-[ * #ta * whd in ⊢ (%→?); #Hta * #tb * whd in ⊢ (%→?); #Htb
- * #tc * whd in ⊢ (%→?); #Htc whd in ⊢ (%→?); #Houtc
- #l0 #x #rs #Hintape cases (true_or_false (〈x,true〉==c)) #Hc
- [ % % [ @(\P Hc) ]
- #a #l1 #x0 #a0 #l2 #Hl1 #Hrs >Hrs in Hintape; #Hintape
- >Hintape in Hta; * #_(* #x1 * whd in ⊢ ((??%?)→?); #H destruct (H) #Hx *)
- #Hta lapply (proj2 … Htb … Hta) -Htb -Hta #Htb
- cases (Htc … Htb) [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
- -Htc * #_ #Htc lapply (Htc l1 〈x0,true〉 (〈a0,false〉::l2) (refl ??) (refl ??) Hl1)
- -Htc #Htc cases (Houtc ???????? Htc) -Houtc
- [ * #Hx0 #Houtc %
- % [ <(\P Hx0) in Hc; #Hx lapply (\P Hx) #Hx' destruct (Hx') %
- | >Houtc >reverse_reverse % ]
- | * #Hx0 #Houtc %2
- % [ <(\P Hc) in Hx0; #Hx0 lapply (\Pf Hx0) @not_to_not #Hx' >Hx' %
- | >Houtc % ]
- | #x #membx @Hl1 <(reverse_reverse …l1) @memb_reverse @membx ]
- | %2 % [ @(\Pf Hc) ]
- >Hintape in Hta; * * #x1 * whd in ⊢ ((??%?)→?); #H destruct (H)
- >Hc #H destruct (H) ]
-| * #ta * whd in ⊢ (%→?); * #Hc #Hta #Helse #ls #c0 #rs #Hintape %2
- >Hintape in Hta; #Hta % [ @(\Pf (Hc …)) >Hintape % | <Hta @Helse ]
-]
+ [* #ta * whd in ⊢ (%→?); * * #cin * #Hcin #Hcintrue #Hta * #tb * whd in ⊢ (%→?); #Htb
+ * #tc * whd in ⊢ (%→?); #Htc * whd in ⊢ (%→%→?); #Houtc #Houtc1
+ #ls #x #rs #Hintape >Hintape in Hcin; whd in ⊢ ((??%?)→?); #H destruct (H) %
+ [#_ cases (dec_marked ? rs) #Hdec
+ [%
+ [#_ #a #l1
+ >Hintape in Hta; #Hta
+ lapply (proj2 ?? Htb … Hta) -Htb -Hta cases rs in Hdec;
+ (* by cases on rs *)
+ [#_ whd in ⊢ ((???%)→?); #Htb >Htb in Htc; #Htc
+ lapply (proj1 ?? Htc (refl …)) -Htc #Htc <Htc in Houtc1; #Houtc1
+ normalize in ⊢ (???%→?); #Hl1 destruct(Hl1) @(Houtc1 (refl …))
+ |#r0 #rs0 #Hdec whd in ⊢ ((???%)→?); #Htb >Htb in Htc; #Htc
+ >reverse_cons >reverse_cons #Hl1
+ cases (proj2 ?? Htc … (refl …))
+ [* >(Hdec …) [ #Hfalse destruct(Hfalse) ] @memb_hd
+ |* #_ -Htc #Htc cut (∃l2.l1 = l2@[〈x,true〉])
+ [generalize in match Hl1; -Hl1 <(reverse_reverse … l1)
+ cases (reverse ? l1)
+ [#Hl1 cut ([a]=〈x,true〉::r0::rs0)
+ [ <(reverse_reverse … (〈x,true〉::r0::rs0))
+ >reverse_cons >reverse_cons <Hl1 %
+ | #Hfalse destruct(Hfalse)]
+ |#a0 #l10 >reverse_cons #Heq
+ lapply (append_l2_injective_r ? (a::reverse ? l10) ???? Heq) //
+ #Ha0 destruct(Ha0) /2/ ]
+ |* #l2 #Hl2 >Hl2 in Hl1; #Hl1
+ lapply (append_l1_injective_r ? (a::l2) … Hl1) // -Hl1 #Hl1
+ >reverse_cons in Htc; #Htc lapply (Htc … (sym_eq … Hl1))
+ [ #x0 #Hmemb @Hdec @memb_cons @Hmemb ]
+ -Htc #Htc >Htc in Houtc1; #Houtc1 >associative_append @Houtc1 %
+ ]
+ ]
+ ]
+ |#l1 #x0 #l2 #_ #Hrs @False_ind
+ @(absurd ?? not_eq_true_false)
+ change with (is_marked ? 〈x0,true〉) in match true;
+ @Hdec >Hrs @memb_append_l2 @memb_hd
+ ]
+ |% [#H @False_ind @(absurd …H Hdec)]
+ (* by cases on l1 *) *
+ [#x0 #l2 #Hdec normalize in ⊢ (%→?); #Hrs >Hrs in Hintape; #Hintape
+ >Hintape in Hta; (* * * #x1 * whd in ⊢ ((??%?)→?); #H destruct (H) #Hx *)
+ #Hta lapply (proj2 … Htb … Hta) -Htb -Hta
+ whd in match (mk_tape ????); whd in match (tail ??); #Htb cases Htc -Htc
+ #_ #Htc cases (Htc … Htb) -Htc
+ [2: * * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
+ * * #Htc >Htb in Htc; -Htb #Htc cases (Houtc … Htc) -Houtc *
+ #H1 #H2 #H3 cases (true_or_false (x==x0)) #eqxx0
+ [>(\P eqxx0) % [2: #H @False_ind /2/] %
+ [#_ #Hl2 >(H2 … Hl2) <(\P eqxx0) [% | @Hcintrue]
+ |#_ #a #a0 #b #l1' #l2' normalize in ⊢ (%→?); #Hdes destruct (Hdes)
+ #Hl2 @(H3 … Hdec … Hl2) <(\P eqxx0) [@Hcintrue | % | @reverse_single]
+ ]
+ |% [% #eqx @False_ind lapply (\Pf eqxx0) #Habs @(absurd … eqx Habs)]
+ #_ @H1 @(\bf ?) @(not_to_not ??? (\Pf eqxx0)) <(\P Hcintrue)
+ #Hdes destruct (Hdes) %
+ ]
+ |#l1hd #l1tl #x0 #l2 #Hdec normalize in ⊢ (%→?); #Hrs >Hrs in Hintape; #Hintape
+ >Hintape in Hta; (* * * #x1 * whd in ⊢ ((??%?)→?); #H destruct (H) #Hx *)
+ #Hta lapply (proj2 … Htb … Hta) -Htb -Hta
+ whd in match (mk_tape ????); whd in match (tail ??); #Htb cases Htc -Htc
+ #_ #Htc cases (Htc … Htb) -Htc
+ [* #Hfalse @False_ind >(Hdec … (memb_hd …)) in Hfalse; #H destruct]
+ * * #_ #Htc lapply (Htc … (refl …) (refl …) ?) -Htc
+ [#x1 #membx1 @Hdec @memb_cons @membx1] #Htc
+ cases (Houtc … Htc) -Houtc *
+ #H1 #H2 #H3 #_ cases (true_or_false (x==x0)) #eqxx0
+ [>(\P eqxx0) % [2: #H @False_ind /2/] %
+ [#_ #Hl2 >(H2 … Hl2) <(\P eqxx0)
+ [>reverse_cons >associative_append % | @Hcintrue]
+ |#_ #a #a0 #b #l1' #l2' normalize in ⊢ (%→?); #Hdes (* destruct (Hdes) *)
+ #Hl2 @(H3 ?????? (reverse … (l1hd::l1tl)) … Hl2) <(\P eqxx0)
+ [@Hcintrue
+ |>reverse_cons >associative_append %
+ |#c0 #memc @Hdec <(reverse_reverse ? (l1hd::l1tl)) @memb_reverse @memc
+ |>Hdes >reverse_cons >reverse_reverse >(\P eqxx0) %
+ ]
+ ]
+ |% [% #eqx @False_ind lapply (\Pf eqxx0) #Habs @(absurd … eqx Habs)]
+ #_ >reverse_cons >associative_append @H1 @(\bf ?)
+ @(not_to_not ??? (\Pf eqxx0)) <(\P Hcintrue) #Hdes
+ destruct (Hdes) %
+ ]
+ ]
+ ]
+ |>(\P Hcintrue) * #Hfalse @False_ind @Hfalse %
+ ]
+ | * #ta * * #Hcur #Hta #Houtc
+ #l0 #x #rs #Hintape >Hintape in Hcur; #Hcur lapply (Hcur ? (refl …)) -Hcur #Hc %
+ [ #Hfalse >Hfalse in Hc; #Hc cases (\Pf Hc) #Hc @False_ind @Hc %
+ | -Hc #Hc <Hintape <Hta @Houtc ] ]