X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Fmatch.ma;h=d7481fa7e67d6b3e522e8d51361f04d2200e06e2;hb=0b5cf203f54518bfa59b05a3567138ea64d1f6f5;hp=f16190491a202810abf3f2e5fb192b2d477a2884;hpb=302d83e414e27210fbfbc1de4d213786d9580e23;p=helm.git

diff --git a/matita/matita/lib/turing/multi_universal/match.ma b/matita/matita/lib/turing/multi_universal/match.ma
index f16190491..d7481fa7e 100644
--- a/matita/matita/lib/turing/multi_universal/match.ma
+++ b/matita/matita/lib/turing/multi_universal/match.ma
@@ -12,350 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "turing/multi_universal/moves.ma".
-include "turing/if_multi.ma".
-include "turing/inject.ma".
-include "turing/basic_machines.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 …)).
-
-(*
-
-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.λis_endc.
- 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 notb (is_endc ai) ∧ 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,is_endc.
-  mk_mTM sig n compare_states (trans_compare_step i j sig n is_endc) 
-    comp0 (λq.q == comp1 ∨ q == comp2).
-
-definition R_comp_step_true ≝ 
-  λi,j,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
-  ∃x.
-   is_endc x = false ∧
-   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.
-
-definition R_comp_step_false ≝ 
-  λi,j:nat.λsig,n,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 ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
-   current ? (nth i ? int (niltape ?)) = None ? ∨
-   current ? (nth j ? int (niltape ?)) = None ?) ∧ outt = int.
-
-lemma comp_q0_q2_null :
-  ∀i,j,sig,n,is_endc,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 is_endc) (mk_mconfig ??? comp0 v) 
-  = mk_mconfig ??? comp2 v.
-#i #j #sig #n #is_endc #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 ] ]
-qed.
-
-lemma comp_q0_q2_neq :
-  ∀i,j,sig,n,is_endc,v.i < S n → j < S n → 
-  ((∃x.nth i ? (current_chars ?? v) (None ?) = Some ? x ∧ is_endc x = true) ∨ 
-    nth i ? (current_chars ?? v) (None ?) ≠ nth j ? (current_chars ?? v) (None ?)) → 
-  step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v) 
-  = mk_mconfig ??? comp2 v.
-#i #j #sig #n #is_endc #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 *
-    [ * #c * >Hai #Heq #Hendc whd in ⊢ (??%?); 
-      >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
-      [ whd in match (trans ????); >Hai >Haj destruct (Heq) 
-        whd in ⊢ (??(???%)?); >Hendc // 
-      | whd in match (trans ????); >Hai >Haj destruct (Heq) 
-        whd in ⊢ (??(???????(???%))?); >Hendc @tape_move_null_action
-      ]
-    | #Hneq
-      whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
-      [ whd in match (trans ????); >Hai >Haj
-        whd in ⊢ (??(???%)?); cut ((¬is_endc ai∧ai==aj)=false)
-        [>(\bf ?) /2 by not_to_not/ cases (is_endc ai) // |#Hcut >Hcut //]
-        | whd in match (trans ????); >Hai >Haj
-          whd in ⊢ (??(???????(???%))?); cut ((¬is_endc ai∧ai==aj)=false)
-          [>(\bf ?) /2 by not_to_not/ cases (is_endc ai) // 
-          |#Hcut >Hcut @tape_move_null_action
-          ]
-        ]
-      ]
-    ]
-]
-qed.
-
-lemma comp_q0_q1 :
-  ∀i,j,sig,n,is_endc,v,a.i ≠ j → i < S n → j < S n → 
-  nth i ? (current_chars ?? v) (None ?) = Some ? a → is_endc a = false →
-  nth j ? (current_chars ?? v) (None ?) = Some ? a → 
-  step sig n (compare_step i j sig n is_endc) (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 #is_endc #v #a #Heq #Hi #Hj #Ha1 #Hnotendc #Ha2
-whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
-[ whd in match (trans ????);
-  >Ha1 >Ha2 whd in ⊢ (??(???%)?); >Hnotendc >(\b ?) //
-| whd in match (trans ????);
-  >Ha1 >Ha2 whd in ⊢ (??(???????(???%))?); >Hnotendc >(\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.
-
-lemma sem_comp_step :
-  ∀i,j,sig,n,is_endc.i ≠ j → i < S n → j < S n → 
-  compare_step i j sig n is_endc ⊨ 
-    [ comp1: R_comp_step_true i j sig n is_endc, 
-             R_comp_step_false i j sig n is_endc ].
-#i #j #sig #n #is_endc #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 % % %2 % #H destruct (H) ] ]
-  | #b #Hb %{2} 
-   cases (true_or_false (is_endc a)) #Haendc
-    [ %
-      [| % [ % 
-        [whd in ⊢  (??%?);  >comp_q0_q2_neq //
-         % %{a} % // <Ha @sym_eq @nth_vec_map
-        | normalize in ⊢ (%→?); #H destruct (H) ]
-      | #_ % // % % % >Ha %{a} % // ]
-      ]
-    |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 ?)) %2 >Ha >Hb
-           @(not_to_not ??? (\Pf Hab)) #H destruct (H) %
-          | normalize in ⊢ (%→?); #H destruct (H) ]
-        | #_ % // % % %2 >Ha >Hb @(not_to_not ??? (\Pf Hab)) #H destruct (H) % ] ]
-      ]
-    ]
-  ]
-]
-qed.
-
-definition compare ≝ λi,j,sig,n,is_endc.
-  whileTM … (compare_step i j sig n is_endc) comp1.
-
-definition R_compare ≝ 
-  λi,j,sig,n,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 ?)) ≠ 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,rs0. 
-    nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
-    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)
-           (midtape sig (reverse ? xs@x::ls0) cj rs1) j)).
-          
-lemma wsem_compare : ∀i,j,sig,n,is_endc.i ≠ j → i < S n → j < S n → 
-  compare i j sig n is_endc ⊫ R_compare i j sig n is_endc.
-#i #j #sig #n #is_endc #Hneq #Hi #Hj #ta #k #outc #Hloop
-lapply (sem_while … (sem_comp_step i j sig n is_endc Hneq Hi Hj) … Hloop) //
--Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ #tc whd in ⊢ (%→?); * * [ * [ *
-  [* #curi * #Hcuri #Hendi #Houtc %
-    [ #_ @Houtc  
-    | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi #Hnthj #Hnotendc 
-      @False_ind
-      >Hnthi in Hcuri; normalize in ⊢ (%→?); #H destruct (H)
-      >(Hnotendc ? (memb_hd … )) in Hendi; #H destruct (H)
-    ]
-  |#Hcicj #Houtc % 
-    [ #_ @Houtc
-    | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi #Hnthj
-      >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H %
-    ]]
-  | #Hci #Houtc %
-    [ #_ @Houtc
-    | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi >Hnthi in Hci;
-      normalize in ⊢ (%→?); #H destruct (H) ] ]
-  | #Hcj #Houtc %
-    [ #_ @Houtc
-    | #ls #x #xs #ci #rs #ls0 #rs0 #_ #Hnthj >Hnthj in Hcj;
-      normalize in ⊢ (%→?); #H destruct (H) ] ]
-  | #tc #td #te * #x * * * #Hendcx #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
-    #IH1 #IH2 %
-    [ >Hci >Hcj * [* #x0 * #H destruct (H) >Hendcx #H destruct (H) 
-    |* [* #H @False_ind [cases H -H #H @H % | destruct (H)] | #H destruct (H)]] 
-    | #ls #c0 #xs #ci #rs #ls0 #rs0 cases xs
-      [ #Hnthi #Hnthj #Hnotendc cases rs0 in Hnthj;
-        [ #Hnthj % % // >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 %2 %2 >nth_change_vec // >Hnthj % ]
-        | #r1 #rs1 #Hnthj %2 %{r1} %{rs1} % // *
-          [ #Hendci >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 % %{ci} % //
-        ]
-      |#Hcir1 >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 %2 % % >nth_change_vec //
-          >nth_change_vec_neq [|@sym_not_eq //]
-          >nth_change_vec // >Hnthi >Hnthj normalize @(not_to_not … Hcir1)
-          #H destruct (H) % ]
-      ]
-    ]
-  |#x0 #xs0 #Hnthi #Hnthj #Hnotendc 
-   cut (c0 = x) [ >Hnthi in Hci; normalize #H destruct (H) // ]
-   #Hcut destruct (Hcut) cases rs0 in Hnthj;
-    [ #Hnthj % % // 
-      cases (IH2 (x::ls) x0 xs0 ci rs (x::ls0) [ ] ???) -IH2
-      [ * #_ #IH2 >IH2 >Hd >change_vec_commute in ⊢ (??%?); //
-        >change_vec_change_vec >change_vec_commute in ⊢ (??%?); //
-        @sym_not_eq //
-      | * #cj * #rs1 * #H destruct (H)
-      | >Hd >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
-        >Hnthi %
-      | >Hd >nth_change_vec // >Hnthj %
-      | #c0 #Hc0 @Hnotendc @memb_cons @Hc0 ]
-    | #r1 #rs1 #Hnthj %2 %{r1} %{rs1} % // #Hcir1
-      cases(IH2 (x::ls) x0 xs0 ci rs (x::ls0) (r1::rs1) ???)
-      [ * #H destruct (H)
-      | * #r1' * #rs1' * #H destruct (H) #Hc1r1 >Hc1r1 //
-        >Hd >change_vec_commute in ⊢ (??%?); //
-        >change_vec_change_vec >change_vec_commute in ⊢ (??%?); //
-          @sym_not_eq //
-      | >Hd >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
-        >Hnthi //
-      | >Hd >nth_change_vec // >Hnthi >Hnthj %
-      | #c0 #Hc0 @Hnotendc @memb_cons @Hc0
-]]]]]
-qed.      
- 
-lemma terminate_compare :  ∀i,j,sig,n,is_endc,t.
-  i ≠ j → i < S n → j < S n → 
-  compare i j sig n is_endc ↓ t.
-#i #j #sig #n #is_endc #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 * * * #Hendcx >nth_change_vec // normalize in ⊢ (%→?);
-   #H1 destruct (H1) #Hxsep >change_vec_change_vec #Ht1 % 
-   #t2 * #x0 * * * #Hendcx0 >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
-  ]
-]
-qed.
-
-lemma sem_compare : ∀i,j,sig,n,is_endc.
-  i ≠ j → i < S n → j < S n → 
-  compare i j sig n is_endc ⊨ R_compare i j sig n is_endc.
-#i #j #sig #n #is_endc #Hneq #Hi #Hj @WRealize_to_Realize /2/
-qed.
-
-(*
-   |conf1   $
-   |confin 0/1 confout move
-
-  match machine step ≝
-    compare;
-    if (cur(src) != $)
-      then
-        parmoveL;
-        moveR(dst);
-      else nop
- *)
+include "turing/multi_universal/compare.ma".
 
 definition Rtc_multi_true ≝ 
   λalpha,test,n,i.λt1,t2:Vector ? (S n).
@@ -394,201 +51,7 @@ axiom comp_list: ∀S:DeqSet. ∀l1,l2:list S.∀is_endc. ∃l,tl1,tl2.
   
 axiom daemon : ∀X:Prop.X.
 
-(*
-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,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 → 
-  is_startc s = true → 
-  (∀c.c ∈ right ? (nth src (tape sig) int (niltape sig)) = true → is_startc c = false) →
-  (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧
-  (∀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,rs,ls0,rs0. 
-    nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
-    nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
-    (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → 
-    (∀cj,rs1.rs0 = cj::rs1 → 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_test_char_multi sig (λa.is_endc a == false) n src (le_S_S_to_le … Hsrc))
-      (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 * * * #c * #Hcurtc #Hcend #Htd >Htd -Htd
- #Htb #s #Hcurta_src #Hstart #Hnotstart % [ %
- [#Hdst_none @daemon 
- | #s1 #Hcurta_dst #Hneqss1
-   lapply Htb lapply Hcurtc -Htb -Hcurtc >(?:tc=ta) 
-   [|@Hcomp1 %2 % % >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) % ]
-   #Hcurtc * #te * * #_ #Hte >Hte [2: %1 %1 %{s} % //] 
-   whd in ⊢ (%→?); * * #_ #Htbdst #Htbelse %
-   [ @(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 >(Htbdst … Hta_mid) >Hta_mid cases rs //
-     | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Htbelse @sym_not_eq // ]
-   | >Hcurtc in Hcurta_src; #H destruct (H) cases (is_endc s) in Hcend;
-     normalize #H destruct (H) // ]
-   ]
- |#ls #x #xs #ci #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc 
-  cases rs00 in Htadst_mid;
-   [(* case rs empty *) #Htadst_mid % [ #cj #rs1 #H destruct (H) ]
-     #_ cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) -Hcomp2 
-     [2: * #x0 * #rs1 * #H destruct (H) ]
-    * #_ #Htc cases Htb #td * * #_ #Htd >Htasrc_mid in Hcurta_src; 
-    normalize in ⊢ (%→?); #H destruct (H)  
-    >Htd [2: %2 >Htc >nth_change_vec // cases (reverse sig ?) //]
-    >Htc * * >nth_change_vec // #Htbdst #_ #Htbelse
-     @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
-      [ >Hidst >nth_change_vec // <Htbdst // cases (reverse sig ?) //
-      |@sym_eq @Htbelse @sym_not_eq //
-      ] 
-    |#cj0 #rs0 #Htadst_mid % [| #H destruct (H) ]
-     #cj #rs1 #H destruct (H) #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 Htb #td * * #Htd #_ >Htasrc_mid in Hcurta_src; normalize in ⊢ (%→?);
-     #H destruct (H)
-     >(Htd 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 >nth_change_vec_neq [|@sym_not_eq // ] @nth_change_vec // ]
-    * * #_ #Htbdst #Htbelse %
-    [ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
-      [ >Hidst >nth_change_vec // >Htadst_mid >(Htbdst ls0 s (xs@cj'::rs0'))
-        [ cases xs //
-        | >nth_change_vec // ]
-      | >nth_change_vec_neq [|@sym_not_eq //]
-        <Htbelse [|@sym_not_eq // ]
-        >nth_change_vec_neq [|@sym_not_eq //]
-        cases (decidable_eq_nat i src) #Hisrc
-        [ >Hisrc >nth_change_vec // >Htasrc_mid //
-        | >nth_change_vec_neq [|@sym_not_eq //]
-          <(Htbelse i) [|@sym_not_eq // ]
-          >Htc >nth_change_vec_neq [|@sym_not_eq // ]
-          >nth_change_vec_neq [|@sym_not_eq // ] //
-        ]
-       ] 
-    | >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //] 
-      >nth_change_vec // whd in ⊢ (??%?→?); 
-      #H destruct (H) cases (is_endc c) in Hcend;
-      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 #Hnotendxs1 #Hneq %{ls_dst} %{rsj} >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 #Hta %
-         [ >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
-           #Hc lapply (Hc ? (refl ??)) #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 >Hmid_dst >Hxsxs1 % ]
-         | #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0) ]
-       | * #cj * #rs2 * #Hrs2 #Hta lapply (Hta ?) 
-         [ cases (Hneq … Hrs1) /2/ #H %2 @(H ?? Hrs2) ]
-         -Hta #Hta >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //] 
-         >nth_change_vec // #Hc lapply (Hc ? (refl ??)) #Hendr1
-         (* lemmatize this proof *) 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 >Hmid_dst >Hxsxs1 % //
-           #rsj0 #c #Hcrsj destruct (Hxsxs1 Hrs2 Hcrsj) @eq_f3 //
-           @eq_f3 // lapply (append_l2_injective ?????? Hrs_src) //
-           #Hendr1 destruct (Hendr1) % ]
-       ]
-     ]
-   (* STOP *)
-   |#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; >Hmid_src #Hc lapply (Hc ? (refl …)) -Hc 
-    >(Hnotend c_src) // normalize #H destruct (H)   
-   ]
-  ]
-]
-qed. 
-*)
 
 definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc.
   compare src dst sig n is_endc ·