X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Fmatch.ma;h=76a1fc30d0a6dbd270d7c926a79b02de98239910;hb=c16905138e385d30856d587f07c396a3cab301ed;hp=b153ef8c7c5929f318a0206d623e571b133a9f32;hpb=315610badd512e271f6e99011721a3b4d3e316fc;p=helm.git

diff --git a/matita/matita/lib/turing/multi_universal/match.ma b/matita/matita/lib/turing/multi_universal/match.ma
index b153ef8c7..76a1fc30d 100644
--- a/matita/matita/lib/turing/multi_universal/match.ma
+++ b/matita/matita/lib/turing/multi_universal/match.ma
@@ -12,257 +12,409 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "turing/turing.ma".
-include "turing/inject.ma".
-include "turing/while_multi.ma".
+include "turing/multi_universal/compare.ma".
+include "turing/multi_universal/par_test.ma".
 
-definition compare_states ≝ initN 3.
 
-definition comp0 : compare_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
-definition comp1 : compare_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
-definition comp2 : compare_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+definition Rtc_multi_true ≝ 
+  λalpha,test,n,i.λt1,t2:Vector ? (S n).
+   (∃c. current alpha (nth i ? t1 (niltape ?)) = Some ? c ∧ test c = true) ∧ t2 = t1.
+   
+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.
 
-(*
-
-0) (x,x) → (x,x)(R,R) → 1
-   (x,y≠x) → None 2
-1) (_,_) → None 1
-2) (_,_) → None 2
-
-*)
-
-definition trans_compare_step ≝ 
- λi,j.λsig:FinSet.λn.
- λp:compare_states × (Vector (option sig) (S n)).
- let 〈q,a〉 ≝ p in
- match pi1 … q with
- [ O ⇒ match nth i ? a (None ?) with
-   [ None ⇒ 〈comp2,null_action ? n〉
-   | Some ai ⇒ match nth j ? a (None ?) with 
-     [ None ⇒ 〈comp2,null_action ? n〉
-     | Some aj ⇒ if ai == aj 
-         then 〈comp1,change_vec ? (S n) 
-                      (change_vec ? (S n) (null_action ? n) (Some ? 〈ai,R〉) i)
-                        (Some ? 〈aj,R〉) j〉
-         else 〈comp2,null_action ? n〉 ]
-   ]
- | S q ⇒ match q with 
-   [ O ⇒ (* 1 *) 〈comp1,null_action ? n〉
-   | S _ ⇒ (* 2 *) 〈comp2,null_action ? n〉 ] ].
-
-definition compare_step ≝ 
-  λi,j,sig,n.
-  mk_mTM sig n compare_states (trans_compare_step i j sig n) 
-    comp0 (λq.q == comp1 ∨ q == comp2).
-
-definition R_comp_step_true ≝ 
-  λi,j,sig,n.λint,outt: Vector (tape sig) (S n).
-  ∃x.
-   current ? (nth i ? int (niltape ?)) = Some ? x ∧
-   current ? (nth j ? int (niltape ?)) = Some ? x ∧
-   outt = change_vec ?? 
-            (change_vec ?? int
-              (tape_move ? (nth i ? int (niltape ?)) (Some ? 〈x,R〉)) i)
-            (tape_move ? (nth j ? int (niltape ?)) (Some ? 〈x,R〉)) j.
+lemma sem_test_char_multi :
+  ∀alpha,test,n,i.i ≤ n → 
+  inject_TM ? (test_char ? test) n i ⊨ 
+  [ tc_true : Rtc_multi_true alpha test n i, Rtc_multi_false alpha test n i ].
+#alpha #test #n #i #Hin #int
+cases (acc_sem_inject … Hin (sem_test_char alpha test) int)
+#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
+[ @Hloop
+| #Hqtrue lapply (Htrue Hqtrue) * * * #c *
+  #Hcur #Htestc #Hnth_i #Hnth_j %
+  [ %{c} % //
+  | @(eq_vec … (niltape ?)) #i0 #Hi0
+    cases (decidable_eq_nat i0 i) #Hi0i
+    [ >Hi0i @Hnth_i
+    | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
+| #Hqfalse lapply (Hfalse Hqfalse) * * #Htestc #Hnth_i #Hnth_j %
+  [ @Htestc
+  | @(eq_vec … (niltape ?)) #i0 #Hi0
+    cases (decidable_eq_nat i0 i) #Hi0i
+    [ >Hi0i @Hnth_i
+    | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
+qed.
 
-definition R_comp_step_false ≝ 
-  λi,j:nat.λsig,n.λint,outt: Vector (tape sig) (S n).
-  (current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
-   current ? (nth i ? int (niltape ?)) = None ? ∨
-   current ? (nth j ? int (niltape ?)) = None ?) ∧ outt = int.
+definition Rm_test_null_true ≝ 
+  λalpha,n,i.λt1,t2:Vector ? (S n).
+   current alpha (nth i ? t1 (niltape ?)) ≠ None ? ∧ t2 = t1.
+   
+definition Rm_test_null_false ≝ 
+  λalpha,n,i.λt1,t2:Vector ? (S n).
+    current alpha (nth i ? t1 (niltape ?)) = None ? ∧ t2 = t1.
 
-lemma comp_q0_q2_null :
-  ∀i,j,sig,n,v.i < S n → j < S n → 
-  (nth i ? (current_chars ?? v) (None ?) = None ? ∨
-   nth j ? (current_chars ?? v) (None ?) = None ?) → 
-  step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v) 
-  = mk_mconfig ??? comp2 v.
-#i #j #sig #n #v #Hi #Hj
-whd in ⊢ (? → ??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (?→??%?);
-* #Hcurrent
-[ @eq_f2
-  [ whd in ⊢ (??(???%)?); >Hcurrent %
-  | whd in ⊢ (??(???????(???%))?); >Hcurrent @tape_move_null_action ]
-| @eq_f2
-  [ whd in ⊢ (??(???%)?); >Hcurrent cases (nth i ?? (None sig)) //
-  | whd in ⊢ (??(???????(???%))?); >Hcurrent
-    cases (nth i ?? (None sig)) [|#x] @tape_move_null_action ] ]
+lemma sem_test_null_multi : ∀alpha,n,i.i ≤ n → 
+  inject_TM ? (test_null ?) n i ⊨ 
+    [ tc_true : Rm_test_null_true alpha n i, Rm_test_null_false alpha n i ].
+#alpha #n #i #Hin #int
+cases (acc_sem_inject … Hin (sem_test_null alpha) int)
+#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
+[ @Hloop
+| #Hqtrue lapply (Htrue Hqtrue) * * #Hcur #Hnth_i #Hnth_j % //
+  @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) #Hi0i
+  [ >Hi0i @sym_eq @Hnth_i | @sym_eq @Hnth_j @sym_not_eq // ] ] 
+| #Hqfalse lapply (Hfalse Hqfalse) * * #Hcur #Hnth_i #Hnth_j %
+  [ @Hcur
+  | @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) // 
+    #Hi0i @sym_eq @Hnth_j @sym_not_eq // ] ] 
 qed.
 
-lemma comp_q0_q2_neq :
-  ∀i,j,sig,n,v.i < S n → j < S n → 
-  nth i ? (current_chars ?? v) (None ?) ≠ nth j ? (current_chars ?? v) (None ?) → 
-  step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v) 
-  = mk_mconfig ??? comp2 v.
-#i #j #sig #n #v #Hi #Hj lapply (refl ? (nth i ?(current_chars ?? v)(None ?)))
-cases (nth i ?? (None ?)) in ⊢ (???%→?);
-[ #Hnth #_ @comp_q0_q2_null // % //
-| #ai #Hai lapply (refl ? (nth j ?(current_chars ?? v)(None ?)))
-  cases (nth j ?? (None ?)) in ⊢ (???%→?);
-  [ #Hnth #_ @comp_q0_q2_null // %2 //
-  | #aj #Haj #Hneq
-    whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
-    [ whd in match (trans ????); >Hai >Haj
-      whd in ⊢ (??(???%)?); >(\bf ?) // @(not_to_not … Hneq) //
-    | whd in match (trans ????); >Hai >Haj
-      whd in ⊢ (??(???????(???%))?); >(\bf ?) /2 by not_to_not/
-      @tape_move_null_action
-] ]
-qed.
+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).
+  
+axiom daemon : ∀X:Prop.X.
 
-lemma comp_q0_q1 :
-  ∀i,j,sig,n,v,a.i ≠ j → i < S n → j < S n → 
-  nth i ? (current_chars ?? v) (None ?) = Some ? a →
-  nth j ? (current_chars ?? v) (None ?) = Some ? a → 
-  step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v) =
-    mk_mconfig ??? comp1 
-     (change_vec ? (S n) 
-       (change_vec ?? v
-         (tape_move ? (nth i ? v (niltape ?)) (Some ? 〈a,R〉)) i)
-       (tape_move ? (nth j ? v (niltape ?)) (Some ? 〈a,R〉)) j).
-#i #j #sig #n #v #a #Heq #Hi #Hj #Ha1 #Ha2
-whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
-[ whd in match (trans ????);
-  >Ha1 >Ha2 whd in ⊢ (??(???%)?); >(\b ?) //
-| whd in match (trans ????);
-  >Ha1 >Ha2 whd in ⊢ (??(???????(???%))?); >(\b ?) //
-  change with (change_vec ?????) in ⊢ (??(???????%)?);
-  <(change_vec_same … v j (niltape ?)) in ⊢ (??%?);
-  <(change_vec_same … v i (niltape ?)) in ⊢ (??%?);
-  >pmap_change >pmap_change >tape_move_null_action
-  @eq_f2 // @eq_f2 // >nth_change_vec_neq //
-]
-qed.
+definition match_test ≝ λsrc,dst.λsig:DeqSet.λn,is_endc.λv:Vector ? n.
+  match (nth src (option sig) v (None ?)) with 
+  [ None ⇒  false 
+  | Some x ⇒  notb ((is_endc x) ∨ (nth dst (DeqOption sig) v (None ?) == None ?))]. 
 
-lemma sem_comp_step :
-  ∀i,j,sig,n.i ≠ j → i < S n → j < S n → 
-  compare_step i j sig n ⊨ 
-    [ comp1: R_comp_step_true i j sig n, 
-             R_comp_step_false i j sig n ].
-#i #j #sig #n #Hneq #Hi #Hj #int
-lapply (refl ? (current ? (nth i ? int (niltape ?))))
-cases (current ? (nth i ? int (niltape ?))) in ⊢ (???%→?);
-[ #Hcuri %{2} %
-  [| % [ %
-    [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ % <Hcuri in ⊢ (???%); 
-      @sym_eq @nth_vec_map
-    | normalize in ⊢ (%→?); #H destruct (H) ]
-  | #_ % // % %2 // ] ]
-| #a #Ha lapply (refl ? (current ? (nth j ? int (niltape ?))))
-  cases (current ? (nth j ? int (niltape ?))) in ⊢ (???%→?);
-  [ #Hcurj %{2} %
-    [| % [ %
-       [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ %2 <Hcurj in ⊢ (???%); 
-         @sym_eq @nth_vec_map
-       | normalize in ⊢ (%→?); #H destruct (H) ]
-       | #_ % >Ha >Hcurj % % % #H destruct (H) ] ]
-  | #b #Hb %{2} cases (true_or_false (a == b)) #Hab
-    [ %
-      [| % [ % 
-        [whd in ⊢  (??%?);  >(comp_q0_q1 … a Hneq Hi Hj) //
-          [>(\P Hab) <Hb @sym_eq @nth_vec_map
-          |<Ha @sym_eq @nth_vec_map ]
-        | #_ whd >(\P Hab) %{b} % // % // <(\P Hab) // ]
-        | * #H @False_ind @H %
-      ] ]
-    | %
-      [| % [ % 
-        [whd in ⊢  (??%?);  >comp_q0_q2_neq //
-         <(nth_vec_map ?? (current …) i ? int (niltape ?))
-         <(nth_vec_map ?? (current …) j ? int (niltape ?)) >Ha >Hb
-         @(not_to_not ??? (\Pf Hab)) #H destruct (H) %
-        | normalize in ⊢ (%→?); #H destruct (H) ]
-      | #_ % // % % >Ha >Hb @(not_to_not ??? (\Pf Hab)) #H destruct (H) % ] ]
+definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc.
+  compare src dst sig n is_endc ·
+     (ifTM ?? (partest sig n (match_test src dst sig ? is_endc))
+      (single_finalTM ??
+        (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst)))
+      (nop …)
+      partest1).
+      
+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,xs0. nth dst ? int (niltape ?) = midtape sig ls0 x rs0 ∧
+      xs = rs0@xs0 ∧
+      current sig (nth dst (tape sig) outt (niltape sig)) = None ?) ∨
+    (∃ls0,rs0. 
+     nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
+     ∀rsj,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_true ≝ 
+  λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
+  ∀s.current sig (nth src (tape sig) int (niltape sig)) = Some ? s → 
+  current sig (nth dst (tape sig) int (niltape sig)) ≠ None ? ∧
+  (is_startc s = true → 
+   (∀c.c ∈ right ? (nth src (tape sig) int (niltape sig)) = true → is_startc c = false) →
+   (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 → s ≠ s1 →  
+    outt = change_vec ?? int 
+          (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈s1,R〉)) dst ∧ is_endc s = false) ∧  
+   (∀ls,x,xs,ci,cj,rs,ls0,rs0. 
+     nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
+     nth dst ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) →
+     (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → 
+      ci ≠ cj →
+      (outt = change_vec ?? int 
+          (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false))). 
+(*    ∧
+    (rs0 = [ ] →
+     outt = change_vec ??
+           (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) src)
+           (mk_tape sig (reverse ? xs@x::ls0) (None ?) [ ]) dst)). *)
+           
+lemma sem_match_step :
+  ∀src,dst,sig,n,is_startc,is_endc.src ≠ dst → src < S n → dst < S n → 
+  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 ].
+#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_partest sig n (match_test src dst sig ? is_endc))
+      (sem_seq … 
+        (sem_parmoveL ???? is_startc Hneq Hsrc Hdst) 
+        (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
+      (sem_nop …)))
+[#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd >Htd -Htd
+ * #te * #Hte #Htb whd 
+ #s #Hcurta_src % 
+ [ lapply (refl ? (current ? (nth dst ? ta (niltape ?)))) 
+   cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
+   [| #c #_ % #Hfalse destruct (Hfalse) ]
+   #Hcurta_dst >Hcomp1 in Htest; [| %2 %2 //]
+   whd in ⊢ (??%?→?); change with (current ? (niltape ?)) in match (None ?);
+    <nth_vec_map >Hcurta_src whd in ⊢ (??%?→?); <nth_vec_map
+    >Hcurta_dst cases (is_endc s) normalize in ⊢ (%→?); #H destruct (H)
+ | #Hstart #Hnotstart %
+   [ #s1 #Hcurta_dst #Hneqss1 -Hcomp2
+     cut (tc = ta) 
+     [@Hcomp1 %2 %1 %1 >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) //] 
+     #H destruct (H) -Hcomp1 cases Hte #_ -Hte #Hte
+     cut (te = ta) [@Hte %1 %1 %{s} % //] -Hte #H destruct (H) %
+     [cases Htb * #_ #Hmove #Hmove1 @(eq_vec … (niltape … ))
+      #i #Hi cases (decidable_eq_nat i dst) #Hidst
+       [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
+         #ls * #rs #Hta_mid >(Hmove … Hta_mid) >Hta_mid cases rs //
+       | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Hmove1 @sym_not_eq // ]
+     | whd in Htest:(??%?); >(nth_vec_map ?? (current sig)) in Hcurta_src; #Hcurta_src
+       >Hcurta_src in Htest; whd in ⊢ (??%?→?);
+       cases (is_endc s) // whd in ⊢ (??%?→?); #H @sym_eq // 
+     ]
+   |#ls #x #xs #ci #cj #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc #Hcicj 
+    cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) [ * #H destruct (H) ]
+    * #cj' * #rs0' * #Hcjrs0 destruct (Hcjrs0) -Hcomp2 #Hcomp2
+    lapply (Hcomp2 (or_intror ?? Hcicj)) -Hcomp2 #Htc %
+    [ cases Hte -Hte #Hte #_ whd in Hte;
+      >Htasrc_mid in Hcurta_src; whd in ⊢ (??%?→?); #H destruct (H) 
+      lapply (Hte ls ci (reverse ? xs) rs s ??? ls0 cj' (reverse ? xs) s rs0' (refl ??) ?) //
+      [ >Htc >nth_change_vec //
+      | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid cases (orb_true_l … Hc0) -Hc0 #Hc0
+        [@memb_append_l2 >(\P Hc0) @memb_hd
+        |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse //
+        ]
+      | >Htc >change_vec_commute // >nth_change_vec // ] -Hte
+      >Htc >change_vec_commute // >change_vec_change_vec 
+      >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
+      >Hte in Htb; * * #_ >reverse_reverse #Htbdst1 #Htbdst2 -Hte @(eq_vec … (niltape ?))
+      #i #Hi cases (decidable_eq_nat i dst) #Hidst
+      [ >Hidst >nth_change_vec // >(Htbdst1 ls0 s (xs@cj'::rs0'))
+        [| >nth_change_vec // ]
+        >Htadst_mid cases xs //
+      | >nth_change_vec_neq [|@sym_not_eq // ]
+        <Htbdst2 [| @sym_not_eq // ] >nth_change_vec_neq [| @sym_not_eq // ]
+        <Htasrc_mid >change_vec_same % ]
+    | >Hcurta_src in Htest; whd in ⊢(??%?→?);
+      >Htc >change_vec_commute //
+      change with (current ? (niltape ?)) in match (None ?);
+      <nth_vec_map >nth_change_vec // whd in ⊢ (??%?→?);
+      cases (is_endc ci) whd in ⊢ (??%?→?); #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
+     cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * * 
+     #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq >Hrs_dst in Hmid_dst; #Hmid_dst
+     cut (∃r1,rs1.rsi = r1::rs1) [@daemon] * #r1 * #rs1 #Hrs1 >Hrs1 in Hrs_src;
+     #Hrs_src >Hrs_src in Hmid_src; #Hmid_src <(\P Hceq) in Hmid_dst; #Hmid_dst
+     lapply (Hcomp2 ??????? Hmid_src Hmid_dst ?) 
+     [ #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
+       [ >(\P Hc0) @Hnotend @memb_hd | @Hnotendxs1 //] ] 
+     *
+     [ * #Hrsj >Hrsj #Hta % %2 >Hta >nth_change_vec //
+       %{ls_dst} %{xs1} cut (∃xs0.xs = xs1@xs0)
+       [lapply Hnotendxs1 -Hnotendxs1 lapply Hrs_src lapply xs elim xs1
+         [ #l #_ #_ %{l} %
+         | #x2 #xs2 #IH * 
+           [ whd in ⊢ (??%%→?); #H destruct (H) #Hnotendxs2
+             >Hnotendxs2 in Hend; [ #H destruct (H) |@memb_hd ]
+           | #x2' #xs2' whd in ⊢ (??%%→?); #H destruct (H)
+             #Hnotendxs2 cases (IH xs2' e0 ?)
+             [ #xs0 #Hxs2 %{xs0} @eq_f //
+             |#c #Hc @Hnotendxs2 @memb_cons // ]
+           ]
+         ] 
+       ] * #xs0 #Hxs0 %{xs0} % [ %
+       [ >Hmid_dst >Hrsj >append_nil %
+       | @Hxs0 ]
+       | cases (reverse ? xs1) // ]
+     | * #cj * #rs2 * #Hrsj #Hta lapply (Hta ?)
+       [ cases (Hneq ?? Hrs1) /2/ #Hr1 %2 @(Hr1 ?? Hrsj) ] -Hta #Hta
+       %2 >Hta in Hc; whd in ⊢ (??%?→?);
+       change with (current ? (niltape ?)) in match (None ?);
+       <nth_vec_map >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
+       whd in ⊢ (??%?→?); #Hc cut (is_endc r1 = true)
+       [ cases (is_endc r1) in Hc; whd in ⊢ (??%?→?); //
+         change with (current ? (niltape ?)) in match (None ?);
+         <nth_vec_map >nth_change_vec // normalize #H destruct (H) ]
+       #Hendr1 cut (xs = xs1)
+       [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1
+         -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs
+         [ * normalize in ⊢ (%→?); //
+           #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1
+           lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H)
+         | #x2 #xs2 #IH *
+           [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc
+             >Hnotendc in Hendr1; [| @memb_cons @memb_hd ]
+             normalize in ⊢ (%→?); #H destruct (H)
+           | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq)
+             #Hnotendc #Hnotendcxs1 @eq_f @IH
+             [ @(cons_injective_r … Heq)
+             | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0
+               [ >(\P Hc0) @memb_hd
+               | @memb_cons @memb_cons // ]
+             | #c #Hc @Hnotendcxs1 @memb_cons // ]
+           ]
+         ]
+       | #Hxsxs1 destruct (Hxsxs1) >Hmid_dst %{ls_dst} %{rsj} % //
+         #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0) 
+         lapply (append_l2_injective … Hrs_src) // #Hrs' destruct (Hrs') %
+       ]
+     ]
+    |#Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls0 * #rs0 #Hdst 
+     @False_ind lapply (Hcomp1 ?) [%2 %1 %1 >Hmid_src normalize
+     @(not_to_not ??? (\Pf Hceq)) #H destruct //] #Hintape >Hintape in Hc;
+     whd in ⊢(??%?→?); >Hmid_src  
+     change with (current ? (niltape ?)) in match (None ?);
+     <nth_vec_map >Hmid_src whd in ⊢ (??%?→?);
+     >(Hnotend c_src) [|@memb_hd]
+     change with (current ? (niltape ?)) in match (None ?);
+     <nth_vec_map >Hmid_src whd in ⊢ (??%?→?); >Hdst normalize #H destruct (H)
+   ]
   ]
 ]
 qed.
 
-definition compare ≝ λi,j,sig,n.
-  whileTM … (compare_step i j sig n) comp1.
+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 ≝ 
+  λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
+(*  (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧ *)
+  ∀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) ∧
+  (is_startc x = true →
+   (∀ls0,x0,rs0.
+    nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
+    (∃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) end rs) src)
+            (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) dst) ∨
+    ∀l,l1.x0::rs0 ≠ l@x::xs@l1)).
 
-definition R_compare ≝ 
-  λi,j,sig,n.λint,outt: Vector (tape sig) (S n).
-  (current sig (nth i (tape sig) int (niltape sig))
-   ≠current sig (nth j (tape sig) int (niltape sig)) → 
-   outt = int) ∧  
-  (∀ls,x,xs,ci,rs,ls0,cj,rs0. 
+(*
+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.
+    (∀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 x (xs@cj::rs0) → ci ≠ cj → 
-    outt = change_vec ?? 
-           (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
-           (midtape sig (reverse ? xs@x::ls0) cj rs0) j).
+    nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
+    (∃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).
+*)
 
-lemma wsem_compare : ∀i,j,sig,n.i ≠ j → i < S n → j < S n → 
-  compare i j sig n ⊫ R_compare i j sig n.
-#i #j #sig #n #Hneq #Hi #Hj #ta #k #outc #Hloop
-lapply (sem_while … (sem_comp_step i j sig n Hneq Hi Hj) … Hloop) //
+(*
+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 → 
+  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 ⊢ (%→?); * * [ *
-  [ #Hcicj #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 #td #te * #x * * #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
-    #IH1 #IH2 %
-    [ >Hci >Hcj * #H @False_ind @H %
-    | #ls #c0 #xs #ci #rs #ls0 #cj #rs0 cases xs
-      [ #Hnthi #Hnthj #Hcicj >IH1 
-        [ >Hd @eq_f3 // 
-          [ @eq_f3 // >(?:c0=x) [ >Hnthi % ]
-            >Hnthi in Hci;normalize #H destruct (H) %
-          | >(?:c0=x) [ >Hnthj % ]
-            >Hnthi in Hci;normalize #H destruct (H) % ]
-        | >Hd >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
-          >nth_change_vec // >Hnthi >Hnthj normalize @(not_to_not ??? Hcicj)
-          #H destruct (H) % ]
-      | #x0 #xs0 #Hnthi #Hnthj #Hcicj
-        >(IH2 (c0::ls) x0 xs0 ci rs (c0::ls0) cj rs0 … Hcicj)
-        [ >Hd >change_vec_commute in ⊢ (??%?); //
-          >change_vec_change_vec >change_vec_commute in ⊢ (??%?); //
-          @sym_not_eq //
-        | >Hd >nth_change_vec // >Hnthj normalize
-          >Hnthi in Hci;normalize #H destruct (H) %
-        | >Hd >nth_change_vec_neq [|@sym_not_eq //] >Hnthi
-          >nth_change_vec // normalize
-          >Hnthi in Hci;normalize #H destruct (H) %
+[ #tc #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend
+  cases (Hfalse … Hmid_src Hnotend Hend) -Hfalse 
+  [(* current dest = None *) *
+    [ * #Hcur_dst #Houtc %
+      [#_ >Houtc //
+      |#Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst; 
+       normalize in ⊢ (%→?); #H destruct (H)
+      ]
+    | * #ls0 * #rs0 * #xs0 * * #Htc_dst #Hrs0 #HNone %
+      [ >Htc_dst normalize in ⊢ (%→?); #H destruct (H)
+      | #Hstart #ls1 #x1 #rs1 >Htc_dst #H destruct (H)
+        >Hrs0 cases xs0
+        [ % %{[ ]} %{[ ]} % [ >append_nil >append_nil %]
+          #cj #ls2 #H destruct (H)
+        | #x2 #xs2 %2 #l #l1 % #Habs lapply (eq_f ?? (length ?) ?? Habs)
+          >length_append whd in ⊢ (??%(??%)→?); >length_append
+          >length_append normalize >commutative_plus whd in ⊢ (???%→?);
+          #H destruct (H) lapply e0 >(plus_n_O (|rs1|)) in ⊢ (??%?→?);
+          >associative_plus >associative_plus 
+          #e1 lapply (injective_plus_r ??? e1) whd in ⊢ (???%→?);
+          #e2 destruct (e2)
         ]
-]]]
-qed.      
- 
-lemma terminate_compare :  ∀i,j,sig,n,t.
-  i ≠ j → i < S n → j < S n → 
-  compare i j sig n  ↓ t.
-#i #j #sig #n #t #Hneq #Hi #Hj
-@(terminate_while … (sem_comp_step …)) //
-<(change_vec_same … t i (niltape ?))
-cases (nth i (tape sig) t (niltape ?))
-[ % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct 
-|2,3: #a0 #al0 % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
-| #ls #c #rs lapply c -c lapply ls -ls lapply t -t elim rs
-  [#t #ls #c % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?);
-   #H1 destruct (H1) #Hxsep >change_vec_change_vec #Ht1 % 
-   #t2 * #x0 * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
-   >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H)
-  |#r0 #rs0 #IH #t #ls #c % #t1 * #x * * >nth_change_vec //
-   normalize in ⊢ (%→?); #H destruct (H) #Hcur
-   >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH
+      ]
+    ]
+  |* #ls0 * #rs0 * #Hmid_dst #HFalse %
+    [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
+    | #Hstart #ls1 #x1 #rs1 >Hmid_dst #H destruct (H)
+     %1 %{[ ]} %{rs0} % [%] #cj #l2 #Hnotnil 
+     >reverse_cons >associative_append @(HFalse ?? Hnotnil)
+    ]
   ]
-]
-qed.
+|#ta #tb #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
+ #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend 
+ lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
+ cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?); 
+  [#Hmid_dst % 
+    [#_ whd in Htrue; >Hmid_src in Htrue; #Htrue
+     cases (Htrue x (refl … )) -Htrue * #Htaneq #_
+     @False_ind >Hmid_dst in Htaneq; /2/
+    |#Hstart #ls0 #x0 #rs0 #Hmid_dst2 >Hmid_dst2 in Hmid_dst; normalize in ⊢ (%→?); 
+     #H destruct (H)
+    ]
+  | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ]
+    #Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcurta_dst; normalize in ⊢ (%→?);
+    #H destruct (H) whd in Htrue; >Hmid_src in Htrue; #Htrue
+    cases (Htrue x (refl …)) -Htrue #_ #Htrue cases (Htrue Hstart ?) -Htrue
+    [2: #z #membz @daemon (*aggiungere l'ipotesi*)]
+    cases (true_or_false (x==c)) #eqx
+    [ #_ #Htrue cases (comp_list ? (xs@end::rs) rs0 is_endc)
+      #x1 * #tl1 * #tl2 * * * #Hxs #Hrs0 #Hnotendx1
+      cases tl1 in Hxs; 
+      [>append_nil #Hx1 @daemon (* absurd by Hx1 e notendx1 *)]
+      #ci -tl1 #tl1 #Hxs #H cases (H … (refl … ))
+      [(* this is absurd, since Htrue conlcudes is_endc ci =false *)
+       #Hend_ci @daemon (* lapply(Htrue … (refl …)) -Htrue *)
+      |cases tl2 in Hrs0;
+        [
+        | #cj #tl2' #Hrs0 #Hcomp lapply (Htrue ls x x1 ci cj tl1 ls0 tl2' ????)
+          [ @(Hcomp ?? (refl ??))
+          | #c0 #Hc0 cases (orb_true_l … Hc0) #Hc0
+            [ @Hnotend >(\P Hc0) @memb_hd
+            | @Hnotendx1 // ]
+          | >Hmid_dst >Hrs0 >(\P eqx) %
+          | >Hxs %
+          | * #Htb >Htb #Hendci %2 >Hrs0 >Hxs
+            cases (IH ls x xs end rs ? Hnotend Hend) [|
+            STOP
+    
 
-lemma sem_compare : ∀i,j,sig,n.
-  i ≠ j → i < S n → j < S n → 
-  compare i j sig n ⊨ R_compare i j sig n.
-#i #j #sig #n #Hneq #Hi #Hj @WRealize_to_Realize /2/
+          
+           >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
+          
+        >Hrs0 in Hmid_dst; #Hmid_dst
+       cases(Htrue ???????? Hmid_dst) -Htrue #Htb #Hendx
+       whd in IH;
+       cases(IH ls x xs end rs ? Hstart Hnotend Hend)
+       [* #H1 #H2 >Htb in H1; >nth_change_vec // 
+        >Hmid_dst cases rs0 [2: #a #tl normalize in ⊢ (%→?); #H destruct (H)] 
+        #_ %2 @daemon (* si dimostra *)
+       |@daemon
+       |>Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src
+       ] 
+    ]
+  ]
+]
 qed.