--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A||
+ \ / This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ V_____________________________________________________________*)
+
+include "turing/universal/move_char_c.ma".
+include "turing/universal/move_char_l.ma".
+include "turing/universal/tuples.ma".
+
+definition init_cell_states ≝ initN 2.
+
+definition init_cell ≝
+ mk_TM STape init_cell_states
+ (λp.let 〈q,a〉 ≝ p in
+ match q with
+ [ O ⇒ match a with
+ [ None ⇒ 〈1, Some ? 〈〈null,false〉,N〉〉
+ | Some _ ⇒ 〈1, None ?〉 ]
+ | S _ ⇒ 〈1,None ?〉 ])
+ O (λq.q == 1).
+
+definition R_init_cell ≝ λt1,t2.
+ (∃c.current STape t1 = Some ? c ∧ t2 = t1) ∨
+ (current STape t1 = None ? ∧ t2 = midtape ? (left ? t1) 〈null,false〉 (right ? t1)).
+
+axiom sem_init_cell : Realize ? init_cell R_init_cell.
+
+definition swap_states : FinSet → FinSet ≝ λalpha:FinSet.FinProd (initN 4) alpha.
+
+definition swap ≝
+ λalpha:FinSet.λd:alpha.
+ mk_TM alpha (mcl_states alpha)
+ (λp.let 〈q,a〉 ≝ p in
+ let 〈q',b〉 ≝ q in
+ match a with
+ [ None ⇒ 〈〈3,d〉,None ?〉
+ | Some a' ⇒
+ match q' with
+ [ O ⇒ (* qinit *)
+ 〈〈1,a'〉,Some ? 〈a',R〉〉
+ | S q' ⇒ match q' with
+ [ O ⇒ (* q1 *)
+ 〈〈2,a'〉,Some ? 〈b,L〉〉
+ | S q' ⇒ match q' with
+ [ O ⇒ (* q2 *)
+ 〈〈3,d〉,Some ? 〈b,N〉〉
+ | S _⇒ (* qacc *)
+ 〈〈3,d〉,None ?〉 ] ] ] ])
+ 〈0,d〉
+ (λq.let 〈q',a〉 ≝ q in q' == 3).
+
+definition R_swap ≝
+ λalpha,t1,t2.
+ ∀a,b,ls,rs.
+ t1 = midtape alpha ls b (a::rs) →
+ t2 = midtape alpha ls a (b::rs).
+
+(*
+lemma swap_q0_q1 :
+ ∀alpha:FinSet.∀d,a,ls,a0,rs.
+ step alpha (swap alpha d)
+ (mk_config ?? 〈0,a〉 (mk_tape … ls (Some ? a0) rs)) =
+ mk_config alpha (states ? (swap alpha d)) 〈1,a0〉
+ (tape_move_right alpha ls a0 rs).
+#alpha #d #a *
+[ #a0 #rs %
+| #a1 #ls #a0 #rs %
+]
+qed.
+
+lemma swap_q1_q2 :
+ ∀alpha:FinSet.∀d,a,ls,a0,rs.
+ step alpha (swap alpha d)
+ (mk_config ?? 〈1,a〉 (mk_tape … ls (Some ? a0) rs)) =
+ mk_config alpha (states ? (swap alpha d)) 〈2,a0〉
+ (tape_move_left alpha ls a rs).
+#alpha #sep #a #ls #a0 * //
+qed.
+
+lemma swap_q2_q3 :
+ ∀alpha:FinSet.∀d,a,ls,a0,rs.
+ step alpha (swap alpha d)
+ (mk_config ?? 〈2,a〉 (mk_tape … ls (Some ? a0) rs)) =
+ mk_config alpha (states ? (swap alpha d)) 〈3,d〉
+ (tape_move_left alpha ls a rs).
+#alpha #sep #a #ls #a0 * //
+qed.
+*)
+
+lemma sem_swap :
+ ∀alpha,d.
+ Realize alpha (swap alpha d) (R_swap alpha).
+#alpha #d #tapein @(ex_intro ?? 4) cases tapein
+[ @ex_intro [| % [ % | #a #b #ls #rs #Hfalse destruct (Hfalse) ] ]
+| #a #al @ex_intro [| % [ % | #a #b #ls #rs #Hfalse destruct (Hfalse) ] ]
+| #a #al @ex_intro [| % [ % | #a #b #ls #rs #Hfalse destruct (Hfalse) ] ]
+| #ls #c #rs cases rs
+ [ @ex_intro [| % [ % | #a #b #ls0 #rs0 #Hfalse destruct (Hfalse) ] ]
+ | -rs #r #rs @ex_intro
+ [|%
+ [%
+ | #r0 #c0 #ls0 #rs0 #Htape destruct (Htape) normalize cases ls0
+ [% | #l1 #ls1 %] ] ] ] ]
+qed.
+
+axiom ssem_move_char_l :
+ ∀alpha,sep.
+ Realize alpha (move_char_l alpha sep) (R_move_char_l alpha sep).
+
+(*
+MOVE TAPE RIGHT:
+
+ ls # current c # table # d? rs
+ ^
+ ls # current c # table # d? rs init
+ ^
+ ls # current c # table # d? rs
+ ^
+ ls # current c # table # d rs ----------------------
+ ^ move_l
+ ls # current c # table # d rs
+ ^ swap
+ ls # current c # table d # rs --------------------
+ ^
+ ls # current c # table d # rs
+ ^
+ ls # current c # d table # rs sub1
+ ^
+ ls # current c # d table # rs
+ ^
+ ls # current c d # table # rs -------------------
+ ^ move_l
+ ls # current c d # table # rs -------------------
+ ^
+ ls # current c d # table # rs
+ ^
+ ls # c current d # table # rs sub1
+ ^
+ ls # c current d # table # rs
+ ^
+ ls c # current d # table # rs ------------------
+ ^
+
+(move_to_grid_r;)
+move_r;
+init_cell;
+move_l;
+swap;
+
+move_l;
+move_char_l;
+---------move_l;
+swap;
+
+move_l;
+
+move_l;
+move_char_l;
+---------move_l;
+swap
+*)
+
+(* l1 # l2 r ---> l1 r # l2
+ ^ ^
+ *)
+definition mtr_aux ≝
+ seq ? (move_l …) (seq ? (move_char_l STape 〈grid,false〉)
+ (swap STape 〈grid,false〉)).
+definition R_mtr_aux ≝ λt1,t2.
+ ∀l1,l2,l3,r. t1 = midtape STape (l2@〈grid,false〉::l1) r l3 → no_grids l2 →
+ t2 = midtape STape l1 r (〈grid,false〉::reverse ? l2@l3).
+
+lemma sem_mtr_aux : Realize ? mtr_aux R_mtr_aux.
+#intape
+cases (sem_seq … (sem_move_l …) (sem_seq … (ssem_move_char_l STape 〈grid,false〉)
+ (sem_swap STape 〈grid,false〉)) intape)
+#k * #outc * #Hloop #HR @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
+#l1 #l2 #l3 #r #Hintape #Hl2
+cases HR -HR #ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hintape) -Hta #Hta
+* #tb * whd in ⊢(%→?); generalize in match Hta; -Hta cases l2 in Hl2;
+[ #_ #Hta #Htb lapply (Htb … Hta) -Htb * #Htb lapply (Htb (refl ??)) -Htb #Htb #_
+ whd in ⊢(%→?); >Htb #Houtc lapply (Houtc … Hta) -Houtc #Houtc @Houtc
+| #c0 #l0 #Hnogrids #Hta #Htb lapply (Htb … Hta) -Htb * #_ #Htb
+ lapply (Htb … (refl ??) ??)
+ [ cases (true_or_false (memb STape 〈grid,false〉 l0)) #Hmemb
+ [ @False_ind lapply (Hnogrids 〈grid,false〉 ?)
+ [ @memb_cons // | normalize #Hfalse destruct (Hfalse) ]
+ | @Hmemb ]
+ | % #Hc0 lapply (Hnogrids c0 ?)
+ [ @memb_hd | >Hc0 normalize #Hfalse destruct (Hfalse) ]
+ | #Htb whd in ⊢(%→?); >Htb #Houtc lapply (Houtc … (refl ??)) -Houtc #Houtc
+ >reverse_cons >associative_append @Houtc
+]]
+qed.
+
+definition move_tape_r ≝
+ seq ? (move_r …) (seq ? init_cell (seq ? (move_l …)
+ (seq ? (swap STape 〈grid,false〉)
+ (seq ? mtr_aux (seq ? (move_l …) (seq ? mtr_aux (move_r …))))))).
+
+definition R_move_tape_r ≝ λt1,t2.
+ ∀rs,n,table,c0,bc0,curconfig,ls0.
+ bit_or_null c0 = true → only_bits_or_nulls curconfig → table_TM n (reverse ? table) →
+ t1 = midtape STape (table@〈grid,false〉::〈c0,bc0〉::curconfig@〈grid,false〉::ls0)
+ 〈grid,false〉 rs →
+ (rs = [] ∧
+ t2 = midtape STape (〈c0,bc0〉::ls0) 〈grid,false〉 (reverse STape curconfig@〈null,false〉::
+ 〈grid,false〉::reverse STape table@[〈grid,false〉])) ∨
+ (∃r0,rs0.rs = r0::rs0 ∧
+ t2 = midtape STape (〈c0,bc0〉::ls0) 〈grid,false〉 (reverse STape curconfig@r0::
+ 〈grid,false〉::reverse STape table@〈grid,false〉::rs0)).
+
+lemma sem_move_tape_r : Realize ? move_tape_r R_move_tape_r.
+#tapein
+cases (sem_seq …(sem_move_r …) (sem_seq … sem_init_cell (sem_seq … (sem_move_l …)
+ (sem_seq … (sem_swap STape 〈grid,false〉) (sem_seq … sem_mtr_aux
+ (sem_seq … (sem_move_l …) (sem_seq … sem_mtr_aux (sem_move_r …))))))) tapein)
+#k * #outc * #Hloop #HR @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
+#rs #n #table #c0 #bc0 #curconfig #ls0 #Hbitnullc0 #Hbitnullcc #Htable #Htapein
+cases HR -HR #ta * whd in ⊢ (%→?); #Hta lapply (Hta … Htapein) -Hta #Hta
+* #tb * whd in ⊢ (%→?); *
+[ * #r0 *
+ generalize in match Hta; generalize in match Htapein; -Htapein -Hta cases rs
+ [ #_ #Hta >Hta normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
+ #r1 #rs1 #Htapein #Hta change with (midtape ?? r1 rs1) in Hta:(???%); >Hta
+ #Hr0 whd in Hr0:(??%?); #Htb * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc #Htc
+ * #td * whd in ⊢ (%→?); #Htd lapply (Htd … Htc) -Htd #Htd
+ * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd ?) [ (*memb_reverse @(no_grids_in_table … Htable)*) @daemon ] -Hte #Hte
+ * #tf * whd in ⊢ (%→?); #Htf lapply (Htf … Hte) -Htf #Htf
+ * #tg * whd in ⊢ (%→?); #Htg lapply (Htg … Htf ?) [ #x #Hx @bit_or_null_not_grid @Hbitnullcc // ] -Htg #Htg
+ whd in ⊢ (%→?); #Houtc lapply (Houtc … Htg) -Houtc #Houtc
+ %2 @(ex_intro ?? r1) @(ex_intro ?? rs1) % [%] @Houtc
+| * generalize in match Hta; generalize in match Htapein; -Htapein -Hta cases rs
+ [|#r1 #rs1 #_ #Hta >Hta normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
+ #Htapein #Hta change with (rightof ???) in Hta:(???%); >Hta
+ #Hr0 whd in Hr0:(??%?); #Htb * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc #Htc
+ * #td * whd in ⊢ (%→?); #Htd lapply (Htd … Htc) -Htd #Htd
+ * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd ?) [(*same as above @(no_grids_in_table … Htable) *) @daemon ] -Hte #Hte
+ * #tf * whd in ⊢ (%→?); #Htf lapply (Htf … Hte) -Htf #Htf
+ * #tg * whd in ⊢ (%→?); #Htg lapply (Htg … Htf ?) [ #x #Hx @bit_or_null_not_grid @Hbitnullcc // ] -Htg #Htg
+ whd in ⊢ (%→?); #Houtc lapply (Houtc … Htg) -Houtc #Houtc
+ % % [% | @Houtc ]
+qed.
+
+(*
+MOVE TAPE LEFT:
+
+ ls # current c # table # d rs
+ ^
+ ls # current c # table # d rs
+ ^
+ ls # current c # table d # rs
+ ^
+ ls # current c # d table # rs
+ ^
+ ls # current c # d table # rs
+ ^
+ ls # current c d # table # rs
+ ^
+ ls # current c d # table # rs
+ ^
+ ls # c current c # table # rs
+ ^
+ ls # c current c # table # rs
+ ^
+ ls c # current c # table # rs
+ ^
+
+move_to_grid_r;
+swap;
+move_char_l;
+move_l;
+swap;
+move_l;
+move_char_l;
+move_l;
+swap
+*)
+axiom move_tape_l : TM STape.
+(* seq ? (move_r …) (seq ? init_cell (seq ? (move_l …)
+ (seq ? (swap STape 〈grid,false〉)
+ (seq ? mtr_aux (seq ? (move_l …) mtr_aux))))). *)
+
+definition R_move_tape_l ≝ λt1,t2.
+ ∀rs,n,table,c0,bc0,curconfig,ls0.
+ bit_or_null c0 = true → only_bits_or_nulls curconfig → table_TM n (reverse ? table) →
+ t1 = midtape STape (table@〈grid,false〉::〈c0,bc0〉::curconfig@〈grid,false〉::ls0)
+ 〈grid,false〉 rs →
+ (ls0 = [] ∧
+ t2 = midtape STape [] 〈grid,false〉
+ (reverse ? curconfig@〈null,false〉::〈grid,false〉::reverse ? table@〈grid,false〉::〈c0,bc0〉::rs)) ∨
+ (∃l1,ls1. ls0 = l1::ls1 ∧
+ t2 = midtape STape ls1 〈grid,false〉
+ (reverse ? curconfig@l1::〈grid,false〉::reverse ? table@〈grid,false〉::〈c0,bc0〉::rs)).
+
+axiom sem_move_tape_l : Realize ? move_tape_l R_move_tape_l.
+
+(*
+ by cases on current:
+ case bit false: move_tape_l
+ case bit true: move_tape_r
+ case null: adv_to_grid_l; move_l; adv_to_grid_l;
+*)
+
+definition lift_tape ≝ λls,c,rs.
+ let 〈c0,b〉 ≝ c in
+ let c' ≝ match c0 with
+ [ null ⇒ None ?
+ | _ ⇒ Some ? c ]
+ in
+ mk_tape STape ls c' rs.
+
+definition sim_current_of_tape ≝ λt.
+ match current STape t with
+ [ None ⇒ 〈null,false〉
+ | Some c0 ⇒ c0 ].
+
+definition mk_tuple ≝ λc,newc,mv.
+ c @ 〈comma,false〉:: newc @ 〈comma,false〉 :: [〈mv,false〉].
+
+inductive match_in_table (c,newc:list STape) (mv:unialpha) : list STape → Prop ≝
+| mit_hd :
+ ∀tb.
+ match_in_table c newc mv (mk_tuple c newc mv@〈bar,false〉::tb)
+| mit_tl :
+ ∀c0,newc0,mv0,tb.
+ match_in_table c newc mv tb →
+ match_in_table c newc mv (mk_tuple c0 newc0 mv0@〈bar,false〉::tb).
+
+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.
+
+definition R_move_tape_r_abstract ≝ λt1,t2.
+ ∀rs,n,table,curc,curconfig,ls.
+ bit_or_null curc = true → only_bits_or_nulls curconfig → table_TM n (reverse ? table) →
+ t1 = midtape STape (table@〈grid,false〉::〈curc,false〉::curconfig@〈grid,false〉::ls)
+ 〈grid,false〉 rs →
+ no_nulls rs →
+ ∀t1'.t1' = lift_tape ls 〈curc,false〉 rs →
+ ∃ls1,rs1,newc.
+ (t2 = midtape STape ls1 〈grid,false〉 (reverse ? curconfig@newc::
+ 〈grid,false〉::reverse ? table@〈grid,false〉::rs1) ∧
+ lift_tape ls1 newc rs1 =
+ tape_move_right STape ls 〈curc,false〉 rs).
+
+lemma lift_tape_not_null :
+ ∀ls,c,rs. is_null (\fst c) = false →
+ lift_tape ls c rs = mk_tape STape ls (Some ? c) rs.
+#ls * #c0 #bc0 #rs cases c0
+[|normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
+//
+qed.
+
+lemma mtr_concrete_to_abstract :
+ ∀t1,t2.R_move_tape_r t1 t2 → R_move_tape_r_abstract t1 t2.
+#t1 #t2 whd in ⊢(%→?); #Hconcrete
+#rs #n #table #curc #curconfig #ls #Hcurc #Hcurconfig #Htable #Ht1
+#Hrsnonulls #t1' #Ht1'
+cases (Hconcrete … Htable Ht1) //
+[ * #Hrs #Ht2
+ @(ex_intro ?? (〈curc,false〉::ls)) @(ex_intro ?? [])
+ @(ex_intro ?? 〈null,false〉) %
+ [ >Ht2 %
+ | >Hrs % ]
+| * #r0 * #rs0 * #Hrs #Ht2
+ @(ex_intro ?? (〈curc,false〉::ls)) @(ex_intro ?? rs0)
+ @(ex_intro ?? r0) %
+ [ >Ht2 %
+ | >Hrs >lift_tape_not_null
+ [ %
+ | @Hrsnonulls >Hrs @memb_hd ] ]
+qed.
+
+definition R_move_tape_l_abstract ≝ λt1,t2.
+ ∀rs,n,table,curc,curconfig,ls.
+ bit_or_null curc = true → only_bits_or_nulls curconfig → table_TM n (reverse ? table) →
+ t1 = midtape STape (table@〈grid,false〉::〈curc,false〉::curconfig@〈grid,false〉::ls)
+ 〈grid,false〉 rs →
+ no_nulls ls →
+ ∀t1'.t1' = lift_tape ls 〈curc,false〉 rs →
+ ∃ls1,rs1,newc.
+ (t2 = midtape STape ls1 〈grid,false〉 (reverse ? curconfig@newc::
+ 〈grid,false〉::reverse ? table@〈grid,false〉::rs1) ∧
+ lift_tape ls1 newc rs1 =
+ tape_move_left STape ls 〈curc,false〉 rs).
+
+lemma mtl_concrete_to_abstract :
+ ∀t1,t2.R_move_tape_l t1 t2 → R_move_tape_l_abstract t1 t2.
+#t1 #t2 whd in ⊢(%→?); #Hconcrete
+#rs #n #table #curc #curconfig #ls #Hcurc #Hcurconfig #Htable #Ht1
+#Hlsnonulls #t1' #Ht1'
+cases (Hconcrete … Htable Ht1) //
+[ * #Hls #Ht2
+ @(ex_intro ?? [])
+ @(ex_intro ?? (〈curc,false〉::rs))
+ @(ex_intro ?? 〈null,false〉) %
+ [ >Ht2 %
+ | >Hls % ]
+| * #l0 * #ls0 * #Hls #Ht2
+ @(ex_intro ?? ls0)
+ @(ex_intro ?? (〈curc,false〉::rs))
+ @(ex_intro ?? l0) %
+ [ >Ht2 %
+ | >Hls >lift_tape_not_null
+ [ %
+ | @Hlsnonulls >Hls @memb_hd ] ]
+qed.
+
+lemma Realize_to_Realize :
+ ∀alpha,M,R1,R2.(∀t1,t2.R1 t1 t2 → R2 t1 t2) → Realize alpha M R1 → Realize alpha M R2.
+#alpha #M #R1 #R2 #Himpl #HR1 #intape
+cases (HR1 intape) -HR1 #k * #outc * #Hloop #HR1
+@(ex_intro ?? k) @(ex_intro ?? outc) % /2/
+qed.
+
+lemma sem_move_tape_l_abstract : Realize … move_tape_l R_move_tape_l_abstract.
+@(Realize_to_Realize … mtl_concrete_to_abstract) //
+qed.
+
+lemma sem_move_tape_r_abstract : Realize … move_tape_r R_move_tape_r_abstract.
+@(Realize_to_Realize … mtr_concrete_to_abstract) //
+qed.
+
+(*
+ t1 = ls # cs c # table # rs
+
+ let simt ≝ lift_tape ls c rs in
+ let simt' ≝ move_left simt' in
+
+ t2 = left simt'# cs (sim_current_of_tape simt') # table # right simt'
+*)
+
+(*
+definition R_move
+
+definition R_exec_move ≝ λt1,t2.
+ ∀ls,current,table1,newcurrent,table2,rs.
+ t1 = midtape STape (current@〈grid,false〉::ls) 〈grid,false〉
+ (table1@〈comma,true〉::newcurrent@〈comma,false〉::move::table2@
+ 〈grid,false〉::rs) →
+ table_TM (table1@〈comma,false〉::newcurrent@〈comma,false〉::move::table2) →
+ t2 = midtape
+*)
+
+(*
+
+step :
+
+if is_true(current) (* current state is final *)
+ then nop
+ else
+ init_match;
+ match_tuple;
+ if is_marked(current) = false (* match ok *)
+ then exec_move;
+ else sink;
+
+*)
+
+
+definition move_tape ≝
+ ifTM ? (test_char ? (λc:STape.c == 〈bit false,false〉))
+ (* spostamento a sinistra: verificare se per caso non conviene spostarsi
+ sulla prima grid invece dell'ultima *)
+ (seq ? (adv_to_mark_r ? (λc:STape.is_grid (\fst c))) move_tape_l)
+ (ifTM ? (test_char ? (λc:STape.c == 〈bit true,false〉))
+ (seq ? (adv_to_mark_r ? (λc:STape.is_grid (\fst c))) move_tape_r)
+ (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
+ (seq ? (move_l …) (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))))
+ tc_true) tc_true.
+
+definition R_move_tape ≝ λt1,t2.
+ ∀rs,n,table1,c,table2,curc,curconfig,ls.
+ bit_or_null curc = true → bit_or_null c = true → only_bits_or_nulls curconfig →
+ table_TM n (reverse ? table1@〈c,false〉::table2) →
+ t1 = midtape STape (table1@〈grid,false〉::〈curc,false〉::curconfig@〈grid,false〉::ls)
+ 〈c,false〉 (table2@〈grid,false〉::rs) →
+ no_nulls ls → no_nulls rs →
+ ∀t1'.t1' = lift_tape ls 〈curc,false〉 rs →
+ ∃ls1,rs1,newc.
+ (t2 = midtape STape ls1 〈grid,false〉 (reverse ? curconfig@newc::
+ 〈grid,false〉::reverse ? table1@〈c,false〉::table2@〈grid,false〉::rs1) ∧
+ ((c = bit false ∧ lift_tape ls1 newc rs1 = tape_move_left STape ls 〈curc,false〉 rs) ∨
+ (c = bit true ∧ lift_tape ls1 newc rs1 = tape_move_right STape ls 〈curc,false〉 rs) ∨
+ (c = null ∧ ls1 = ls ∧ rs1 = rs ∧ 〈curc,false〉 = newc))).
+
+lemma sem_move_tape : Realize ? move_tape R_move_tape.
+#intape
+cases (sem_if ? (test_char ??) … tc_true (sem_test_char ? (λc:STape.c == 〈bit false,false〉))
+ (sem_seq … (sem_adv_to_mark_r ? (λc:STape.is_grid (\fst c))) sem_move_tape_l_abstract)
+ (sem_if ? (test_char ??) … tc_true (sem_test_char ? (λc:STape.c == 〈bit true,false〉))
+ (sem_seq … (sem_adv_to_mark_r ? (λc:STape.is_grid (\fst c))) sem_move_tape_r_abstract)
+ (sem_seq … (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
+ (sem_seq … (sem_move_l …) (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))))) intape)
+#k * #outc * #Hloop #HR
+@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
+#rs #n #table1 #c #table2 #curc #curconfig #ls
+#Hcurc #Hc #Hcurconfig #Htable #Hintape #Hls #Hrs #t1' #Ht1'
+generalize in match HR; -HR *
+[ * #ta * whd in ⊢ (%→?); #Hta cases (Hta 〈c,false〉 ?)
+ [| >Hintape % ] -Hta #Hceq #Hta lapply (\P Hceq) -Hceq #Hceq destruct (Hta Hceq)
+ * #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hintape) -Htb -Hintape
+ [ * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
+ * #_ #Htb lapply (Htb … (refl ??) (refl ??) ?)
+ [ @daemon ] -Htb >append_cons <associative_append #Htb
+ whd in ⊢ (%→?); #Houtc lapply (Houtc … Htb … Ht1') //
+ [ >reverse_append >reverse_append >reverse_reverse @Htable |]
+ -Houtc -Htb * #ls1 * #rs1 * #newc * #Houtc #Hnewtape
+ @(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? newc) %
+ [ >Houtc >reverse_append >reverse_append >reverse_reverse
+ >associative_append >associative_append %
+ | % % % // ]
+| * #ta * whd in ⊢ (%→?); #Hta cases (Hta 〈c,false〉 ?)
+ [| >Hintape % ] -Hta #Hcneq cut (c ≠ bit false)
+ [ lapply (\Pf Hcneq) @not_to_not #Heq >Heq % ] -Hcneq #Hcneq #Hta destruct (Hta)
+ *
+ [ * #tb * whd in ⊢ (%→?);#Htb cases (Htb 〈c,false〉 ?)
+ [| >Hintape % ] -Htb #Hceq #Htb lapply (\P Hceq) -Hceq #Hceq destruct (Htb Hceq)
+ * #tc * whd in ⊢ (%→?); #Htc cases (Htc … Hintape) -Htc -Hintape
+ [ * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
+ * #_ #Htc lapply (Htc … (refl ??) (refl ??) ?)
+ [ @daemon ] -Htc >append_cons <associative_append #Htc
+ whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc … Ht1') //
+ [ >reverse_append >reverse_append >reverse_reverse @Htable |]
+ -Houtc -Htc * #ls1 * #rs1 * #newc * #Houtc #Hnewtape
+ @(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? newc) %
+ [ >Houtc >reverse_append >reverse_append >reverse_reverse
+ >associative_append >associative_append %
+ | % %2 % // ]
+ | * #tb * whd in ⊢ (%→?); #Htb cases (Htb 〈c,false〉 ?)
+ [| >Hintape % ] -Htb #Hcneq' cut (c ≠ bit true)
+ [ lapply (\Pf Hcneq') @not_to_not #Heq >Heq % ] -Hcneq' #Hcneq' #Htb destruct (Htb)
+ * #tc * whd in ⊢ (%→?); #Htc cases (Htc … Hintape)
+ [ * >(bit_or_null_not_grid … Hc) #Hfalse destruct (Hfalse) ] -Htc
+ * #_ #Htc lapply (Htc … (refl ??) (refl ??) ?) [@daemon] -Htc #Htc
+ * #td * whd in ⊢ (%→?); #Htd lapply (Htd … Htc) -Htd -Htc
+ whd in ⊢ (???%→?); #Htd whd in ⊢ (%→?); #Houtc lapply (Houtc … Htd) -Houtc *
+ [ * >(bit_or_null_not_grid … Hcurc) #Hfalse destruct (Hfalse) ]
+ * #_ #Houtc lapply (Houtc … (refl ??) (refl ??) ?) [@daemon] -Houtc #Houtc
+ @(ex_intro ?? ls) @(ex_intro ?? rs) @(ex_intro ?? 〈curc,false〉) %
+ [ @Houtc
+ | %2 % // % // % //
+ generalize in match Hcneq; generalize in match Hcneq';
+ cases c in Hc; normalize //
+ [ * #_ normalize [ #Hfalse @False_ind cases Hfalse /2/ | #_ #Hfalse @False_ind cases Hfalse /2/ ]
+ |*: #Hfalse destruct (Hfalse) ]
+ ]
+ ]
+]
+qed.
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