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 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) %
+ ]
+ ]
+ | //
]
- | //
]
]
]
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) →
+ 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) →
#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
+ | (* 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
| >Hs0 %
| (* da decidere se aggiungere un'assunzione o utilizzare Hmatch *) @daemon
| (* Htable *) @daemon
- | (* Htable, Hmatch → |config| = n *) @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
- * #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)
+ | (* 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))
- lapply (Hte … Htd)
-
- (* univocità delle tuple in table *)
- cut (newc = 〈s1,false〉::newconfig@[〈c1,false〉]) [@daemon] #Hnewc
- c00 ≝ c0
- curconfig0 ≝ curconfig
- s00 ≝ s0
- ls0 ≝ ls
- table10 ≝ (table1@〈s0,false〉::curconfig@〈c0,false〉)
- s10 ≝ s1
- newconfig0 ≝ newconfig
- c10 ≝ c1
- mv0 ≝ mv1
- table20 ≝ table2
- rs0 ≝ rs
-
- lapply (Hte … Htd)
- | * #td * whd in ⊢ (%→%→?); >Htc #Htd
- cases (Htd ? (refl ??)) normalize in ⊢ (%→?);
- #Hfalse destruct (Hfalse)
- ]
-
-
+ [ >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
+ #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 //
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