- (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 %
- [ #s1 #Hcurta_dst #Hneqss1
- lapply Htb lapply Hcurtc -Htb -Hcurtc >(?:tc=ta)
- [|@Hcomp1 % % >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) % ]
- #Hcurtc * #te * * #_ #Hte >Hte // 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 #cj #rs0 #Htasrc_mid #Htadst_mid #Hcicj
- lapply (Hcomp2 … Htasrc_mid Htadst_mid Hcicj) -Hcomp2 #Hcomp2
- 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 ??)) //
- [| >Hcomp2 >nth_change_vec //
- | @daemon
- | >Hcomp2 >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 //]
- STOP.
-
- >nth_change_vec in Htbdst; // #Htbdst >(Htbdst
- … Htadst_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) // ]
-|#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;
- [#Hcomp1 #_ %1 % [%1 %2 // | @Hcomp1 %1 %2 %]
- |#c_src lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
- cases (current … (nth dst ? intape (niltape ?)))
- [#_ #Hcomp1 #_ %1 % [%2 % | @Hcomp1 %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 #_
- #Hmid_src cases (Hmid_src c_src (refl …)) -Hmid_src
- #ls_src * #rs_src #Hmid_src %2
- cases (comp_list … rs_src rs_dst) #xs * #rsi * #rsj * *
- #Hrs_src #Hrs_dst #Hneq
- %{ls_src} %{ls_dst} %{rsi} %{rsj} %{c_src} %{xs}
- #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
- lapply(Hcomp2 … Hmid_src Hmid_dst ?)
- [@(Hneq … Hend Hc_dst)]
- -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 //
- |@Hmid_src]
- |@Hmid_dst]
- |#_ #Hcomp1 #_ %1 %
- [% % @(not_to_not ??? (\Pf Hceq)) #H destruct (H) //
- |@Hcomp1 %1 %1 @(not_to_not ??? (\Pf Hceq)) #H destruct (H) //
+ (sem_rewind_strong ???? Hneq Hsrc Hdst)
+ (sem_move_multi … R ?))
+ (sem_nop …))) [/2/]
+[ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?))))
+ cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%);
+ [ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %]
+ whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
+ >nth_current_chars >Hcurta_src normalize in ⊢ (%→?); #H destruct (H)
+ | #s #Hs lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
+ cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
+ [ #Hcurta_dst #Hcomp #_ * #td * >Hcomp [| %2 %]
+ whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
+ >nth_current_chars >nth_current_chars >Hs >Hcurta_dst
+ normalize in ⊢ (%→?); #H destruct (H)
+ | #s0 #Hs0
+ cases (current_to_midtape … Hs) #ls * #rs #Hmidta_src >Hmidta_src
+ cases (current_to_midtape … Hs0) #ls0 * #rs0 #Hmidta_dst >Hmidta_dst
+ cases (true_or_false (s == s0)) #Hss0
+ [ lapply (\P Hss0) -Hss0 #Hss0 destruct (Hss0)
+ #_ #Hcomp cases (Hcomp ????? (refl ??) (refl ??)) -Hcomp [ *
+ [ * #rs' * #_ #Hcurtc_dst * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
+ >nth_current_chars >nth_current_chars >Hcurtc_dst
+ cases (current ? (nth src …))
+ [normalize in ⊢ (%→?); #H destruct (H)
+ | #x >nth_change_vec [|@Hdst] cases (reverse ? rs0)
+ [ normalize in ⊢ (%→?); #H destruct (H)
+ | #r1 #rs1 normalize in ⊢ (%→?); #H destruct (H) ] ]
+ | * #rs0' * #_ #Hcurtc_src * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
+ >(?:nth src ? (current_chars ?? tc) (None ?) = None ?)
+ [|>nth_current_chars >Hcurtc_src >nth_change_vec_neq
+ [>nth_change_vec [cases (append ???) // | @Hsrc]
+ |@(not_to_not … Hneq) //
+ ]]
+ normalize in ⊢ (%→?); #H destruct (H) ]
+ | * #xs * #ci * #cj * #rs'' * #rs0' * * * #Hcicj #Hrs #Hrs0
+ #Htc * #td * * #Hmatch #Htd destruct (Htd) * #te * * *
+ >Htc >change_vec_commute [|//] >nth_change_vec [|//]
+ >change_vec_commute [|@sym_not_eq //] >nth_change_vec [|//]
+ cases (lists_length_split ? ls ls0) #lsa * #lsb * * #Hlen #Hlsalsb
+ destruct (Hlsalsb) *
+ [ #Hte #_ #_ <(reverse_reverse … ls) in Hte; <(reverse_reverse … lsa)
+ cut (|reverse ? lsa| = |reverse ? ls|) [ // ] #Hlen'
+ @(list_cases2 … Hlen')
+ [ #H1 #H2 >H1 >H2 -H1 -H2 normalize in match (reverse ? [ ]); #Hte #_
+ lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
+ >change_vec_commute [|//] >change_vec_change_vec
+ >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
+ >Hte whd in ⊢ (%→?); >change_vec_change_vec >nth_change_vec [|//]
+ >reverse_reverse #Htb
+ cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta (midtape sig [ ] s0 (xs@ci::rs'')) src) (mk_tape sig (s0::lsb) (option_hd sig (xs@cj::rs0')) (tail sig (xs@cj::rs0'))) dst)
+ [ >Htb @eq_f3 // cases (xs@cj::rs0') // ]
+ -Htb #Htb >Htb whd >nth_change_vec [|//]
+ >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec [|//]
+ >right_mk_tape [|cases xs [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H)]
+ normalize in match (left ??);
+ >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand
+ whd in match (option_cons ???); >Hrs0
+ normalize in ⊢ (?(?%)%); //
+ | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
+ >reverse_cons >reverse_cons #Hte
+ lapply (Hte ci hdb (reverse ? xs@s0::reverse ? tlb) rs'' ?
+ lsb cj hda (reverse ? xs@s0::reverse ? tla) rs0' ??)
+ [ /2 by cons_injective_l, nil/
+ | >length_append >length_append @eq_f @(eq_f ?? S)
+ >H1 in Hlen'; >H2 whd in ⊢ (??%%→?); #Hlen'
+ >length_reverse >length_reverse destruct (Hlen') //
+ | /2 by refl, trans_eq/ ] -Hte
+ #Hte #_ whd in ⊢ (%→?); #Htb
+ cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
+ (mk_tape sig (hda::lsb) (option_hd ? (reverse sig (reverse sig xs@s0::reverse sig tla)@cj::rs0')) (tail ? (reverse sig (reverse sig xs@s0::reverse sig tla)@cj::rs0'))) dst)
+ (midtape ? [ ] hdb (reverse sig (reverse sig xs@s0::reverse sig tlb)@ci::rs'')) src)
+ [ >Htb >Hte >nth_change_vec // >change_vec_change_vec >change_vec_commute [|//]
+ >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
+ >change_vec_change_vec >change_vec_commute [|//]
+ @eq_f3 // cases (reverse sig (reverse sig xs@s0::reverse sig tla)@cj::rs0') // ]
+ -Htb #Htb >Htb whd
+ >nth_change_vec [|//] >nth_change_vec_neq [|//] >nth_change_vec [|//]
+ >right_mk_tape
+ [| cases (reverse sig (reverse sig xs@s0::reverse sig tla))
+ [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
+ >Hmidta_src >Hmidta_dst
+ whd in match (left ??); whd in match (left ??); whd in match (right ??);
+ >current_mk_tape <opt_cons_tail_expand whd in match (option_cons ???);
+ >Hrs0 >length_append whd in ⊢ (??(??%)); >length_append >length_reverse
+ >length_append >commutative_plus in match (|reverse ??| + ?);
+ whd in match (|?::?|); >length_reverse >length_reverse
+ <(length_reverse ? ls) <Hlen' >H1 normalize // ]
+ | #_ #Hte #_ <(reverse_reverse … ls0) in Hte; <(reverse_reverse … lsa)
+ cut (|reverse ? lsa| = |reverse ? ls0|) [ // ] #Hlen'
+ @(list_cases2 … Hlen')
+ [ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #Hte
+ lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
+ >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
+ >change_vec_change_vec #Hte #_
+ >Hte whd in ⊢ (%→?); >nth_change_vec [|//] >reverse_reverse #Htb
+ cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta (mk_tape ? [s0] (option_hd ? (xs@cj::rs0')) (tail ? (xs@cj::rs0'))) dst)
+ (midtape ? lsb s0 (xs@ci::rs'')) src)
+ [ >Htb >change_vec_change_vec >change_vec_commute [|//]
+ @eq_f3 // <Hrs0 cases rs0 // ]
+ -Htb #Htb >Htb whd >nth_change_vec [|//]
+ >nth_change_vec_neq [|//] >nth_change_vec [|//]
+ >right_mk_tape
+ [| cases xs [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
+ normalize in match (left ??);
+ >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand >Hrs0
+ >length_append normalize >length_append >length_append
+ <(reverse_reverse ? lsa) >H1 normalize //
+ | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
+ >reverse_cons >reverse_cons #Hte
+ lapply (Hte cj hdb (reverse ? xs@s0::reverse ? tlb) rs0' ?
+ lsb ci hda (reverse ? xs@s0::reverse ? tla) rs'' ??)
+ [ /2 by cons_injective_l, nil/
+ | >length_append >length_append @eq_f @(eq_f ?? S)
+ >H1 in Hlen'; >H2 whd in ⊢ (??%%→?); #Hlen'
+ >length_reverse >length_reverse destruct (Hlen') //
+ | /2 by refl, trans_eq/ ] -Hte
+ #Hte #_ whd in ⊢ (%→?); >Hte >nth_change_vec_neq [|//] >nth_change_vec [|//] #Htb
+ cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
+ (mk_tape sig [hdb] (option_hd ? (reverse sig (reverse sig xs@s0::reverse sig tlb)@cj::rs0')) (tail ? (reverse sig (reverse sig xs@s0::reverse sig tlb)@cj::rs0'))) dst)
+ (midtape ? lsb hda (reverse sig (reverse sig xs@s0::reverse sig tla)@ci::rs'')) src)
+ [ >Htb >change_vec_change_vec >change_vec_commute [|//]
+ >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
+ >change_vec_change_vec >change_vec_commute [|//]
+ @eq_f3 // cases (reverse sig (reverse sig xs@s0::reverse sig tlb)@cj::rs0') // ]
+ -Htb #Htb >Htb whd
+ >nth_change_vec [|//] >nth_change_vec_neq [|//] >nth_change_vec [|//]
+ >right_mk_tape
+ [| cases (reverse sig (reverse sig xs@s0::reverse sig tlb))
+ [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
+ >Hmidta_src >Hmidta_dst
+ whd in match (left ??); whd in match (left ??); whd in match (right ??);
+ >current_mk_tape <opt_cons_tail_expand
+ whd in match (option_cons ???);
+ >Hrs0 >length_append whd in ⊢ (??(??%)); >length_append >length_reverse
+ >length_append >commutative_plus in match (|reverse ??| + ?);
+ whd in match (|?::?|); >length_reverse >length_reverse
+ <(length_reverse ? lsa) >Hlen' >H2 >length_append
+ normalize //
+ ]
+ ]