*)
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
cases HR -HR
-#ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hintape) -Hta -Hintape #Hta
-* #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Htb -Hta
+#ta * whd in ⊢ (%→?); * #Hta #_ lapply (Hta … Hintape) -Hta -Hintape #Hta
+* #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
- [* whd in ⊢ ((??%?)→?); #Habs destruct (Habs)]
-* #_ #Htf lapply (Htf (reverse ? l) 〈c,true〉 ls (refl …) (refl …) ?)
- [#x #membl @Hnomarks @daemon] -Htf #Htf >Htf >reverse_reverse %
+* #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
+* #tf * whd in ⊢ (%→?); * #_ #Htf lapply (Htf … Hte) -Htf -Hte #Htf
+whd in ⊢ (%→?); * #_ #Htg cases (Htg … Htf) -Htg -Htf
+#_ #Htg cases (Htg (refl …)) -Htg #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.
#l1 #l2 #c #ls #d #rs #Hl1marks #Hl1grids #Hl2marks #Hc #Hintape
cases HR -HR
#ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hl1grids Hc Hintape) -Hta -Hintape #Hta
-* #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta) -Htb -Hta
+* #tb * whd in ⊢ (%→?); * #_ #Htb lapply (Htb … Hta) -Htb -Hta
generalize in match Hl1marks; -Hl1marks cases (list_last ? l1)
[#eql1 >eql1 #Hl1marks whd in ⊢ ((???%)→?); whd in ⊢ ((???(????%))→?); #Htb
- * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb *
+ * #tc * whd in ⊢ (%→?); * #_ #Htc lapply (Htc … Htb) -Htc -Htb *
[* whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)]
- * #_ #Htc lapply (Htc … (refl …) (refl …) ?)
+ * * #_ #Htc #_ lapply (Htc … (refl …) (refl …) ?)
[#x #membx @Hl2marks @membx]
- #Htc whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc -Htc #Houtc
+ #Htc whd in ⊢ (%→?); * #Houtc #_ cases (Houtc (reverse ? l2@〈grid,false〉::〈c,true〉::〈grid,false〉::ls) comma)
+ -Houtc #Houtc lapply (Houtc … Htc) -Houtc -Htc #Houtc #_
>Houtc %
|* #c1 * #tl #eql1 >eql1 #Hl1marks >reverse_append >reverse_single
whd in ⊢ ((???%)→?); whd in ⊢ ((???(????%))→?);
>associative_append whd in ⊢ ((???(????%))→?); #Htb
- * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb *
+ * #tc * whd in ⊢ (%→?); * #_ #Htc lapply (Htc … Htb) -Htc -Htb *
[* >Hl1marks [#Htemp destruct (Htemp)] @memb_append_l2 @memb_hd]
- * #_ >append_cons <associative_append #Htc lapply (Htc … (refl …) (refl …) ?)
+ *
* #_ >append_cons <associative_append #Htc lapply (Htc … (refl …) (refl …) ?)
[#x #membx cases (memb_append … membx) -membx #membx
[cases (memb_append … membx) -membx #membx
[@Hl1marks @memb_append_l1 @daemon
]
|@Hl2marks @membx
]]
- #Htc whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc -Htc #Houtc
+ -Htc #Htc #_ whd in ⊢ (%→?); * #Houtc #_ cases (Houtc (reverse (FinProd FSUnialpha FinBool) ((reverse STape tl@[〈grid,false〉])@l2)
+ @c1::〈c,true〉::〈grid,false〉::ls) comma)
+ -Houtc #Houtc lapply (Houtc … Htc) -Houtc -Htc #Houtc #_
>Houtc >reverse_append >reverse_append >reverse_single
>reverse_reverse >associative_append >associative_append
>associative_append %
]
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 Pre_exec_action ≝ λt1.
+ ∃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) ∧ bit_or_null c1 = true ∧
+ |curconfig| = |newconfig| ∧
+ 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).
+
+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〉]) →
*)
-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 #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
+(*lemma GRealize_to_Realize :
+ ∀sig,M,R.GRealize sig M (λx.True) R → Realize sig M R.
+#sig #M #R #HR #t @HR //
+qed.*)
+
+lemma sem_exec_action : GRealize ? exec_action Pre_exec_action R_exec_action.
+@(sem_seq_app_guarded … (Realize_to_GRealize … sem_init_copy) ???)
+[3:@(sem_seq_app_guarded … sem_copy)
+ [3:@(sem_seq_app_guarded … (Realize_to_GRealize … (sem_move_r STape)))
+ [3:@(Realize_to_GRealize … (sem_move_tape …))
+ |@(λt.True)
+ |4://
+ |5:@sub_reflexive]
+ |@(λt.True)
+ |4://
+ |5:@sub_reflexive]
+|4:#t1 #t2
+ * #n * #curconfig * #ls * #rs * #c0 * #c1 * #s0 * #s1 * #table1 * #newconfig
+ * #mv * #table2 * * * * * * *
+ #Htable #Hcurconfig1 #Hcurconfig2 #Hnewconfig #Hc1 #Hlen #Htape #Ht1
+ whd in ⊢ (%→?); >Ht1 #HR
+ lapply (HR (〈c0,false〉::curconfig) table1 s0 ls s1
+ (newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs) ???? (refl ??))
+ [(*Hcurconfig2*) @daemon
+ |(*by Htable *) @daemon
+ |(*Hcurconfig2*) @daemon
+ |(*Hcurconfig1*) @daemon ]
+ -HR #Ht2 whd
+ %{(〈grid,false〉::ls)} %{s0} %{s1} %{c0} %{c1} %{(〈mv,false〉::table2@〈grid,false〉::rs)}
+ %{newconfig} %{(〈comma,false〉::reverse ? table1)} %{curconfig} >Ht2
+ % [ % [ % [ % [ % [ % [ % [ %
+ [ %
+ |(*by Htabel*) @daemon ]
+ |(*by Htable*) @daemon ]
+ |// ]
+ |>Hlen % ]
+ |@Hcurconfig2 ]
+ |@Hnewconfig ]
+ |cases Htape * * // ]
+ | // ]
+ |1,2:skip]
+#ta #outtape #HR #n #curconfig #ls #rs #c0 #c1 #s0 #s1 #table1 #newconfig #mv
+#table2 #Htable #Hcurconfig1 #Hcurconfig2 #Hnewconfig #Hbitnullc1 #Hlen #Htape
+#Hta #ta' #Hta'
+cases HR -HR #tb * whd in ⊢ (%→?); #Htb
+lapply (Htb (〈c0,false〉::curconfig) table1 s0 ls s1
+ (newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs)
+ ???? Hta) -Htb
[ (*Hcurconfig2*) @daemon
| (*Htable*) @daemon
-| (*bit_or_null c0 = true *) @daemon
-| (*Hcurconfig1*) @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:|*:(* bit_or_null c0,c1; |curconfig| = |newconfig|*) @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〉::〈c1,false〉::reverse ? newconfig@〈s1,false〉::〈comma,false〉::reverse ? table1)
- 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 %
- | %
- | % ]
+| (* by Htape bit_or_null c0 = true, moreover Hcurconfig2 *) @daemon
+| (*Hcurconfig1*) @daemon ]
+#Htb * #tc * whd in ⊢ (%→?); #Htc
+lapply (Htc (〈grid,false〉::ls) s0 s1 c0 c1 (〈mv,false〉::table2@〈grid,false〉::rs) newconfig (〈comma,false〉::reverse ? table1) curconfig Htb ????????) -Htc
+[9:|*:(* bit_or_null c0,c1; |curconfig| = |newconfig|*) @daemon ]
+#Htc * #td * whd in ⊢ (%→?); * #_ #Htd lapply (Htd … Htc) -Htd whd in ⊢(???(??%%%)→?);#Htd
+whd in ⊢ (%→?); #Houtc whd in Htd:(???%); whd in Htd:(???(??%%%));
+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 ????????)
+[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:>Htd @(eq_f3 … (midtape ?))
+ [ @eq_f @eq_f >associative_append >associative_append %
| %
- || >reverse_cons >reverse_cons >reverse_append >reverse_reverse
- >reverse_cons >reverse_cons >reverse_reverse
- >associative_append >associative_append >associative_append
- >associative_append >associative_append
- @Htable
- | (* well formedness of table *) @daemon
- | (* Hnewconfig *) @daemon
- | (* bit_or_null mv = true (well formedness of table) *) @daemon
- | -Houtc * #ls1 * #rs1 * #newc * #Hnewtapelegal * #Houtc *
- [ *
- [ * #Hmv #Htapemove
- @(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' >Htapemove
- (* mv = bit false -→ c1 = bit ? *)
- cut (∃c1'.c1 = bit c1') [ @daemon ] * #c1' #Hc1
- >Hc1 >tape_move_left_eq >(legal_tape_left … Htape)
- >(legal_tape_right … Htape) %
- ]
- | //
- ]
- | * #Hmv #Htapemove
- @(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' >Htapemove
- cut (∃c1'.c1 = bit c1') [ @daemon ] * #c1' #Hc1
- >Hc1 >tape_move_right_eq >(legal_tape_left … Htape)
- >(legal_tape_right … Htape) %
- ]
- | //
+ | % ]
+| %
+|| >reverse_cons >reverse_cons >reverse_append >reverse_reverse
+ >reverse_cons >reverse_cons >reverse_reverse
+ >associative_append >associative_append >associative_append
+ >associative_append >associative_append
+ @Htable
+| (* well formedness of table *) @daemon
+| (* Hnewconfig *) @daemon
+| (* bit_or_null mv = true (well formedness of table) *) @daemon
+| -Houtc * #ls1 * #rs1 * #newc * #Hnewtapelegal * #Houtc *
+ [ *
+ [ * #Hmv #Htapemove
+ @(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 >Hta' >Htapemove
+ (* mv = bit false -→ c1 = bit ? *)
+ cut (∃c1'.c1 = bit c1') [ @daemon ] * #c1' #Hc1
+ >Hc1 >tape_move_left_eq >(legal_tape_left … Htape)
+ >(legal_tape_right … Htape) %
]
+ | //
]
- | * * * #Hmv #Hlseq #Hrseq #Hnewc
+ | * #Hmv #Htapemove
@(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? newc) %
[ %
[ >Houtc -Houtc >reverse_append
>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 >Hta' >Htapemove
+ cut (∃c1'.c1 = bit c1') [ @daemon ] * #c1' #Hc1
+ >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 >Hta' 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) %
+ ]
+ ]
+ | //
+ ]
]
]
qed.
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,table,s0,s1,c0,c1,ls,rs,curconfig,newconfig,mv.
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.
+(*axiom sem_match_tuple : Realize ? match_tuple R_match_tuple.*)
+definition us_acc : states ? uni_step ≝ (inr … (inl … (inr … start_nop))).
+
+definition Pre_uni_step ≝ λt1.
+ ∃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〉::table@〈grid,false〉::rs).
+
lemma sem_uni_step :
- accRealize ? uni_step (inr … (inl … (inr … start_nop)))
+ accGRealize ? uni_step us_acc Pre_uni_step
R_uni_step_true R_uni_step_false.
+@daemon (* this no longer works: TODO *) (*
+@(acc_sem_if_app_guarded STape … (sem_test_char ? (λc:STape.\fst c == bit false))
+ ? (test_char_inv …) (sem_nop …) …)
+[| @(sem_seq_app_guarded … (Realize_to_GRealize … sem_init_match) ???)
+ [ 5: @sub_reflexive
+ | 3: @(sem_seq_app_guarded … sem_match_tuple
+ (Realize_to_GRealize … (sem_if ????????? (sem_test_char … (λc.¬is_marked FSUnialpha c))
+ (sem_seq … sem_exec_action (sem_move_r …))
+ (sem_nop …))))
+ [@(λx.True)
+ |//
+ |@sub_reflexive]
+ ||| #t1 #t2 * #n * #table * #s0 * #s1 * #c0 * #c1 * #ls * #rs * #curconfig
+ * #newconfig * #mv * * * *
+ #Hlen1 #Htable #Hmatch #Hlegal #Ht1
+ whd in ⊢ (%→?);
+ cut (∃tup,table0.table = tup@table0 ∧ tuple_TM (S n) tup)
+ [@daemon]
+ * #tup * #table0 * #Htableeq * #qin * #cin * #qout * #cout * #mv0
+ * * * * * * * * * *
+ #Hqinnomarks #_ #Hqinbits #_ #_ #_ #_ #_ #Hqinlen #_ #Htupeq
+ cut (∃d,qin0.qin = 〈d,false〉::qin0)
+ [ lapply Hqinlen lapply Hqinnomarks -Hqinlen -Hqinnomarks cases qin
+ [ #_ normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
+ | * #d #bd #qin0 #Hqinnomarks #_ %{d} %{qin0}
+ >(?:bd=false) [%]
+ @(Hqinnomarks 〈d,bd〉) @memb_hd ] ]
+ * #d * #qin0 #Hqineq
+ #Ht2
+ lapply (Ht2 (〈grid,false〉::ls) (curconfig@[〈c0,false〉])
+ (qin0@〈cin,false〉::〈comma,false〉::qout@〈cout,false〉::〈comma,false〉::〈mv0,false〉::table0@〈grid,false〉::rs) s0 d ???)
+ [ >Ht1 @eq_f >associative_append @eq_f @eq_f @eq_f
+ >Htableeq >Htupeq >associative_append whd in ⊢ (??%?);
+ @eq_f >Hqineq >associative_append @eq_f whd in ⊢ (??%?);
+ @eq_f whd in ⊢ (??%?); @eq_f
+ >associative_append %
+ | @daemon
+ | @daemon
+ ]
+ #Ht2 % [| % [| % [| % [ @Ht2 ]
+ %2
+ (* ls0 = ls
+ c = s0
+ l1 = curconfig@[〈c0,false〉]
+ l2 = [〈bar,false〉]
+ c10 = d
+ l3 = qin0@[〈cin,false〉]
+ l4 = qout@〈cout,false〉::〈comma,false〉::〈mv0,false〉::table0
+ rs00 = rs
+ n0 = S n ?
+ *)
+ %{ls} %{s0} %{(curconfig@[〈c0,false〉])}
+ %{([〈bar,false〉])} %{d} %{(qin0@[〈cin,false〉])}
+ %{(qout@〈cout,false〉::〈comma,false〉::〈mv0,false〉::table0)}
+ %{rs} %{n} @daemon (* TODO *)
+ ]
+ ]
+ ]
+ ]
+ | #intape #outtape
+ #ta whd in ⊢ (%→?); #Hta #HR
+ #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; * * * #c #bc *
+ whd in ⊢ (??%?→?); #HSome destruct (HSome) #Hc #Hta % [@(\P Hc)]
+ cases HR -HR
+ #tb * whd in ⊢ (%→?); #Htb
+ lapply (Htb (〈grid,false〉::ls) (curconfig@[〈c0,false〉]) (table@〈grid,false〉::rs) c t0 ???)
+ [ >Hta >associative_append %
+ | @daemon
+ | @daemon
+ | -Hta -Htb #Htb *
+ #tc * whd in ⊢ (%→?); #Htc cases (Htc … Htable … Htb) -Htb -Htc
+ [| * #Hcurrent #Hfalse @False_ind
+ (* absurd by Hmatch *) @daemon
+ | >(\P Hc) %
+ | (* Htable (con lemma) *) @daemon
+ | (* 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 * * #c2 * whd in ⊢ (??%?→?); #Hc2 destruct (Hc2)
+ #_ #Htd
+ cut (newc = 〈s1,false〉::newconfig@[〈c1,false〉]) [@daemon] #Hnewc
+ >Hnewc cut (mv1 = 〈mv,false〉)
+ [@daemon] #Hmv1
+ * #te * whd in ⊢ (%→?); #Hte
+ cut (td = midtape STape (〈c0,false〉::reverse STape curconfig@〈c,false〉::〈grid,false〉::ls)
+ 〈grid,false〉
+ ((table1@〈bar,false〉::〈c,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 >Hmv1 >Hnewc @eq_f @eq_f @eq_f @eq_f @eq_f @eq_f
+ whd in ⊢ (??%?); >associative_append %
+ ]
+ ]
+ -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 in Htableeq; >Hnewc
+ >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 ((〈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
+ >Hmv1 %
+ | @Hliftte
+ ]
+ | //
+ ]
+ ]
+ ]
+ | * #td * whd in ⊢ (%→%→?); >Htc * #Htd
+ lapply (Htd ? (refl ??)) normalize in ⊢ (%→?);
+ #Hfalse destruct (Hfalse)
+ ]
+ ]
+| #t1 #t2 #t3 whd in ⊢ (%→%→?); #Ht1 #Ht2
+ #b #Hb >Hb in Ht1; * #Hc #Ht1 lapply (Hc ? (refl ??)) -Hc #Hb' %
+ // cases b in Hb'; normalize #H1 //
+]
+*)
+qed.
+
+(*
@(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_move (sem_move_r …))
+ (sem_seq … sem_exec_action (sem_move_r …))
(sem_nop …))))
(sem_nop …)
…)
[ #intape #outtape
#ta whd in ⊢ (%→?); #Hta #HR
#n #fulltable #s0 #s1 #c0 #c1 #ls #rs #curconfig #newconfig #mv
- #Htable_len cut (∃t0,table. fulltable =〈t0,false〉::table) [(* 0 < |table| *) @daemon]
+ #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 % //
+ >Hintape in Hta; * * * #c #bc *
+ whd in ⊢ (??%?→?); #HSome destruct (HSome) #Hc #Hta % [@(\P Hc)]
cases HR -HR
#tb * whd in ⊢ (%→?); #Htb
- lapply (Htb (〈grid,false〉::ls) (curconfig@[〈c0,false〉]) (table@〈grid,false〉::rs) s0 t0 ???)
+ lapply (Htb (〈grid,false〉::ls) (curconfig@[〈c0,false〉]) (table@〈grid,false〉::rs) c t0 ???)
[ >Hta >associative_append %
| @daemon
| @daemon
#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
+ | >(\P Hc) %
+ | (* Htable (con lemma) *) @daemon
+ | (* Hmatch *) @daemon
| (* Htable *) @daemon
| (* Htable, Hmatch → |config| = n
necessaria modifica in R_match_tuple, le dimensioni non corrispondono
- *) @daemon ]
+ *) @daemon
+ ]
* #table1 * #newc * #mv1 * #table2 * #Htableeq #Htc *
- [ * #td * whd in ⊢ (%→?); >Htc -Htc #Htd
- cases (Htd ? (refl ??)) #_ -Htd
+ [ * #td * whd in ⊢ (%→?); >Htc -Htc * * #c2 * whd in ⊢ (??%?→?); #Hc2 destruct (Hc2)
+ #_ #Htd
cut (newc = 〈s1,false〉::newconfig@[〈c1,false〉]) [@daemon] #Hnewc
- >Hnewc #Htd
- cut (∃mv2. mv1 = [〈mv2,false〉])
- [@daemon] * #mv2 #Hmv1
+ >Hnewc cut (mv1 = 〈mv,false〉)
+ [@daemon] #Hmv1
* #te * whd in ⊢ (%→?); #Hte
- cut (td = midtape STape (〈c0,false〉::reverse STape curconfig@〈s0,false〉::〈grid,false〉::ls)
+ cut (td = midtape STape (〈c0,false〉::reverse STape curconfig@〈c,false〉::〈grid,false〉::ls)
〈grid,false〉
- ((table1@〈s0,false〉::curconfig@[〈c0,false〉])@〈comma,true〉::〈s1,false〉::
- newconfig@〈c1,false〉::〈comma,false〉::〈mv2,false〉::table2@〈grid,false〉::rs))
+ ((table1@〈bar,false〉::〈c,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 >Hmv1 >Hnewc @eq_f @eq_f @eq_f @eq_f @eq_f @eq_f
+ whd in ⊢ (??%?); >associative_append %
]
]
-Htd #Htd lapply (Hte … (S n) … Htd … Ht1') -Htd -Hte
>associative_append >associative_append
#Htableeq <Htableeq // ]
* #ls1 * #rs1 * #c2 * * #Hte #Hliftte #Hlegalte
- whd in ⊢ (%→?); #Houttape lapply (Houttape … Hte) -Houttape #Houttape
+ 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 ⊢ (???%);
+ 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
>Hmv1 %
- | >(?: mv = mv2) [| (*Hmatch, Htableeq*) @daemon ]
- @Hliftte
+ | @Hliftte
]
| //
]
]
]
- | * #td * whd in ⊢ (%→%→?); >Htc #Htd
- cases (Htd ? (refl ??)) normalize in ⊢ (%→?);
+ | * #td * whd in ⊢ (%→%→?); >Htc * #Htd
+ lapply (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 //
+ #b #Hb >Hb in Ht1; * #Hc #Ht1 lapply (Hc ? (refl ??)) -Hc #Hb' %
+ // cases b in Hb'; normalize #H1 //
]
qed.
-
+*)
\ No newline at end of file
projects / helm.git / blobdiff