(* /move_tape *)
else sink;
-*)
\ No newline at end of file
+*)
+
+definition init_match ≝
+ seq ? (mark ?)
+ (seq ? (adv_to_mark_r ? (λc:STape.is_grid (\fst c)))
+ (seq ? (move_r ?)
+ (seq ? (mark ?)
+ (seq ? (move_l ?)
+ (adv_to_mark_l ? (is_marked ?)))))).
+
+definition R_init_match ≝ λt1,t2.
+ ∀ls,l,rs,c,d. no_grids (〈c,false〉::l) → no_marks l →
+ t1 = midtape STape ls 〈c,false〉 (l@〈grid,false〉::〈d,false〉::rs) →
+ t2 = midtape STape ls 〈c,true〉 (l@〈grid,false〉::〈d,true〉::rs).
+
+lemma sem_init_match : Realize ? init_match R_init_match.
+#intape
+cases (sem_seq ????? (sem_mark ?)
+ (sem_seq ????? (sem_adv_to_mark_r ? (λc:STape.is_grid (\fst c)))
+ (sem_seq ????? (sem_move_r ?)
+ (sem_seq ????? (sem_mark ?)
+ (sem_seq ????? (sem_move_l ?)
+ (sem_adv_to_mark_l ? (is_marked ?)))))) intape)
+#k * #outc * #Hloop #HR
+@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
+#ls #l #rs #c #d #Hnogrids #Hnomarks #Hintape
+cases HR -HR
+#ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hintape) -Hta -Hintape #Hta
+* #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Htb -Hta
+ [* #Hgridc @False_ind @(absurd … Hgridc) @eqnot_to_noteq
+ @(Hnogrids 〈c,false〉) @memb_hd ]
+* #Hgrdic #Htb lapply (Htb l 〈grid,false〉 (〈d,false〉::rs) (refl …) (refl …) ?)
+ [#x #membl @Hnogrids @memb_cons @membl] -Htb #Htb
+* #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb #Htc
+* #td * whd in ⊢ (%→?); #Htd lapply (Htd … Htc) -Htd -Htc #Htd
+* #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Hte -Htd #Hte
+whd in ⊢ (%→?); #Htf cases (Htf … Hte) -Htf -Hte
+ [* whd in ⊢ ((??%?)→?); #Habs destruct (Habs)]
+* #_ #Htf lapply (Htf (reverse ? l) 〈c,true〉 ls (refl …) (refl …) ?)
+ [#x #membl @Hnomarks @daemon] -Htf #Htf >Htf >reverse_reverse %
+qed.
+
+
+(* init_copy
+
+ init_current_on_match; (* no marks in current *)
+ move_r;
+ adv_to_mark_r;
+ adv_mark_r;
+
+*)
+
+definition init_copy ≝
+ seq ? init_current_on_match
+ (seq ? (move_r ?)
+ (seq ? (adv_to_mark_r ? (is_marked ?))
+ (adv_mark_r ?))).
+
+definition R_init_copy ≝ λt1,t2.
+ ∀l1,l2,c,ls,d,rs.
+ no_marks l1 → no_grids l1 →
+ no_marks l2 → is_grid c = false →
+ t1 = midtape STape (l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉 (l2@〈comma,true〉::〈d,false〉::rs) →
+ t2 = midtape STape (〈comma,false〉::(reverse ? l2)@〈grid,false〉::l1@〈c,true〉::〈grid,false〉::ls) 〈d,true〉 rs.
+
+lemma list_last: ∀A.∀l:list A.
+ l = [ ] ∨ ∃a,l1. l = l1@[a].
+#A #l <(reverse_reverse ? l) cases (reverse A l)
+ [%1 //
+ |#a #l1 %2 @(ex_intro ?? a) @(ex_intro ?? (reverse ? l1)) //
+ ]
+qed.
+
+lemma sem_init_copy : Realize ? init_copy R_init_copy.
+#intape
+cases (sem_seq ????? sem_init_current_on_match
+ (sem_seq ????? (sem_move_r ?)
+ (sem_seq ????? (sem_adv_to_mark_r ? (is_marked ?))
+ (sem_adv_mark_r ?))) intape)
+#k * #outc * #Hloop #HR
+@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
+#l1 #l2 #c #ls #d #rs #Hl1marks #Hl1grids #Hl2marks #Hc #Hintape
+cases HR -HR
+#ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hl1grids Hc Hintape) -Hta -Hintape #Hta
+* #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta) -Htb -Hta
+generalize in match Hl1marks; -Hl1marks cases (list_last ? l1)
+ [#eql1 >eql1 #Hl1marks whd in ⊢ ((???%)→?); whd in ⊢ ((???(????%))→?); #Htb
+ * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb *
+ [* whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)]
+ * #_ #Htc lapply (Htc … (refl …) (refl …) ?)
+ [#x #membx @Hl2marks @membx]
+ #Htc whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc -Htc #Houtc
+ >Houtc %
+ |* #c1 * #tl #eql1 >eql1 #Hl1marks >reverse_append >reverse_single
+ whd in ⊢ ((???%)→?); whd in ⊢ ((???(????%))→?);
+ >associative_append whd in ⊢ ((???(????%))→?); #Htb
+ * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb *
+ [* >Hl1marks [#Htemp destruct (Htemp)] @memb_append_l2 @memb_hd]
+ * #_ >append_cons <associative_append #Htc lapply (Htc … (refl …) (refl …) ?)
+ [#x #membx cases (memb_append … membx) -membx #membx
+ [cases (memb_append … membx) -membx #membx
+ [@Hl1marks @memb_append_l1 @daemon
+ |>(memb_single … membx) %
+ ]
+ |@Hl2marks @membx
+ ]]
+ #Htc whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc -Htc #Houtc
+ >Houtc >reverse_append >reverse_append >reverse_single
+ >reverse_reverse >associative_append >associative_append
+ >associative_append %
+qed.
+
+(* OLD
+definition init_copy ≝
+ seq ? (adv_mark_r ?)
+ (seq ? init_current_on_match
+ (seq ? (move_r ?)
+ (adv_to_mark_r ? (is_marked ?)))).
+
+definition R_init_copy ≝ λt1,t2.
+ ∀l1,l2,c,l3,d,rs.
+ no_marks l1 → no_grids l1 →
+ no_marks l2 → no_grids l2 → is_grid c = false → is_grid d =false →
+ t1 = midtape STape (l1@〈grid,false〉::l2@〈c,false〉::〈grid,false〉::l3) 〈comma,true〉 (〈d,false〉::rs) →
+ t2 = midtape STape (〈comma,false〉::l1@〈grid,false〉::l2@〈c,true〉::〈grid,false〉::l3) 〈d,true〉 rs.
+
+lemma list_last: ∀A.∀l:list A.
+ l = [ ] ∨ ∃a,l1. l = l1@[a].
+#A #l <(reverse_reverse ? l) cases (reverse A l)
+ [%1 //
+ |#a #l1 %2 @(ex_intro ?? a) @(ex_intro ?? (reverse ? l1)) //
+ ]
+qed.
+
+lemma sem_init_copy : Realize ? init_copy R_init_copy.
+#intape
+cases (sem_seq ????? (sem_adv_mark_r ?)
+ (sem_seq ????? sem_init_current_on_match
+ (sem_seq ????? (sem_move_r ?)
+ (sem_adv_to_mark_r ? (is_marked ?)))) intape)
+#k * #outc * #Hloop #HR
+@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
+#l1 #l2 #c #l3 #d #rs #Hl1marks #Hl1grids #Hl2marks #Hl2grids #Hc #Hd #Hintape
+cases HR -HR
+#ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hintape) -Hta -Hintape #Hta
+* #tb * whd in ⊢ (%→?);
+>append_cons #Htb lapply (Htb (〈comma,false〉::l1) l2 c … Hta)
+ [@Hd |@Hc |@Hl2grids
+ |#x #membx cases (orb_true_l … membx) -membx #membx
+ [>(\P membx) // | @Hl1grids @membx]
+ ] -Htb #Htb
+* #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb
+>reverse_append >reverse_cons cases (list_last ? l2)
+ [#Hl2 >Hl2 >associative_append whd in ⊢ ((???(??%%%))→?); #Htc
+ whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd -Htc
+ [* whd in ⊢ ((??%?)→?); #Habs destruct (Habs)]
+ * #_ #Htf lapply (Htf … (refl …) (refl …) ?)
+ [#x >reverse_cons #membx cases (memb_append … membx) -membx #membx
+ [@Hl1marks @daemon |>(memb_single … membx) //]
+ -Htf
+ |#Htf >Htf >reverse_reverse >associative_append %
+ ]
+ |* #a * #l21 #Heq >Heq >reverse_append >reverse_single
+ >associative_append >associative_append >associative_append whd in ⊢ ((???(??%%%))→?); #Htc
+ whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd -Htc
+ [* >Hl2marks [#Habs destruct (Habs) |>Heq @memb_append_l2 @memb_hd]]
+ * #_ <associative_append <associative_append #Htf lapply (Htf … (refl …) (refl …) ?)
+ [#x >reverse_cons #membx cases (memb_append … membx) -membx #membx
+ [cases (memb_append … membx) -membx #membx
+ [@Hl2marks >Heq @memb_append_l1 @daemon
+ |>(memb_single … membx) //]
+ |cases (memb_append … membx) -membx #membx
+ [@Hl1marks @daemon |>(memb_single … membx) //]
+ ]
+ | #Htf >Htf >reverse_append >reverse_reverse
+ >reverse_append >reverse_reverse >associative_append
+ >reverse_single >associative_append >associative_append
+ >associative_append %
+ ]
+ ]
+qed. *)
+
+include "turing/universal/move_tape.ma".
+
+definition exec_move ≝
+ seq ? init_copy
+ (seq ? copy
+ (seq ? (move_r …) move_tape)).
+
+definition R_exec_move ≝ λt1,t2.
+ ∀n,curconfig,ls,rs,c0,c1,s0,s1,table1,newconfig,mv,table2.
+ table_TM n (table1@〈comma,false〉::〈s1,false〉::newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2) →
+ no_marks curconfig → only_bits (curconfig@[〈s0,false〉]) → only_bits (〈s1,false〉::newconfig) →
+ no_nulls ls → no_nulls rs →
+ t1 = midtape STape (〈c0,false〉::curconfig@〈s0,false〉::〈grid,false〉::ls) 〈grid,false〉
+ (table1@〈comma,true〉::〈s1,false〉::newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs) →
+ ∀t1'.t1' = lift_tape ls 〈c0,false〉 rs →
+ ∃ls1,rs1,c2.
+ t2 = midtape STape ls1 〈grid,false〉
+ (〈s1,false〉::newconfig@〈c2,false〉::〈grid,false〉::
+ table1@〈comma,false〉::〈s1,false〉::newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs1) ∧
+ lift_tape ls1 〈c2,false〉 rs1 =
+ tape_move STape t1' (Some ? 〈〈c2,false〉,move_of_unialpha mv〉).
+
+lemma sem_exec_move : Realize ? exec_move R_exec_move.
+#intape
+cases (sem_seq … sem_init_copy
+ (sem_seq … sem_copy
+ (sem_seq … (sem_move_r …) sem_move_tape )) intape)
+#k * #outc * #Hloop #HR
+@(ex_intro ?? k) @(ex_intro ?? outc) % [ @Hloop ] -Hloop
+#n #curconfig #ls #rs #c0 #c1 #s0 #s1 #table1 #newconfig #mv #table2
+#Htable #Hcurconfig1 #Hcurconfig2 #Hnewconfig #Hls #Hrs #Hintape #t1' #Ht1'
+cases HR -HR #ta * whd in ⊢ (%→?); #Hta
+lapply (Hta (〈c0,false〉::curconfig) table1 s0 ls s1
+ (newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs) … Hintape) -Hta
+[ (*Hcurconfig2*) @daemon
+| (*Htable*) @daemon
+| (*FIXME*) @daemon
+| (*FIXME*) @daemon
+| #Hta * #tb * whd in ⊢ (%→?); #Htb
+ lapply (Htb (〈grid,false〉::ls) s0 s1 c0 c1 (〈mv,false〉::table2@〈grid,false〉::rs) newconfig (〈comma,false〉::reverse ? table1) curconfig Hta ????????) -Htb
+ [9:|*:(* rivedere le precondizioni *) @daemon ]
+ #Htb * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc whd in ⊢(???(??%%%)→?);#Htc
+ whd in ⊢ (%→?); #Houtc whd in Htc:(???%); whd in Htc:(???(??%%%));
+ lapply (Houtc rs n
+ (〈comma,false〉::〈s1,false〉::reverse ? newconfig@@〈comma,false〉::reverse ? table1)
+ mv table2 (merge_char curc d1) (merge_config curconfig (reverse ? newconfig1)) ls ?????
+ Hls Hrs ??)
+ [3: >Htc @(eq_f3 … (midtape ?))
+ [ @eq_f >associative_append >Hnewconfig
+ >reverse_cons >associative_append @eq_f
+ whd in ⊢ (???%); @eq_f whd in ⊢ (???%); @eq_f
+ <Heqcurconfig <reverse_cons >Hnewconfig1 >reverse_append
+ >merge_cons %
+ | %
+ | % ]
+ | %
+ || >reverse_cons >reverse_append >reverse_reverse >reverse_cons
+ >reverse_reverse
+ >associative_append >associative_append >associative_append
+ normalize @Htable
+ | (* only bits or nulls c1,c2 → only bits or nulls (merge c1 c2) *) @daemon
+ | (* add to hyps? *) @daemon
+ | (* bit_or_null c1,c2 → bit_or_null (merge_char c1 c2) *) @daemon
+ | -Houtc * #ls1 * #rs1 * #newc * #Houtc *
+ [ *
+ [ * #Hmv #Htapemove
+ @(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? (\fst newc))
+ %
+ [ >Houtc >reverse_merge_config [| @daemon ]
+ >reverse_reverse @eq_f
+ >reverse_cons >reverse_append >reverse_cons
+ >reverse_reverse >reverse_reverse
+ @daemon
+ | >Hmv >Ht1' whd in Htapemove:(???%); whd in ⊢ (???%);
+ whd in match (lift_tape ???) in ⊢ (???%);
+ >Htapemove
+
+
+
+ >Heqcurconfig
+>append_cons in Hintape; >associative_append
+cases (Hta
+
+ cases (Hta … Hintape) -Hta
+[ * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
+* #_ #Hta lapply (Hta ? 〈comma,true〉 … (refl ??) (refl ??) ?)
+[ @daemon ] -Hta #Hta
+* #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta)
+
+definition move_of_unialpha ≝
+ λc.match c with
+ [ bit x ⇒ match x with [ true ⇒ R | false ⇒ L ]
+ | _ ⇒ N ].
+
+definition R_uni_step ≝ λt1,t2.
+ ∀n,table,c,c1,ls,rs,curs,curc,news,newc,mv.
+ table_TM n table →
+ match_in_table (〈c,false〉::curs@[〈curc,false〉])
+ (〈c1,false〉::news@[〈newc,false〉]) mv table →
+ t1 = midtape STape (〈grid,false〉::ls) 〈c,false〉
+ (curs@〈curc,false〉::〈grid,false〉::table@〈grid,false〉::rs) →
+ ∀t1',ls1,rs1.t1' = lift_tape ls 〈curc,false〉 rs →
+ (t2 = midtape STape (〈grid,false〉::ls1) 〈c1,false〉
+ (news@〈newc,false〉::〈grid,false〉::table@〈grid,false〉::rs1) ∧
+ lift_tape ls1 〈newc,false〉 rs1 =
+ tape_move STape t1' (Some ? 〈〈newc,false〉,move_of_unialpha mv〉)).
+
+definition no_nulls ≝
+ λl:list STape.∀x.memb ? x l = true → is_null (\fst x) = false.
+
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