X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Fmatch.ma;h=cc7c63b49b97009b798532d1f2d471627b36f272;hb=d64a1790db147a15917f3c999dc5b35211dc5b56;hp=91b887a2768720d12776cd74f09fb67577dff5b4;hpb=943bfd67310c34c05cbba627272864eb10800143;p=helm.git

diff --git a/matita/matita/lib/turing/multi_universal/match.ma b/matita/matita/lib/turing/multi_universal/match.ma
index 91b887a27..cc7c63b49 100644
--- a/matita/matita/lib/turing/multi_universal/match.ma
+++ b/matita/matita/lib/turing/multi_universal/match.ma
@@ -336,58 +336,22 @@ 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.
-
-axiom comp_list: ∀S:DeqSet. ∀l1,l2:list S.∀is_endc. ∃l,tl1,tl2. 
-  l1 = l@tl1 ∧ l2 = l@tl2 ∧ (∀c.c ∈ l = true → is_endc c = false) ∧
-  ∀a,tla. tl1 = a::tla → is_endc a = true ∨ (∀b,tlb.tl2 = b::tlb → a≠b).
-
-   
-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,x,x0,rs,rs0.
-     nth src ? int (niltape ?) = midtape sig ls x rs → 
-     nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 → 
-     x ≠ x0 ∨
-     (x = x0 ∧
-      ∀xs,end,rs',rs0'.rs = xs@end::rs' → rs0 = xs@rs0' → 
-      (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
-      is_endc end = false ∨
-      (is_endc end = true ∧
-      outt = change_vec ??
-           (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
-           (mk_tape sig (reverse ? xs@x::ls0) (option_hd ? rs0) (tail ? rs0)) dst))).
    
-     ∀ls,ls0,rs,rs0,x,xs,end. 
-     (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
-     nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) → 
-     nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) → 
-     is_endc end = false ∨
-     (is_endc end = true ∧
-      outt = change_vec ??
-           (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
-           (mk_tape sig (reverse ? xs@x::ls0) (option_hd ? rs0) (tail ? rs0)) 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).
@@ -438,9 +402,7 @@ lemma sem_match_step :
   match_step src dst sig n is_startc is_endc ⊨ 
     [ inr ?? (inr ?? (inl … (inr ?? start_nop))) : 
       R_match_step_true src dst sig n is_startc is_endc, 
-      R_match_step_false src dst sig n is_endc ]. 
-@daemon
-(*
+      R_match_step_false src dst sig n is_endc ].
 #src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst 
 @(acc_sem_seq_app sig n … (sem_compare src dst sig n is_endc Hneq Hsrc Hdst)
     (acc_sem_if ? n … (sem_test_char_multi sig (λa.is_endc a == false) n src (le_S_S_to_le … Hsrc))
@@ -512,7 +474,8 @@ 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
@@ -524,14 +487,14 @@ 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
@@ -543,67 +506,32 @@ lemma sem_match_step :
       ]
     ]
   ] 
-*)
 qed.
 
 definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc.
   whileTM … (match_step src dst sig n is_startc is_endc) 
     (inr ?? (inr ?? (inl … (inr ?? start_nop)))).
 
-(*
 definition R_match_m ≝ 
   λi,j,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
-  ∀ls,x,rs,ls0,x0,rs0.
-    nth i ? int (niltape ?) = midtape sig ls x rs →
-    nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
-    
-  ,xs,ci,rs,ls0,x0,rs0. 
-    is_startc x = true → is_endc ci = true → 
-    (∀c0.c0 ∈ (x::xs) = true → is_endc c0 = false) → 
-    nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
-    nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
-   
   (((∃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 → 
-    (∀c0.c0 ∈ (x::xs) = true → is_endc c0 = false) → 
+    (∀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).
-           
-lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
-src ≠ dst → src < S n → dst < S n → 
-  match_m src dst sig n is_startc is_endc ⊫ R_match_m src dst sig n is_startc is_endc.
-#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 ⊢ (%→?); * 
-  [ * * [ *
-    [ * #cur_src * #H1 #H2 #Houtc %   
-      [ #_ @Houtc
-      | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #_ #Hnotendc #Hnthsrc
-        @False_ind >Hnthsrc in H1;normalize
-        #H1 destruct (H1) >(Hnotendc ? (memb_hd …)) in H2; #H2 destruct (H2)
-      ]
-    | #Hci #Houtc %
-      [ #_ @Houtc
-      | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hstart #Hend_ci #Hnotend 
-        #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) ] ]
-  | * #ls * #ls0 * #rs * #rs0 * #x * #xs #Houtc %
-    [ Houtc ?? x x (refl ??) (refl ??))
-     #Hcases 
-      cut (∃end,rsi.rs = end::rsi ∧ nth src ? tc (niltape ?) = midtape ? ls x (xs@rs))
-      [ cases (nth src ? tc (niltape ?)) in
+    (∃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 → 
@@ -611,107 +539,47 @@ 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 %
-      ]
-  | #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 ⊢ (%→?); * * [ *
-
-*)
-
-definition R_match_m ≝ 
-  λi,j,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
-  (((∃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. 
-    is_startc x = true → is_endc ci = true → 
-    (∀c0.c0 ∈ (x::xs) = true → is_endc c0 = false) → 
-    nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
-    nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
-    (∃x1. is_endc x1 = false ∧ current ? (nth i ? outt (niltape ?)) = Some ? x1) ∨
-     (∃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)).
-           
-lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
-src ≠ dst → src < S n → dst < S n → 
-  match_m src dst sig n is_startc is_endc ⊫ R_match_m src dst sig n is_startc is_endc.
-#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 ⊢ (%→?); * #HR1 #HR2 % [ @HR1 ]
-  #ls #x #xs #ci #rs #ls0 #x0 #rs0 #Hstartc #Hendc #Hnotendc #Hsrctc #Hdsttc
-  cases (comp_list ? (x::xs@ci::rs) (x0::rs0) is_endc)
-  #l0 * #l1 * #l2 * * * #Heqsrc #Heqdst #Hnotendsrc #Hor
-  cut (∃x1,l1'.l1 = x1::l1') [@daemon] * #x1 * #l1' #Hl1
-  cases (Hor ?? Hl1) -Hor
-  [
-  cases HR2 -HR2 #HR2 [% @HR2]
-  |cut (is_endc x1 = false) [@daemon] #Hx1
-  
-  
-  [ * * [ *
-    [ * #cur_src * #H1 #H2 #Houtc %   
-      [ #_ @Houtc
-      | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #_ #Hnotendc #Hnthsrc
-        @False_ind >Hnthsrc in H1;normalize
-        #H1 destruct (H1) >(Hnotendc ? (memb_hd …)) in H2; #H2 destruct (H2)
+      | #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 #Hstart #Hend_ci #Hnotend 
+      | #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 #_ #_ #_ #_ #Hnthj >Hnthj in Hcj;
-        normalize in ⊢ (%→?); #H destruct (H) ] ]
-  | * #ls * #ls0 * #rs * #rs0 * #x * #xs #Houtc %
-    [ Houtc ?? x x (refl ??) (refl ??))
-     #Hcases 
-      cut (∃end,rsi.rs = end::rsi ∧ nth src ? tc (niltape ?) = midtape ? ls x (xs@rs))
-      [ cases (nth src ? tc (niltape ?)) in Hcases;
-        [
-
-
-lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
-src ≠ dst → src < S n → dst < S n → 
-  match_m src dst sig n is_startc is_endc ⊫ R_match_m src dst sig n is_startc is_endc.
-#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 ⊢ (%→?); * 
-  [ * * [ *
-    [ * #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 %
-      ]
-  | #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 ⊢ (%→?); * * [ *
+      | #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) // 
+      ] 
+    ]
+  ]
+| 
+    
+    
+     cases (comp_list ? (x1::xs1@ci::rsi) (x2::rs2) is_endc)
+     #l * #tl1 * #tl2 * * * #H1 #H2 #H3 #H4