+ ]
+ | >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 ·
+ (ifTM ?? (inject_TM ? (test_char ? (λa.is_endc a == false)) n src)
+ (ifTM ?? (inject_TM ? (test_null ?) n src)
+ (single_finalTM ??
+ (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst)))
+ (nop …) tc_true)
+ (nop …)
+ tc_true).
+
+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 ? ∧
+ (∀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 ?? (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
+(* test_null versione multi? *)
+@(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))
+ (acc_sem_if ? n … (sem_test_null 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 //