include "turing/universal/move_tape.ma".
definition exec_move ≝
- seq ? (adv_to_mark_r … (is_marked ?))
- (seq ? init_copy
- (seq ? copy
- (seq ? (move_r …)
- (seq ? move_tape (move_r …))))).
-
-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 ].
-
-(*
- 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
+ seq ? init_copy
+ (seq ? copy
+ (seq ? (move_r …) move_tape)).
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 mk_tuple ≝ λc,newc,mv.
- c @ 〈comma,false〉:: newc @ 〈comma,false〉 :: [〈mv,false〉].
+ ∀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
-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).
+ 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
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 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.
\ No newline at end of file
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