- >length_tail >Hrs0 >length_append >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 * * #_ >nth_change_vec // >reverse_reverse
- #H lapply (H … (refl ??)) -H #Htb1 #Htb2
- (* 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) *)
- 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)
- [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
- >nth_change_vec_neq // >nth_change_vec //
- >right_mk_tape normalize in match (left ??);
- >Hmidta_src >Hmidta_dst >length_tail >Hrs0 >length_append
- >commutative_plus in match (pred ?); normalize
- >length_append >(?:|lsa| = O)
- [ normalize <plus_n_Sm <plus_n_Sm // | <(length_reverse ? lsa) >H1 % ]
- | #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 * * #_ >Hte >nth_change_vec_neq // >nth_change_vec // #Htb1 #Htb2
- 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)
- [@daemon] -Htb1 -Htb2 #Htb >Htb whd
- >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
- >right_mk_tape >Hmidta_src >Hmidta_dst
- whd in match (left ??); whd in match (left ??); whd in match (right ??);
- >length_tail >Hrs0 >length_append >length_append >length_reverse
- >length_append >commutative_plus in match (|reverse ??| + ?);
- whd in match (|?::?|); >length_reverse >length_reverse >Hlen
- <(length_reverse ? ls0) >H2 whd in match (|?::?|);
- >length_append normalize //
- ]
- ]
-
- #Hte1 #H2 #H3 #H4 >Hmidta_src >Hmidta_dst #Hte
- lapply (Hte … (refl ??) … (refl ??)) * * #_ #Htb1 #Htb2
- #s1 #rs1 >Hmidta_src #H destruct (H)
- lapply (Hte … Hmidta_src … Hmidta_dst) -Hte #Hte destruct (Hte) %
- [ @(eq_vec … (niltape ?)) #i #Hi
- cases (true_or_false ((dst : DeqNat) == i)) #Hdsti
- [ <(\P Hdsti) >(Htb1 … Hmidta_dst) >nth_change_vec // >Hmidta_dst
- cases rs0 [|#r2 #rs2] %
- | <Htb2 [|@(\Pf Hdsti)] >nth_change_vec_neq [| @(\Pf Hdsti)] % ]
- | >Hs0 %{s0} % // #H destruct (H) @False_ind cases (Hss0) /2/ ] *)
- ]
- ]
- ]
+ >current_mk_tape <opt_cons_tail_expand >length_append
+ >length_reverse >length_reverse <(length_reverse ? ls) <Hlen'
+ >H1 normalize // ]
+ | #_ <(reverse_reverse … ls0) <(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 ??)) -Hte #Hte destruct (Hte)
+ * * #_ >Hmidta_dst #Htb1 lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2
+ cut (tb = change_vec (tape sig) (S n) ta (mk_tape ? (s0::ls0) (option_hd ? rs0) (tail ? rs0)) dst)
+ [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
+ >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst
+ >current_mk_tape >right_mk_tape normalize in ⊢ (??%); <opt_cons_tail_expand
+ normalize //
+ | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
+ >reverse_cons >reverse_cons #Hte #_ #_
+ lapply (Hte s0 hdb (reverse ? tlb) rs0 ?
+ lsb s hda (reverse ? tla) rs ??)
+ [ /2 by cons_injective_l, nil/
+ | >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) //
+ normalize #H destruct (H) //
+ | /2 by refl, trans_eq/ ] -Hte
+ #Hte * * #_ >Hte >nth_change_vec_neq // >nth_change_vec // #Htb1 #Htb2
+ cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
+ (mk_tape sig [hdb] (option_hd ? (reverse sig (reverse sig tlb)@s0::rs0)) (tail ? (reverse sig (reverse sig tlb)@s0::rs0))) dst)
+ (midtape ? lsb hda (reverse sig (reverse sig tla)@s::rs)) src)
+ [@daemon] -Htb1 -Htb2 #Htb >Htb whd
+ >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
+ >right_mk_tape >Hmidta_src >Hmidta_dst
+ whd in match (left ??); whd in match (left ??); whd in match (right ??);
+ >current_mk_tape <opt_cons_tail_expand >length_append
+ normalize in ⊢ (??%); >length_append >reverse_reverse
+ <(length_reverse ? lsa) >Hlen' >H2 normalize //
+ ]
+ ]
+ ]
+ ]
+ ]