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_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 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_move : Realize ? exec_move R_exec_move.
#intape
cases (sem_seq … sem_init_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 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) %
]
]
| //
]
]
- | * * * #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
+ ]
+]
+qed.
+
+(*
+if is_false(current) (* current state is not final *)
+ then init_match;
+ match_tuple;
+ if is_marked(current) = false (* match ok *)
+ then
+ exec_move
+ 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
+ (ifTM ? (test_char ? (λc.¬is_marked ? c))
+ (seq ? exec_move (move_r …))
+ (nop ?) (* sink *)
+ 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 (S n) (〈s0,false〉::curconfig) 〈c0,false〉
+ (〈s1,false〉::newconfig) 〈c1,false〉 〈mv,false〉 (〈t0,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) →
+ ∀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) ∧
+ lift_tape ls1 〈c2,false〉 rs1 =
+ tape_move STape t1' (map_move c1 mv) ∧ legal_tape ls1 〈c2,false〉 rs1).
+
+definition R_uni_step_false ≝ λt1,t2.
+ ∀b. current STape t1 = Some ? 〈bit b,false〉 → b = true ∧ t2 = t1.
+
+axiom sem_match_tuple : Realize ? match_tuple R_match_tuple.
+
+lemma sem_uni_step :
+ accRealize ? uni_step (inr … (inl … (inr … 0)))
+ 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 …))
+ (sem_nop …))))
+ (sem_nop …)
+ …)
+[ #intape #outtape
+ #ta whd in ⊢ (%→?); #Hta #HR
+ #n #t0 #table #s0 #s1 #c0 #c1 #ls #rs #curconfig #newconfig #mv
+ #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
+ lapply (Htb (〈grid,false〉::ls) (curconfig@[〈c0,false〉]) (table@〈grid,false〉::rs) s0 t0 ???)
+ [ >Hta >associative_append %
+ | (* utilizzare Hmatch
+ cases (match_in_table_to_tuple … Hmatch Htable)
+ ma serve l'iniettività di mk_tuple...
+ *) @daemon
+ | (* idem *) @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 *) @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
+ * #te * whd in ⊢ (%→?); #Hte
+ 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))
+ [ >Htd @eq_f3 //
+ [ >reverse_append >reverse_single %
+ | >associative_append >associative_append normalize
+ >associative_append >associative_append <Hmv1 %
+ ]
+ ]
+ -Htd #Htd lapply (Hte … (S n) … Htd … Ht1') -Htd -Hte
+ [ //
+ | (*|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;
+ >associative_append >associative_append normalize
>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 % ]
+ #Htableeq <Htableeq // ]
+ * #ls1 * #rs1 * #c2 * * #Hte #Hliftte #Hlegalte
+ whd in ⊢ (%→?); #Houttape lapply (Houttape … Hte) -Houttape #Houttape
+ 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 ⊢ (???%);
+ >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
]
- | //
+ | //
+ ]
]
+ ]
+ | * #td * whd in ⊢ (%→%→?); >Htc #Htd
+ cases (Htd ? (refl ??)) normalize in ⊢ (%→?);
+ #Hfalse destruct (Hfalse)
]
+ ]
+| #t1 #t2 #t3 whd in ⊢ (%→%→?); #Ht1 #Ht2
+ #b #Hb cases (Ht1 ? Hb) #Hb' #Ht3 >Ht2 % //
+ cases b in Hb'; normalize #H1 //
]
qed.
-
-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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