[ None ⇒ 〈null,false〉
| Some c0 ⇒ c0 ].
-definition mk_tuple ≝ λc,newc,mv.
- c @ 〈comma,false〉:: newc @ 〈comma,false〉 :: [〈mv,false〉].
-
-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).
definition move_of_unialpha ≝
λc.match c with
definition R_move_tape_l_abstract ≝ λt1,t2.
∀rs,n,table,curc,curconfig,ls.
- bit_or_null curc = true → only_bits_or_nulls curconfig → table_TM n (reverse ? table) →
+ is_bit curc = true → only_bits_or_nulls curconfig → table_TM n (reverse ? table) →
t1 = midtape STape (table@〈grid,false〉::〈curc,false〉::curconfig@〈grid,false〉::ls)
〈grid,false〉 rs →
- no_nulls ls → no_marks ls →
legal_tape ls 〈curc,false〉 rs →
∀t1'.t1' = lift_tape ls 〈curc,false〉 rs →
∃ls1,rs1,newc.
∀t1,t2.R_move_tape_l t1 t2 → R_move_tape_l_abstract t1 t2.
#t1 #t2 whd in ⊢(%→?); #Hconcrete
#rs #n #table #curc #curconfig #ls #Hcurc #Hcurconfig #Htable #Ht1
-#Hlsnonulls #Hlsnomarks #Htape #t1' #Ht1'
+* * * #Hnomarks #Hbits #Hcurc #Hlegal #t1' #Ht1'
cases (Hconcrete … Htable Ht1) //
[ * #Hls #Ht2
@(ex_intro ?? [])
[ %
[ >Ht2 %
| >Hls % ]
- | % % % ]
-| * * #l0 #bl0 * #ls0 * #Hls
- cut (bl0 = false) [@(Hlsnomarks 〈l0,bl0〉) >Hls @memb_hd]
+ | % [ % [ %
+ [ #x #Hx cases (orb_true_l … Hx) #Hx'
+ [ >(\P Hx') %
+ | @Hnomarks >Hls @Hx' ]
+ | #x #Hx cases (orb_true_l … Hx) #Hx'
+ [ >(\P Hx') //
+ | @Hbits >Hls @Hx' ]]
+ | % ]
+ | % %2 % ]
+ ]
+| * * #l0 #bl0 * #ls0 * #Hls
+ cut (bl0 = false)
+ [ @(Hnomarks 〈l0,bl0〉) @memb_cons @memb_append_l1 >Hls @memb_hd]
#Hbl0 >Hbl0 in Hls; #Hls #Ht2
- @(ex_intro ?? ls0)
- @(ex_intro ?? (〈curc,false〉::rs))
+ @(ex_intro ?? ls0) @(ex_intro ?? (〈curc,false〉::rs))
@(ex_intro ?? l0) %
[ %
[ >Ht2 %
| >Hls >lift_tape_not_null
[ %
- | @(Hlsnonulls 〈l0,false〉) >Hls @memb_hd ] ]
- | @legal_tape_conditions % % % #Hl0 >Hl0 in Hls; #Hls
- lapply (Hlsnonulls 〈null,false〉 ?)
- [ >Hls @memb_hd | normalize #H destruct (H) ]
- ]
-qed.
-
+ | @bit_not_null @(Hbits 〈l0,false〉) >Hls @memb_append_l1 @memb_hd ] ]
+ | % [ % [ %
+ [ #x #Hx cases (orb_true_l … Hx) #Hx'
+ [ >(\P Hx') %
+ | cases (memb_append … Hx') #Hx'' @Hnomarks
+ [ >Hls @memb_cons @memb_cons @(memb_append_l1 … Hx'')
+ | cases (orb_true_l … Hx'') #Hx'''
+ [ >(\P Hx''') @memb_hd
+ | @memb_cons @(memb_append_l2 … Hx''')]
+ ]
+ ]
+ | whd in ⊢ (?%); #x #Hx cases (memb_append … Hx) #Hx'
+ [ @Hbits >Hls @memb_cons @(memb_append_l1 … Hx')
+ | cases (orb_true_l … Hx') #Hx''
+ [ >(\P Hx'') //
+ | @Hbits @(memb_append_l2 … Hx'')
+ ]]]
+ | whd in ⊢ (??%?); >(Hbits 〈l0,false〉) //
+ @memb_append_l1 >Hls @memb_hd ]
+ | % % % #Hl0 lapply (Hbits 〈l0,false〉?)
+ [ @memb_append_l1 >Hls @memb_hd
+ | >Hl0 normalize #Hfalse destruct (Hfalse)
+ ] ] ] ]
+qed.
+
lemma Realize_to_Realize :
∀alpha,M,R1,R2.(∀t1,t2.R1 t1 t2 → R2 t1 t2) → Realize alpha M R1 → Realize alpha M R2.
#alpha #M #R1 #R2 #Himpl #HR1 #intape
tc_true) tc_true.
definition R_move_tape ≝ λt1,t2.
- ∀rs,n,table1,c,table2,curc,curconfig,ls.
- bit_or_null curc = true → bit_or_null c = true → only_bits_or_nulls curconfig →
- table_TM n (reverse ? table1@〈c,false〉::table2) →
+ ∀rs,n,table1,mv,table2,curc,curconfig,ls.
+ bit_or_null mv = true → only_bits_or_nulls curconfig →
+ (is_bit mv = true → is_bit curc = true) →
+ table_TM n (reverse ? table1@〈mv,false〉::table2) →
t1 = midtape STape (table1@〈grid,false〉::〈curc,false〉::curconfig@〈grid,false〉::ls)
- 〈c,false〉 (table2@〈grid,false〉::rs) →
- no_nulls ls → no_nulls rs → no_marks ls → no_marks rs →
+ 〈mv,false〉 (table2@〈grid,false〉::rs) →
legal_tape ls 〈curc,false〉 rs →
∀t1'.t1' = lift_tape ls 〈curc,false〉 rs →
∃ls1,rs1,newc.
legal_tape ls1 〈newc,false〉 rs1 ∧
(t2 = midtape STape ls1 〈grid,false〉 (reverse ? curconfig@〈newc,false〉::
- 〈grid,false〉::reverse ? table1@〈c,false〉::table2@〈grid,false〉::rs1) ∧
- ((c = bit false ∧ lift_tape ls1 〈newc,false〉 rs1 = tape_move_left STape ls 〈curc,false〉 rs) ∨
- (c = bit true ∧ lift_tape ls1 〈newc,false〉 rs1 = tape_move_right STape ls 〈curc,false〉 rs) ∨
- (c = null ∧ ls1 = ls ∧ rs1 = rs ∧ curc = newc))).
+ 〈grid,false〉::reverse ? table1@〈mv,false〉::table2@〈grid,false〉::rs1) ∧
+ ((mv = bit false ∧ lift_tape ls1 〈newc,false〉 rs1 = tape_move_left STape ls 〈curc,false〉 rs) ∨
+ (mv = bit true ∧ lift_tape ls1 〈newc,false〉 rs1 = tape_move_right STape ls 〈curc,false〉 rs) ∨
+ (mv = null ∧ ls1 = ls ∧ rs1 = rs ∧ curc = newc))).
lemma sem_move_tape : Realize ? move_tape R_move_tape.
#intape
(sem_seq … (sem_move_l …) (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))))) intape)
#k * #outc * #Hloop #HR
@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
-#rs #n #table1 #c #table2 #curc #curconfig #ls
-#Hcurc #Hc #Hcurconfig #Htable #Hintape #Hls #Hrs #Hls1 #Hrs1 #Htape #t1' #Ht1'
+#rs #n #table1 #mv #table2 #curc #curconfig #ls
+#Hmv #Hcurconfig #Hmvcurc #Htable #Hintape #Htape #t1' #Ht1'
generalize in match HR; -HR *
-[ * #ta * whd in ⊢ (%→?); #Hta cases (Hta 〈c,false〉 ?)
+[ * #ta * whd in ⊢ (%→?); #Hta cases (Hta 〈mv,false〉 ?)
[| >Hintape % ] -Hta #Hceq #Hta lapply (\P Hceq) -Hceq #Hceq destruct (Hta Hceq)
* #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hintape) -Htb -Hintape
[ * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
* #_ #Htb lapply (Htb … (refl ??) (refl ??) ?)
[ @daemon ] -Htb >append_cons <associative_append #Htb
whd in ⊢ (%→?); #Houtc lapply (Houtc … Htb … Ht1') //
- [ >reverse_append >reverse_append >reverse_reverse @Htable |]
+ [ >reverse_append >reverse_append >reverse_reverse @Htable
+ | /2/
+ ||]
-Houtc -Htb * #ls1 * #rs1 * #newc * * #Houtc #Hnewtape #Hnewtapelegal
@(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? newc) %
[ //
>associative_append >associative_append %
| % % % // ]
]
-| * #ta * whd in ⊢ (%→?); #Hta cases (Hta 〈c,false〉 ?)
- [| >Hintape % ] -Hta #Hcneq cut (c ≠ bit false)
+| * #ta * whd in ⊢ (%→?); #Hta cases (Hta 〈mv,false〉 ?)
+ [| >Hintape % ] -Hta #Hcneq cut (mv ≠ bit false)
[ lapply (\Pf Hcneq) @not_to_not #Heq >Heq % ] -Hcneq #Hcneq #Hta destruct (Hta)
*
- [ * #tb * whd in ⊢ (%→?);#Htb cases (Htb 〈c,false〉 ?)
+ [ * #tb * whd in ⊢ (%→?);#Htb cases (Htb 〈mv,false〉 ?)
[| >Hintape % ] -Htb #Hceq #Htb lapply (\P Hceq) -Hceq #Hceq destruct (Htb Hceq)
* #tc * whd in ⊢ (%→?); #Htc cases (Htc … Hintape) -Htc -Hintape
[ * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
* #_ #Htc lapply (Htc … (refl ??) (refl ??) ?)
[ @daemon ] -Htc >append_cons <associative_append #Htc
whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc … Ht1') //
- [ >reverse_append >reverse_append >reverse_reverse @Htable |]
+ [ >reverse_append >reverse_append >reverse_reverse @Htable
+ | /2/ |]
-Houtc -Htc * #ls1 * #rs1 * #newc * * #Houtc #Hnewtape #Hnewtapelegal
@(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? newc) %
[ //
>associative_append >associative_append %
| % %2 % // ]
]
- | * #tb * whd in ⊢ (%→?); #Htb cases (Htb 〈c,false〉 ?)
- [| >Hintape % ] -Htb #Hcneq' cut (c ≠ bit true)
+ | * #tb * whd in ⊢ (%→?); #Htb cases (Htb 〈mv,false〉 ?)
+ [| >Hintape % ] -Htb #Hcneq' cut (mv ≠ bit true)
[ lapply (\Pf Hcneq') @not_to_not #Heq >Heq % ] -Hcneq' #Hcneq' #Htb destruct (Htb)
* #tc * whd in ⊢ (%→?); #Htc cases (Htc … Hintape)
- [ * >(bit_or_null_not_grid … Hc) #Hfalse destruct (Hfalse) ] -Htc
+ [ * >(bit_or_null_not_grid … Hmv) #Hfalse destruct (Hfalse) ] -Htc
* #_ #Htc lapply (Htc … (refl ??) (refl ??) ?) [@daemon] -Htc #Htc
* #td * whd in ⊢ (%→?); #Htd lapply (Htd … Htc) -Htd -Htc
whd in ⊢ (???%→?); #Htd whd in ⊢ (%→?); #Houtc lapply (Houtc … Htd) -Houtc *
- [ * >(bit_or_null_not_grid … Hcurc) #Hfalse destruct (Hfalse) ]
+ [ * cases Htape * * #_ #_ #Hcurc #_
+ >(bit_or_null_not_grid … Hcurc) #Hfalse destruct (Hfalse) ]
* #_ #Houtc lapply (Houtc … (refl ??) (refl ??) ?) [@daemon] -Houtc #Houtc
@(ex_intro ?? ls) @(ex_intro ?? rs) @(ex_intro ?? curc) %
[ //
[ @Houtc
| %2 % // % // % //
generalize in match Hcneq; generalize in match Hcneq';
- cases c in Hc; normalize //
+ cases mv in Hmv; normalize //
[ * #_ normalize [ #Hfalse @False_ind cases Hfalse /2/ | #_ #Hfalse @False_ind cases Hfalse /2/ ]
|*: #Hfalse destruct (Hfalse) ]
]
projects / helm.git / blobdiff