[ >Hrs3 in Heq1; @daemon ] #Hl4
@(ex_intro … rs32) @(ex_intro … rs3') % [@Hl4]
>Htapeout @eq_f2
- [@daemon
+ [(* by Hoption, H2 *) @daemon
|(*>Hrs3 *)>append_cons
> (?:l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs
= (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@[〈bar,false〉])@〈d2,true〉::rs3'@〈grid,false〉::rs)
* normalize [ #b ] #H // destruct (H)
qed.
-definition R_match_tuple ≝ λt1,t2.
- ∀ls,c,l1,c1,l2,rs,n.
- is_bit c = true → only_bits l1 → is_bit c1 = true → n = |l1| →
- table_TM (S n) (〈c1,false〉::l2) →
- t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉
- (l1@〈grid,false〉::〈c1,true〉::l2@〈grid,false〉::rs) →
- (* facciamo match *)
- (∃l3,newc,mv,l4.
- 〈c1,false〉::l2 = l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv@l4 ∧
- t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
- (l3@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv@l4@〈grid,false〉::rs))
- ∨
- (* non facciamo match su nessuna tupla;
- non specifichiamo condizioni sul nastro di output, perché
- non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
- (current ? t2 = Some ? 〈grid,true〉 ∧
- ∀l3,newc,mv,l4.
- 〈c1,false〉::l2 ≠ l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv@l4).
-
(* possible variante ?
definition weakR_match_tuple ≝ λt1,t2.
(∀ls,cur,rs,b. t1 = midtape STape ls 〈grid,b〉 rs → t2 = t1) ∧
〈c1,false〉::l3 ≠ l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)).
*)
-definition weakR_match_tuple ≝ λt1,t2.
+definition R_match_tuple0 ≝ λt1,t2.
∀ls,cur,rs.
t1 = midtape STape ls cur rs →
(is_grid (\fst cur) = true → t2 = t1) ∧
axiom table_bit_after_bar :
∀n,l1,c,l2.table_TM n (l1@〈bar,false〉::〈c,false〉::l2) → is_bit c = true.
-lemma wsem_match_tuple : WRealize ? match_tuple weakR_match_tuple.
+lemma wsem_match_tuple : WRealize ? match_tuple R_match_tuple0.
#intape #k #outc #Hloop
lapply (sem_while … sem_match_tuple_step intape k outc Hloop) [%] -Hloop
* #ta * #Hstar @(star_ind_l ??????? Hstar)
>Hcur in Hgrid; #Hgrid >(is_grid_true … Hgrid) normalize in ⊢ (%→?);
#Hc destruct (Hc)
]
-| #tb #tc #td whd in ⊢ (%→?); #Htc
+| (* in the interesting case, we execute a true iteration, then we restart the
+ while cycle, finally we end with a false iteration *)
+ #tb #tc #td whd in ⊢ (%→?); #Htc
#Hstar1 #IH whd in ⊢ (%→?); #Hright lapply (IH Hright) -IH whd in ⊢ (%→?); #IH
#ls #cur #rs #Htb %
- [ #Hcur cases (Htc … Htb) * #Hfalse @False_ind @Hfalse @(is_grid_true … Hcur)
- |#c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs #Hc #Hc1 #Hl1bitnull #Hl1marks
+ [ (* cur can't be true because we assume at least one iteration *)
+ #Hcur cases (Htc … Htb) * #Hfalse @False_ind @Hfalse @(is_grid_true … Hcur)
+ | (* current and a tuple are marked *)
+ #c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs #Hc #Hc1 #Hl1bitnull #Hl1marks
#Hl1len #Htable cases (Htc … Htb) -Htc -Htb * #_ #Htc
+ (* expose the marked tuple in table *)
cut (∃la,lb,mv,lc.l3 = la@〈comma,false〉::lb@〈comma,false〉::mv::lc ∧
S n = |la| ∧ only_bits_or_nulls la)
[@daemon] * #la * #lb * #mv * #lc * * #Hl3 #Hlalen #Hlabitnull
|4: whd in ⊢ (??%?); >Hc1 %
|3: whd in ⊢ (??%?); >Hc %
|-Htc *
- [ * #Heq #Htc % %{[]} %{lb} %{mv} %{lc} destruct (Heq) %
+ [ (* case 1: match successful *)
+ * #Heq #Htc % %{[]} %{lb} %{mv} %{lc} destruct (Heq) %
[%
| cases (IH … Htc) -IH #Houtc #_ >(Houtc (refl ??))
>Htc @eq_f normalize >associative_append normalize
>associative_append normalize %
]
- | * #Hdiff * #c2 * #l5 * #l6 * #Heqlblc #Htc
+ | (* case 2: tuples don't match, we still have other tuples to try *)
+ * #Hdiff * #c2 * #l5 * #l6 * #Heqlblc #Htc
cases (IH ??? … Htc) -IH #_ #IH
+ (* by induction hypothesis *)
lapply (IH ? l1 c2 (l2@〈bar,false〉::〈c1,false〉::la@〈comma,false〉::l5) l6 ? rs0 n (refl ??) (refl ??) ???????)
[ generalize in match Htable;
>associative_append normalize
>associative_append normalize
>associative_append %
|-IH *
- [ * #l7 * #newc * #mv0 * #l8 * #Hl7l8 #Houtc %
+ [ (* the while finally matches a tuple *)
+ * #l7 * #newc * #mv0 * #l8 * #Hl7l8 #Houtc %
>Heqlblc @(ex_intro ?? (〈bar,false〉::〈c1,false〉::la@〈comma,false〉::l5@l7))
%{newc} %{mv0} %{l8} %
[ normalize >Hl7l8 >associative_append normalize
>associative_append normalize >associative_append
normalize >associative_append %
]
- | * #Houtc #Hdiff %2 %
+ | (* the while fails finding a tuple: there are no matches in the whole table *)
+ * #Houtc #Hdiff1 %2 %
[ @Houtc
| #l50 #newc #mv0 #l51 >Heqlblc
@daemon
]
]
]
- | * * #Hdiff #Hnobars generalize in match (refl ? tc);
+ | (* match failed and there is no next tuple: the next while cycle will just exit *)
+ * * #Hdiff #Hnobars generalize in match (refl ? tc);
cases tc in ⊢ (???% → %);
[ #_ normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse)
|2,3: #x #xs #_ normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ]
#ls1 #cur1 #rs1 #Htc normalize in ⊢ (??%?→?); #Hcur1
cases (IH … Htc) -IH #IH #_ %2 %
[ destruct (Hcur1) >IH [ >Htc % | % ]
- | #l4 #newc #mv0 #l5 (* difficile (sempre che sia dimostrabile)
- dobbiamo veramente considerare di fare la table in modo più
- strutturato
- *)
+ | #l4 #newc #mv0 #l5
+ (* no_bars except the first one, where the tuple does not match ⇒
+ no match *)
@daemon
]
]
]
qed.
+definition R_match_tuple ≝ λt1,t2.
+ ∀ls,c,l1,c1,l2,rs,n.
+ is_bit c = true → is_bit c1 = true →
+ only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
+ table_TM (S n) (〈bar,false〉::〈c1,false〉::l2) →
+ t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉
+ (l1@〈grid,false〉::〈bar,false〉::〈c1,true〉::l2@〈grid,false〉::rs) →
+ (* facciamo match *)
+ (∃l3,newc,mv,l4.
+ 〈bar,false〉::〈c1,false〉::l2 = l3@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4 ∧
+ t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
+ (l3@〈bar,false〉::〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l4@〈grid,false〉::rs))
+ ∨
+ (* non facciamo match su nessuna tupla;
+ non specifichiamo condizioni sul nastro di output, perché
+ non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
+ (current ? t2 = Some ? 〈grid,true〉 ∧
+ ∀l3,newc,mv,l4.
+ 〈bar,false〉::〈c1,false〉::l2 ≠ l3@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4).
+
+(* we still haven't proved termination *)
+axiom sem_match_tuple0 : Realize ? match_tuple R_match_tuple0.
+
+axiom Realize_to_Realize :
+ ∀alpha,M,R1,R2.(∀t1,t2.R1 t1 t2 → R2 t1 t2) → Realize alpha M R1 → Realize alpha M R2.
+
+lemma sem_match_tuple : Realize ? match_tuple R_match_tuple.
+generalize in match sem_match_tuple0; @Realize_to_Realize
+#t1 #t2 #HR #ls #c #l1 #c1 #l2 #rs #n #Hc #Hc1 #Hl1bitsnulls #Hl1marks #Hl1len #Htable #Ht1
+cases (HR … Ht1) -HR #_ #HR
+@(HR ??? [] … (refl ??) (refl ??) (refl ??) Hc Hc1 Hl1bitsnulls Hl1marks
+ Hl1len Htable)
+qed.
\ No newline at end of file
*)
include "turing/universal/copy.ma".
+include "turing/universal/move_tape.ma".
(*
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 ?)))))).
-
+ (seq ? (move_r ?)
+ (seq ? (mark ?)
+ (seq ? (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.
]
qed. *)
-include "turing/universal/move_tape.ma".
-
definition exec_move ≝
seq ? init_copy
(seq ? copy
[ #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
[| * #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
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
cut (newc = 〈s1,false〉::newconfig@[〈c1,false〉]) [@daemon] #Hnewc
- >Hnewc #Htd
- cut (∃mv2. mv1 = [〈mv2,false〉])
- [@daemon] * #mv2 #Hmv1
+ >Hnewc #Htd cut (mv1 = 〈mv,false〉)
+ [@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〉::〈mv2,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
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
]
| //
]
#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 / commitdiff