::rsi → rs0 = c::rsj →
     (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) ∧ is_endc end = true ∧
-    nth src ? int (niltape ?) = midtape sig ls x (xs@rs) ∧
-    nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
+    nth dst ? int (niltape ?) = midtape sig ls0 x (xs@c::rsj) ∧
     outt = change_vec ??
            (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rsi) src)
-           (midtape sig (reverse ? xs@x::ls0) c rsj) dst.
+           (midtape sig (reverse ? xs@x::ls0) c rsj) dst).
+*)
 
 definition R_match_step_true ≝ 
   λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
@@ -457,6 +478,51 @@ lemma sem_match_step :
      normalize #H destruct (H) // ]
   ]
 |#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb 
+ whd in ⊢ (%→?); #Hout >Hout >Htb whd
+ #ls #c_src #xs #end #rs #Hmid_src #Hnotend #Hend
+ lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
+ cases (current … (nth dst ? intape (niltape ?))) in Hcomp1;
+  [#Hcomp1 #_ %1 % [% | @Hcomp1 %2 %2 % ]
+  |#c_dst cases (true_or_false (c_src == c_dst)) #Hceq
+    [#_ #Hmid_dst cases (Hmid_dst c_dst (refl …)) -Hmid_dst
+     #ls_dst * #rs_dst #Hmid_dst %2
+     cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * * 
+     #Hrs_src #Hrs_dst #Hnotendc #Hneq    
+     %{ls_dst} %{rsj} % 
+      [<Hrs_dst >(\P Hceq) // ]]     
+         #rsi0 #rsj0 #end #c #Hend #Hc_dst
+         >Hrs_src in Hmid_src; >Hend #Hmid_src
+         >Hrs_dst in Hmid_dst; >Hc_dst <(\P Hceq) #Hmid_dst
+         cut (is_endc end = true ∨ end ≠ c)
+         [cases (Hneq … Hend) /2/ -Hneq #Hneq %2 @(Hneq … Hc_dst) ] #Hneq
+         lapply (Hcomp2 … Hmid_src Hmid_dst ? Hneq)
+          [#c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
+            [ >(\P Hc0) //
+            | @Hnotendc // ] 
+          ]
+         -Hcomp2 #Hcomp2 <Hcomp2
+         % // % [ 
+           >Hcomp2 in Hc; >nth_change_vec_neq [|@sym_not_eq //]
+            >nth_change_vec // #H lapply (H ? (refl …)) 
+            cases (is_endc end) [|normalize #H destruct (H) ]
+            #_ % // #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
+              [ >(\P Hc0) // | @Hnotendc // ]
+         |@Hmid_dst] 
+         ]
+      |#_ #Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls * #rs #Hsrc
+       %1 % 
+        [% % %{c_src} % // lapply (Hc c_src) -Hc >Hcomp1
+         [| %2 % % @(not_to_not ??? (\Pf Hceq)) #H destruct (H) // ]
+         cases (is_endc c_src) //
+         >Hsrc #Hc lapply (Hc (refl ??)) normalize #H destruct (H)
+        |@Hcomp1 %2 %1 %1 @(not_to_not ??? (\Pf Hceq)) #H destruct (H) //
+        ]
+      ]
+    ]
+  ] 
+qed.
+
+#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb 
  whd in ⊢ (%→?); #Hout >Hout >Htb whd
  lapply (current_to_midtape sig (nth src ? intape (niltape ?)))
  cases (current … (nth src ? intape (niltape ?))) in Hcomp1; 
@@ -473,7 +539,8 @@ lemma sem_match_step :
        [ % % [ % % %{c_src} % // | @Hcomp1 % %{c_src} % // ]
        | %2 cases (comp_list … rs_src rs_dst is_endc) #xs * #rsi * #rsj * * * 
          #Hrs_src #Hrs_dst #Hnotendc #Hneq    
-         %{ls_src} %{ls_dst} %{rsi} %{rsj} %{c_src} %{xs} 
+         %{ls_src} %{ls_dst} %{rsi} %{rsj} %{c_src} %{xs} % 
+           [% [% // <Hrs_src //|<Hrs_dst >(\P Hceq) // ]]     
          #rsi0 #rsj0 #end #c #Hend #Hc_dst
          >Hrs_src in Hmid_src; >Hend #Hmid_src
          >Hrs_dst in Hmid_dst; >Hc_dst <(\P Hceq) #Hmid_dst
@@ -485,14 +552,14 @@ lemma sem_match_step :
             | @Hnotendc // ] 
           ]
          -Hcomp2 #Hcomp2 <Hcomp2
-         % // % [ % 
-         [>Hcomp2 in Hc; >nth_change_vec_neq [|@sym_not_eq //]
-          >nth_change_vec // #H lapply (H ? (refl …)) 
-          cases (is_endc end) [|normalize #H destruct (H) ]
-          #_ % // #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
-          [ >(\P Hc0) // | @Hnotendc // ]
-         |@Hmid_src]
-         |@Hmid_dst] ]
+         % // % [ 
+           >Hcomp2 in Hc; >nth_change_vec_neq [|@sym_not_eq //]
+            >nth_change_vec // #H lapply (H ? (refl …)) 
+            cases (is_endc end) [|normalize #H destruct (H) ]
+            #_ % // #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
+              [ >(\P Hc0) // | @Hnotendc // ]
+         |@Hmid_dst] 
+         ]
       |#_ #Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls * #rs #Hsrc
        %1 % 
         [% % %{c_src} % // lapply (Hc c_src) -Hc >Hcomp1
@@ -515,14 +582,23 @@ definition R_match_m ≝
   (((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
     current ? (nth i ? int (niltape ?)) = None ? ∨
     current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
-  (∀ls,x,xs,ci,rs,ls0,x0,rs0. 
+  (∀ls,x,xs,ci,rs,ls0,x0,rs0.
+    (∀x. is_startc x ≠ is_endc x) → 
     is_startc x = true → is_endc ci = true → 
+    (∀z. memb ? z (x::xs) = true → is_endc x = false) →
     nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
     nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
-    ∃l,cj,l1.x0::rs0 = l@x::xs@cj::l1 ∧
-    outt = change_vec ?? 
-           (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
-           (midtape sig ((reverse ? (l@x::xs))@ls0) cj l1) j).
+    (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
+     ∀cj,l2.l1=cj::l2 →
+     outt = change_vec ?? 
+            (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
+            (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) j) ∨
+    ∀l,l1.x0::rs0 ≠ l@x::xs@l1).
+
+(*
+axiom sub_list_dec: ∀A.∀l,ls:list A. 
+  ∃l1,l2. l = l1@ls@l2 ∨ ∀l1,l2. l ≠ l1@ls@l2.
+*)
 
 lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
 src ≠ dst → src < S n → dst < S n → 
@@ -530,23 +606,48 @@ src ≠ dst → src < S n → dst < S n →
 #src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
 lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst) … Hloop) //
 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ #tc whd in ⊢ (%→?); * 
+[ #tc whd in ⊢ (%→%); * 
   [ * * [ *
     [ * #cur_src * #H1 #H2 #Houtc %   
       [ #_ @Houtc
-      | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #_ #Hnthi #Hnthj
-        >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H %
+      | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hdiff #Hstartc #Hendc #Hnotend #Hnthi 
+        @False_ind
+        >Hnthi in H1; whd in ⊢ (??%?→?); #H destruct (H) cases (Hdiff cur_src)
+        #Habs @Habs //
       ]
-  | #Hci #Houtc %
-    [ #_ @Houtc
-    | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi >Hnthi in Hci;
-      normalize in ⊢ (%→?); #H destruct (H) ] ]
-  | #Hcj #Houtc %
-    [ #_ @Houtc
-    | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #Hnthj >Hnthj in Hcj;
-      normalize in ⊢ (%→?); #H destruct (H) ] ]
-
-
-
-[ #tc whd in ⊢ (%→?); * * [ *
+    | #Hci #Houtc %
+      [ #_ @Houtc
+      | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hdiff #Hstartc #Hendc #Hnotend 
+        #Hnthi >Hnthi in Hci; normalize in ⊢ (%→?); #H destruct (H) ] ]
+    | #Hcj #Houtc %
+      [ #_ @Houtc
+      | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hdiff #Hstartc #Hendc #_ #_ #Hnthj >Hnthj in Hcj;
+        normalize in ⊢ (%→?); #H destruct (H) ] 
+    ]
+  |* #ls * #ls0 * #rs * #rs0 * #x0 * #xs * * * #Hsrc #Hx0 #Hdst #H %
+    [>Hsrc * 
+      [* [* #x * whd in ⊢ (??%?→?); #Habs destruct (Habs) >Hx0 #Habs destruct (Habs)
+                |whd in ⊢ (??%?→?); #Habs destruct (Habs) ]
+         |>Hdst whd in ⊢ (??%?→?); #Habs destruct (Habs)  ]
+    |#ls1 #x1 #xs1 #ci #rsi #ls2 #x2 #rs2 
+     #Hdiff #Hstart #Hend #Hnotend 
+     >Hsrc #Hsrc1 destruct (Hsrc1) >Hdst #Hdst1 destruct (Hdst1) 
+     %1 %{[ ]} %{rs0} normalize in ⊢ (%→?); #Heq #cj #l2 #Hl1
+     cut (xs=xs1) 
+       [@(append_l1_injective_r … rs0 rs0 (refl …)) @(cons_injective_r …Heq)] 
+     #eqxs <eqxs 
+     whd in match (append ? [ ] (x2::xs)); >reverse_cons >associative_append
+     normalize in match (append ? [x2] ls2);
+     cases (H rsi l2 ci cj ? Hl1) 
+      [* #_ #_ #H3 @H3  
+      |>eqxs in e0; #e0 @(append_l2_injective … e0) // 
+      ] 
+    ]
+  ]
+|#tc #td #te #Hd #Hstar #IH #He lapply (IH He) -IH *
+    #IH1 #IH2 % [@IH1]
+    
+    
+     cases (comp_list ? (x1::xs1@ci::rsi) (x2::rs2) is_endc)
+     #l * #tl1 * #tl2 * * * #H1 #H2 #H3 #H4