*)
include "turing/universal/copy.ma".
+include "turing/universal/move_tape.ma".
+include "turing/universal/match_machines.ma".
(*
*)
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 ?)))))).
-
+ mark ? · adv_to_mark_r ? (λc:STape.is_grid (\fst c)) · move_r ? ·
+ move_r ? · mark ? · 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).
+ t1 = midtape STape ls 〈c,false〉 (l@〈grid,false〉::〈bar,false〉::〈d,false〉::rs) →
+ t2 = midtape STape ls 〈c,true〉 (l@〈grid,false〉::〈bar,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)
+ (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
* #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 …) ?)
+* #Hgrdic #Htb lapply (Htb l 〈grid,false〉 (〈bar,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
+* #tf * whd in ⊢ (%→?); #Htf lapply (Htf … Hte) -Htf -Hte #Htf
+whd in ⊢ (%→?); #Htg cases (Htg … Htf) -Htg -Htf
[* whd in ⊢ ((??%?)→?); #Habs destruct (Habs)]
-* #_ #Htf lapply (Htf (reverse ? l) 〈c,true〉 ls (refl …) (refl …) ?)
- [#x #membl @Hnomarks @daemon] -Htf #Htf >Htf >reverse_reverse %
+* #_ #Htg lapply (Htg (〈grid,false〉::reverse ? l) 〈c,true〉 ls (refl …) (refl …) ?)
+ [#x #membl @Hnomarks @daemon] -Htg #Htg >Htg >reverse_cons >reverse_reverse
+ >associative_append %
qed.
-
(* init_copy
init_current_on_match; (* no marks in current *)
*)
definition init_copy ≝
- seq ? init_current_on_match
- (seq ? (move_r ?)
- (seq ? (adv_to_mark_r ? (is_marked ?))
- (adv_mark_r ?))).
+ init_current_on_match · move_r ? ·
+ adv_to_mark_r ? (is_marked ?) · adv_mark_r ?.
definition R_init_copy ≝ λt1,t2.
∀l1,l2,c,ls,d,rs.
]
qed. *)
-include "turing/universal/move_tape.ma".
-
-definition exec_move ≝
- seq ? init_copy
- (seq ? copy
- (seq ? (move_r …) move_tape)).
+definition exec_action ≝
+ init_copy · copy · move_r … · move_tape.
definition map_move ≝
λc,mv.match c with [ null ⇒ None ? | _ ⇒ Some ? 〈c,false,move_of_unialpha mv〉 ].
mv ≠ null → c1 ≠ null
dal fatto che c1 e mv sono contenuti nella table
*)
-definition R_exec_move ≝ λt1,t2.
+definition R_exec_action ≝ 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_marks curconfig → only_bits (curconfig@[〈s0,false〉]) →
+ only_bits (〈s1,false〉::newconfig) → bit_or_null c1 = true →
|curconfig| = |newconfig| →
- no_nulls ls → no_nulls rs → no_marks ls → no_marks rs →
legal_tape ls 〈c0,false〉 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) →
]
qed.
-lemma sem_exec_move : Realize ? exec_move R_exec_move.
+lemma merge_char_null : ∀c.merge_char null c = c.
+* //
+qed.
+
+lemma merge_char_cases : ∀c1,c2.merge_char c1 c2 = c1 ∨ merge_char c1 c2 = c2.
+#c1 *
+[ #c1' %2 %
+| % %
+| *: %2 % ]
+qed.
+
+(* lemma merge_char_c_bit :
+ ∀c1,c2.is_bit c2 = true → merge_char c1 c2 = c2.
+#c1 *
+[ #c2' #_ %
+|*: normalize #Hfalse destruct (Hfalse) ]
+qed.
+
+lemma merge_char_c_bit :
+ ∀c1,c2.is_null c2 = true → merge_char c1 c2 = c1.
+#c1 *
+[ #c2' #_ %
+|*: normalize #Hfalse destruct (Hfalse) ]
+qed.
+
+*)
+
+lemma sem_exec_action : Realize ? exec_action R_exec_action.
#intape
cases (sem_seq … sem_init_copy
(sem_seq … sem_copy
#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 #Hlen #Hls #Hrs #Hls1 #Hrs1 #Htape #Hintape #t1' #Ht1'
+#Htable #Hcurconfig1 #Hcurconfig2 #Hnewconfig #Hc1 #Hlen #Htape #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
whd in ⊢ (%→?); #Houtc whd in Htc:(???%); whd in Htc:(???(??%%%));
lapply (Houtc rs n
(〈comma,false〉::〈c1,false〉::reverse ? newconfig@〈s1,false〉::〈comma,false〉::reverse ? table1)
- mv table2 (merge_char c0 c1) (reverse ? newconfig@[〈s1,false〉]) ls ?????
- Hls Hrs Hls1 Hrs1 ???)
- [3: cases (true_or_false (c0 == null)) #Hc0
- [ >(\P Hc0) in Htape; #Htape cases (legal_tape_cases … Htape)
- [ * #Hfalse @False_ind /2/
- | * #_ *
- [ #Hlsnil @legal_tape_conditions % %2 //
- | #Hrsnil @legal_tape_conditions %2 //
- ]
- ]
- | @legal_tape_conditions % % @merge_char_not_null @(\Pf Hc0) ]
+ mv table2 (merge_char c0 c1) (reverse ? newconfig@[〈s1,false〉]) ls ????????)
+ [3: cases Htape -Htape * * #Hnomarks #Hbits #Hc0 #Hlsrs % [ % [ %
+ [ #x #Hx cases (orb_true_l … Hx) #Hx'
+ [ >(\P Hx') %
+ | @Hnomarks @memb_cons // ]
+ | @Hbits ]
+ | cases (merge_char_cases c0 c1) #Hmerge >Hmerge // ]
+ | cases (true_or_false (c0 == null)) #Hc0'
+ [ cases Hlsrs -Hlsrs
+ [ *
+ [ >(\P Hc0') * #Hfalse @False_ind /2/
+ | #Hlsnil % %2 // ]
+ | #Hrsnil %2 // ]
+ | % % @merge_char_not_null @(\Pf Hc0') ] ]
|4:>Htc @(eq_f3 … (midtape ?))
[ @eq_f @eq_f >associative_append >associative_append %
| %
>associative_append >associative_append >associative_append
>associative_append >associative_append
@Htable
+ | (* well formedness of table *) @daemon
| (* Hnewconfig *) @daemon
- | (* bit_or_null mv = true *) @daemon
- | (* bit_or_null c1,c2 → bit_or_null (merge_char c1 c2) *) @daemon
+ | (* bit_or_null mv = true (well formedness of table) *) @daemon
| -Houtc * #ls1 * #rs1 * #newc * #Hnewtapelegal * #Houtc *
[ *
[ * #Hmv #Htapemove
| >Hmv >Ht1' >Htapemove
(* mv = bit false -→ c1 = bit ? *)
cut (∃c1'.c1 = bit c1') [ @daemon ] * #c1' #Hc1
- >Hc1 >tape_move_left_eq cases (legal_tape_cases … Htape)
- [ #Hc0 >lift_tape_not_null /2/
- | * #Hc0 *
- [ #Hls >Hc0 >Hls whd in ⊢ (??%%); cases rs;
- [%| #r0 #rs0 %]
- | #Hrs >Hc0 >Hrs cases ls; [%| #l0 #ls0 %]
- ]
- ]
+ >Hc1 >tape_move_left_eq >(legal_tape_left … Htape)
+ >(legal_tape_right … Htape) %
]
| //
]
>associative_append >associative_append %
|>Hmv >Ht1' >Htapemove
cut (∃c1'.c1 = bit c1') [ @daemon ] * #c1' #Hc1
- >Hc1 >tape_move_right_eq cases (legal_tape_cases … Htape)
- [ #Hc0 >lift_tape_not_null /2/
- | * #Hc0 *
- [ #Hls >Hc0 >Hls whd in ⊢ (??%%); cases rs;
- [%| #r0 #rs0 %]
- | #Hrs >Hc0 >Hrs cases ls; [%| #l0 #ls0 %]
- ]
+ >Hc1 >tape_move_right_eq >(legal_tape_left … Htape)
+ >(legal_tape_right … Htape) %
]
+ | //
]
- | //
]
- ]
- | * * * #Hmv #Hlseq #Hrseq #Hnewc
- @(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? newc) %
- [ %
- [ >Houtc -Houtc >reverse_append
- >reverse_reverse >reverse_single @eq_f
- >reverse_cons >reverse_cons >reverse_append >reverse_cons
- >reverse_cons >reverse_reverse >reverse_reverse
- >associative_append >associative_append
- >associative_append >associative_append
- >associative_append >associative_append %
- |>Hmv >Ht1' cases c1 in Hnewc;
- [ #c1' whd in ⊢ (??%?→?);#Hnewc <Hnewc
- >Hlseq >Hrseq cases Htape * #Hleft #Hright #_
- whd in ⊢ (??%%); >Hleft >Hright %
- | whd in ⊢ (??%?→?); #Hnewc >Hnewc >Hlseq >Hrseq %
- |*: whd in ⊢ (??%?→?);#Hnewc <Hnewc
- >Hlseq >Hrseq cases Htape * #Hleft #Hright #_
- whd in ⊢ (??%%); >Hleft >Hright % ]
+ | * * * #Hmv #Hlseq #Hrseq #Hnewc
+ @(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? newc) %
+ [ %
+ [ >Houtc -Houtc >reverse_append
+ >reverse_reverse >reverse_single @eq_f
+ >reverse_cons >reverse_cons >reverse_append >reverse_cons
+ >reverse_cons >reverse_reverse >reverse_reverse
+ >associative_append >associative_append
+ >associative_append >associative_append
+ >associative_append >associative_append %
+ |>Hmv >Ht1' cases c1 in Hnewc;
+ [ #c1' whd in ⊢ (??%?→?);#Hnewc <Hnewc
+ >Hlseq >Hrseq whd in ⊢ (??%%);
+ >(legal_tape_left … Htape) >(legal_tape_right … Htape) %
+ | whd in ⊢ (??%?→?); #Hnewc >Hnewc >Hlseq >Hrseq %
+ |*: whd in ⊢ (??%?→?);#Hnewc <Hnewc
+ >Hlseq >Hrseq whd in ⊢ (??%%);
+ >(legal_tape_left … Htape) >(legal_tape_right … Htape) %
+ ]
+ ]
+ | //
]
- | //
]
]
]
match_tuple;
if is_marked(current) = false (* match ok *)
then
- exec_move
+ exec_action
move_r;
else sink;
else nop;
definition uni_step ≝
ifTM ? (test_char STape (λc.\fst c == bit false))
- (single_finalTM ? (seq ? init_match
- (seq ? match_tuple
+ (single_finalTM ?
+ (init_match · match_tuple ·
(ifTM ? (test_char ? (λc.¬is_marked ? c))
- (seq ? exec_move (move_r …))
- (nop ?) (* sink *)
- tc_true))))
- (nop ?)
- tc_true.
+ (exec_action · move_r …)
+ (nop ?) tc_true)))
+ (nop ?) tc_true.
definition R_uni_step_true ≝ λt1,t2.
- ∀n,t0,table,s0,s1,c0,c1,ls,rs,curconfig,newconfig,mv.
- table_TM (S n) (〈t0,false〉::table) →
- match_in_table (〈s0,false〉::curconfig@[〈c0,false〉])
- (〈s1,false〉::newconfig@[〈c1,false〉]) mv (〈t0,false〉::table) →
+ ∀n,table,s0,s1,c0,c1,ls,rs,curconfig,newconfig,mv.
+ 0 < |table| → table_TM (S n) table →
+ match_in_table (S n) (〈s0,false〉::curconfig) 〈c0,false〉
+ (〈s1,false〉::newconfig) 〈c1,false〉 〈mv,false〉 table →
legal_tape ls 〈c0,false〉 rs →
t1 = midtape STape (〈grid,false〉::ls) 〈s0,false〉
- (curconfig@〈c0,false〉::〈grid,false〉::〈t0,false〉::table@〈grid,false〉::rs) →
+ (curconfig@〈c0,false〉::〈grid,false〉::table@〈grid,false〉::rs) →
∀t1'.t1' = lift_tape ls 〈c0,false〉 rs →
s0 = bit false ∧
∃ls1,rs1,c2.
(t2 = midtape STape (〈grid,false〉::ls1) 〈s1,false〉
- (newconfig@〈c2,false〉::〈grid,false〉::〈t0,false〉::table@〈grid,false〉::rs1) ∧
+ (newconfig@〈c2,false〉::〈grid,false〉::table@〈grid,false〉::rs1) ∧
lift_tape ls1 〈c2,false〉 rs1 =
tape_move STape t1' (map_move c1 mv) ∧ legal_tape ls1 〈c2,false〉 rs1).
axiom sem_match_tuple : Realize ? match_tuple R_match_tuple.
+definition us_acc : states ? uni_step ≝ (inr … (inl … (inr … start_nop))).
+
lemma sem_uni_step :
- accRealize ? uni_step (inr … (inl … (inr … 0)))
+ accRealize ? uni_step us_acc
R_uni_step_true R_uni_step_false.
-@(acc_sem_if_app … (sem_test_char ? (λc:STape.\fst c == bit false))
- (sem_seq … sem_init_match
- (sem_seq … sem_match_tuple
- (sem_if … (sem_test_char ? (λc.¬is_marked ? c))
- (sem_seq … sem_exec_move (sem_move_r …))
+@(acc_sem_if_app STape … (sem_test_char ? (λc:STape.\fst c == bit false))
+ (sem_seq … sem_init_match
+ (sem_seq … sem_match_tuple
+ (sem_if … (* ????????? (sem_test_char … (λc.¬is_marked FSUnialpha c)) *)
+ (sem_seq … sem_exec_action (sem_move_r …))
(sem_nop …))))
(sem_nop …)
…)
+[@sem_test_char||]
[ #intape #outtape
#ta whd in ⊢ (%→?); #Hta #HR
- #n #t0 #table #s0 #s1 #c0 #c1 #ls #rs #curconfig #newconfig #mv
+ #n #fulltable #s0 #s1 #c0 #c1 #ls #rs #curconfig #newconfig #mv
+ #Htable_len cut (∃t0,table. fulltable =〈bar,false〉::〈t0,false〉::table) [(* 0 < |table| *) @daemon]
+ * #t0 * #table #Hfulltable >Hfulltable -fulltable
#Htable #Hmatch #Htape #Hintape #t1' #Ht1'
>Hintape in Hta; #Hta cases (Hta ? (refl ??)) -Hta
#Hs0 lapply (\P Hs0) -Hs0 #Hs0 #Hta % //
cases HR -HR
- #tb * whd in ⊢ (%→?); #Htb
+ #tb * whd in ⊢ (%→?); #Htb
lapply (Htb (〈grid,false〉::ls) (curconfig@[〈c0,false〉]) (table@〈grid,false〉::rs) s0 t0 ???)
[ >Hta >associative_append %
- | (* da decidere se aggiungere un'assunzione o utilizzare Hmatch *) @daemon
- | (* idem *) @daemon
+ | @daemon
+ | @daemon
| -Hta -Htb #Htb *
#tc * whd in ⊢ (%→?); #Htc cases (Htc … Htable … Htb) -Htb -Htc
[| * #Hcurrent #Hfalse @False_ind
(* absurd by Hmatch *) @daemon
| >Hs0 %
- | (* da decidere se aggiungere un'assunzione o utilizzare Hmatch *) @daemon
+ | (* Htable (con lemma) *) @daemon
+ | (* Hmatch *) @daemon
| (* Htable *) @daemon
- | (* Htable, Hmatch → |config| = n *) @daemon ]
+ | (* Htable, Hmatch → |config| = n
+ necessaria modifica in R_match_tuple, le dimensioni non corrispondono
+ *) @daemon
+ ]
* #table1 * #newc * #mv1 * #table2 * #Htableeq #Htc *
[ * #td * whd in ⊢ (%→?); >Htc -Htc #Htd
cases (Htd ? (refl ??)) #_ -Htd
cut (newc = 〈s1,false〉::newconfig@[〈c1,false〉]) [@daemon] #Hnewc
- >Hnewc #Htd
- cut (mv1 = 〈\fst mv1,false〉)
- [ >(eq_pair_fst_snd … mv1) @eq_f (*Htable, Htableeq*) @daemon ] #Hmv1
+ >Hnewc #Htd cut (mv1 = 〈mv,false〉)
+ [@daemon] #Hmv1
* #te * whd in ⊢ (%→?); #Hte
- (* mv1 ha tipo lista ma dovrebbe avere tipo unialpha *)
cut (td = midtape STape (〈c0,false〉::reverse STape curconfig@〈s0,false〉::〈grid,false〉::ls)
〈grid,false〉
- ((table1@〈s0,false〉::curconfig@[〈c0,false〉])@〈comma,true〉::〈s1,false〉::
- newconfig@〈c1,false〉::〈comma,false〉::〈\fst mv1,false〉::table2@〈grid,false〉::rs))
+ ((table1@〈bar,false〉::〈s0,false〉::curconfig@[〈c0,false〉])@〈comma,true〉::〈s1,false〉::
+ newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs))
[ >Htd @eq_f3 //
[ >reverse_append >reverse_single %
| >associative_append >associative_append normalize
- >associative_append >associative_append <Hmv1 %
+ >associative_append >associative_append >Hmv1 %
]
]
-Htd #Htd lapply (Hte … (S n) … Htd … Ht1') -Htd -Hte
[ //
- |2,3,4,5: (* dovrebbe scomparire (lo metteremo nel legal_tape) *) @daemon
| (*|curconfig| = |newconfig|*) @daemon
+ | (* Htable → bit_or_null c1 = true *) @daemon
| (* only_bits (〈s1,false〉::newconfig) *) @daemon
| (* only_bits (curconfig@[〈s0,false〉]) *) @daemon
| (* no_marks (reverse ? curconfig) *) @daemon
- | <Hmv1 >Hnewc in Htableeq;
+ | >Hmv1 in Htableeq; >Hnewc
>associative_append >associative_append normalize
>associative_append >associative_append
#Htableeq <Htableeq // ]
whd in Houttape:(???%); whd in Houttape:(???(??%%%));
@ex_intro [| @(ex_intro ?? rs1) @ex_intro [| % [ %
[ >Houttape @eq_f @eq_f @eq_f @eq_f
- change with ((〈t0,false〉::table)@?) in ⊢ (???%);
+ change with ((〈bar,false〉::〈t0,false〉::table)@?) in ⊢ (???%);
>Htableeq >associative_append >associative_append
>associative_append normalize >associative_append
>associative_append normalize >Hnewc <Hmv1
- >associative_append normalize >associative_append %
- | >(?: mv = \fst mv1) [| (*Hmatch, Htableeq*) @daemon ]
- @Hliftte
+ >associative_append normalize >associative_append
+ >Hmv1 %
+ | @Hliftte
]
| //
]
#b #Hb cases (Ht1 ? Hb) #Hb' #Ht3 >Ht2 % //
cases b in Hb'; normalize #H1 //
]
-qed.
+qed.
\ No newline at end of file
projects / helm.git / blobdiff