*)
definition mark_next_tuple ≝
- seq ? (adv_to_mark_r ? bar_or_grid)
+ adv_to_mark_r ? bar_or_grid ·
(ifTM ? (test_char ? (λc:STape.is_bar (\fst c)))
(move_right_and_mark ?) (nop ?) tc_true).
-
+
definition R_mark_next_tuple ≝
λt1,t2.
∀ls,c,rs1,rs2.
@(ex_intro ?? j) @ex_intro [|% [@Hloop] ]
-Hloop
#ls #c #rs1 #rs2 #Hrs #Hrs1 #Hrs1' #Hc
- cases (Hleft … Hrs)
+ cases (proj2 ?? Hleft … Hrs)
[ * #Hfalse >Hfalse in Hc; #Htf destruct (Htf)
- | * #_ #Hta cases (tech_split STape (λc.is_bar (\fst c)) rs1)
- [ #H1 lapply (Hta rs1 〈grid,false〉 rs2 (refl ??) ? ?)
+ | * * #_ #Hta #_ cases (tech_split STape (λc.is_bar (\fst c)) rs1)
+ [ #H1 %2 % [@H1]
+ lapply (Hta rs1 〈grid,false〉 rs2 (refl ??) ? ?)
[ * #x #b #Hx whd in ⊢ (??%?); >(Hrs1' … Hx) >(H1 … Hx) %
| %
| -Hta #Hta cases Hright
- [ * #tb * whd in ⊢ (%→?); #Hcurrent
- @False_ind cases (Hcurrent 〈grid,false〉 ?)
- [ normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
- | >Hta % ]
- | * #tb * whd in ⊢ (%→?); #Hcurrent
- cases (Hcurrent 〈grid,false〉 ?)
- [ #_ #Htb whd in ⊢ (%→?); #Houtc
- %2 %
- [ @H1
- | >Houtc >Htb >Hta % ]
- | >Hta % ]
+ [ * #tb * whd in ⊢ (%→?); * * #c1 * >Hta
+ whd in ⊢ ((??%?)→?); #H destruct (H) whd in ⊢ ((??%?)→?); #H destruct
+ | * #tb * whd in ⊢ (%→?); * #_ #Htb >Htb >Hta
+ whd in ⊢ (%→?); #H @H
]
]
- | * #rs3 * #c0 * #rs4 * * #Hc0 #Hsplit #Hrs3
- % @(ex_intro ?? rs3) @(ex_intro ?? rs4)
+ |* #rs3 * #c0 * #rs4 * * #Hc0 #Hsplit #Hrs3
+ % @(ex_intro ?? rs3) @(ex_intro ?? rs4)
lapply (Hta rs3 c0 (rs4@〈grid,false〉::rs2) ???)
- [ #x #Hrs3' whd in ⊢ (??%?); >Hsplit in Hrs1;>Hsplit in Hrs3;
+ [#x #Hrs3' whd in ⊢ (??%?); >Hsplit in Hrs1;>Hsplit in Hrs3;
#Hrs3 #Hrs1 >(Hrs1 …) [| @memb_append_l1 @Hrs3'|]
>(Hrs3 … Hrs3') @Hrs1' >Hsplit @memb_append_l1 //
- | whd in ⊢ (??%?); >Hc0 %
- | >Hsplit >associative_append % ] -Hta #Hta
- cases Hright
- [ * #tb * whd in ⊢ (%→?); #Hta'
- whd in ⊢ (%→?); #Htb
- cases (Hta' c0 ?)
- [ #_ #Htb' >Htb' in Htb; #Htb
- generalize in match Hsplit; -Hsplit
- cases rs4 in Hta;
- [ #Hta #Hsplit >(Htb … Hta)
- >(?:c0 = 〈bar,false〉)
- [ @(ex_intro ?? grid) @(ex_intro ?? false)
- % [ % [ %
- [(* Hsplit *) @daemon |(*Hrs3*) @daemon ] | % ] | % ]
- | (* Hc0 *) @daemon ]
- | #r5 #rs5 >(eq_pair_fst_snd … r5)
- #Hta #Hsplit >(Htb … Hta)
- >(?:c0 = 〈bar,false〉)
- [ @(ex_intro ?? (\fst r5)) @(ex_intro ?? (\snd r5))
- % [ % [ % [ (* Hc0, Hsplit *) @daemon | (*Hrs3*) @daemon ] | % ]
- | % ] | (* Hc0 *) @daemon ] ] | >Hta % ]
- | * #tb * whd in ⊢ (%→?); #Hta'
- whd in ⊢ (%→?); #Htb
- cases (Hta' c0 ?)
- [ #Hfalse @False_ind >Hfalse in Hc0;
- #Hc0 destruct (Hc0)
- | >Hta % ]
-]]]]
+ |whd in ⊢ (??%?); >Hc0 %
+ |>Hsplit >associative_append %
+ ]-Hta #Hta
+ cases Hright -Hright
+ [* whd in ⊢ (%→?); #tb * * * #c1 * >Hta -Hta
+ whd in ⊢ (??%?→?); #H destruct (H) #Hc1 #Htb
+ whd in ⊢ (%→?); #Houtc
+ cut (c1=〈bar,false〉)
+ [lapply Hc1 lapply Hsplit cases c1 #c1l #c1r #Hsplit
+ cases c1l normalize
+ [#b #H destruct |2,3,5:#H destruct]
+ #_ @eq_f @(Hrs1 … 〈c1l,c1r〉) >Hsplit @memb_append_l2 @memb_hd]
+ #Hcut lapply Hsplit -Hsplit
+ cases rs4 in Htb;
+ [#Htb lapply(Houtc … Htb) -Houtc #Houtc #Hsplit
+ @(ex_intro ?? grid) @(ex_intro ?? false) %
+ [% [ % [<Hcut @Hsplit |@Hrs3 ] | % ]
+ |>Houtc >Hcut %
+ ]
+ |* #r5l #r5r #rs5 #Htb
+ lapply(Houtc … Htb) -Houtc #Houtc #Hsplit
+ @(ex_intro ?? r5l) @(ex_intro ?? r5r) %
+ [%[%[<Hcut @Hsplit| @Hrs3] | % ]
+ |>Houtc >Hcut %
+ ]
+ ]
+ |* whd in ⊢ (%→?); #tb * *
+ #H @False_ind >Hta in H; #H lapply(H c0 (refl …))
+ >Hc0 #H destruct
+ ]
+ ]
+ ]
+]
qed.
definition init_current_on_match ≝
- (seq ? (move_l ?)
- (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
- (seq ? (move_r ?) (mark ?)))).
+ move_l ? · adv_to_mark_l ? (λc:STape.is_grid (\fst c)) · move_r ? · mark ?.
definition R_init_current_on_match ≝ λt1,t2.
∀l1,l2,c,rs. no_grids l1 → is_grid c = false →
#k * #outc * #Hloop #HR
@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
#l1 #l2 #c #rs #Hl1 #Hc #Hintape
-cases HR -HR #ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hintape) -Hta -Hintape
+cases HR -HR #ta * whd in ⊢ (%→?); * #_ #Hta lapply (Hta … Hintape) -Hta -Hintape
generalize in match Hl1; cases l1
[#Hl1 whd in ⊢ ((???(??%%%))→?); #Hta
- * #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Hta
- [* >Hc #Htemp destruct (Htemp) ]
- * #_ #Htc lapply (Htc [ ] 〈grid,false〉 ? (refl ??) (refl …) Hl1)
- whd in ⊢ ((???(??%%%))→?); -Htc #Htc
- * #td * whd in ⊢ (%→?); #Htd lapply (Htd … Htc) -Htc -Htd
- whd in ⊢ ((???(??%%%))→?); #Htd
- whd in ⊢ (%→?); #Houtc lapply (Houtc … Htd) -Houtc #Houtc
+ * #tb * whd in ⊢ (%→?); * #_ #Htb cases (Htb … Hta) -Htb -Hta #_
+ (* [* >Hc #Htemp destruct (Htemp) ] *)
+ #Htb cases (Htb … Hc) -Htb #Htb #_
+ lapply (Htb [ ] 〈grid,false〉 ? (refl ??) (refl …) Hl1)
+ whd in ⊢ ((???(??%%%))→?); -Htb #Htb
+ * #tc * whd in ⊢ (%→?); * #_ #Htc lapply (Htc … Htb) -Htb -Htc
+ whd in ⊢ ((???(??%%%))→?); #Htc
+ whd in ⊢ (%→?); * #Houtc #_ lapply (Houtc … Htc) -Houtc #Houtc
>Houtc %
|#d #tl #Htl whd in ⊢ ((???(??%%%))→?); #Hta
- * #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Htb
- [* >(Htl … (memb_hd …)) #Htemp destruct (Htemp)]
- * #Hd >append_cons #Htb lapply (Htb … (refl ??) (refl …) ?)
+ * #tb * whd in ⊢ (%→?); * #_ #Htb cases (Htb … Hta) -Htb
+ #_ (* [* >(Htl … (memb_hd …)) #Htemp destruct (Htemp)] *)
+ #Htb cases (Htb ?) -Htb [2: @Htl @memb_hd]
+ >append_cons #Htb #_ lapply (Htb … (refl ??) (refl …) ?)
[#x #membx cases (memb_append … membx) -membx #membx
[@Htl @memb_cons @membx | >(memb_single … membx) @Hc]]-Htb #Htb
- * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htb -Htc
+ * #tc * whd in ⊢ (%→?); * #_ #Htc lapply (Htc … Htb) -Htb -Htc
>reverse_append >associative_append whd in ⊢ ((???(??%%%))→?); #Htc
- whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc #Houtc
+ whd in ⊢ (%→?); * #Houtc #_ lapply (Houtc … Htc) -Houtc #Houtc
>Houtc >reverse_cons >associative_append %
]
qed.
*)
definition init_current ≝
- seq ? (adv_to_mark_l ? (is_marked ?))
- (seq ? (clear_mark ?)
- (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
- (seq ? (move_r ?) (mark ?)))).
+ adv_to_mark_l ? (is_marked ?) ·clear_mark ? ·
+ adv_to_mark_l ? (λc:STape.is_grid (\fst c)) · move_r ? · mark ?.
definition R_init_current ≝ λt1,t2.
∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 → is_grid c = false →
(sem_seq ????? (sem_move_r ?) (sem_mark ?)))) intape)
#k * #outc * #Hloop #HR
@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop]
-cases HR -HR #ta * whd in ⊢ (%→?); #Hta
-* #tb * whd in ⊢ (%→?); #Htb
-* #tc * whd in ⊢ (%→?); #Htc
-* #td * whd in ⊢ (%→%→?); #Htd #Houtc
+cases HR -HR #ta * whd in ⊢ (%→?); * #_ #Hta
+* #tb * whd in ⊢ (%→?); * #_ #Htb
+* #tc * whd in ⊢ (%→?); * #_ #Htc
+* #td * whd in ⊢ (%→%→?); * #_ #Htd * #Houtc #_
#l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hc0 #Hintape
-cases (Hta … Hintape) [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
--Hta * #_ #Hta lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%]
--Hta #Hta lapply (Htb … Hta) -Htb #Htb cases (Htc … Htb) [ >Hc -Hc * #Hc destruct (Hc) ]
--Htc * #_ #Htc lapply (Htc … (refl ??) (refl ??) ?) [@Hl2]
+cases (Hta … Hintape) #_ -Hta #Hta cases (Hta (refl …)) -Hta
+#Hta #_ lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%]
+-Hta #Hta lapply (Htb … Hta) -Htb #Htb cases (Htc … Htb) #_ #Htc
+cases (Htc Hc) -Htc #Htc #_ lapply (Htc … (refl ??) (refl ??) ?) [@Hl2]
-Htc #Htc lapply (Htd … Htc) -Htd
>reverse_append >reverse_cons
>reverse_cons in Hc0; cases (reverse … l2)
definition match_tuple_step ≝
ifTM ? (test_char ? (λc:STape.¬ is_grid (\fst c)))
(single_finalTM ?
- (seq ? compare
+ (compare ·
(ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
(nop ?)
- (seq ? mark_next_tuple
+ (mark_next_tuple ·
(ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
- (mark ?) (seq ? (move_l ?) init_current) tc_true)) tc_true)))
+ (mark ?) (move_l ? · init_current) tc_true)) tc_true)))
(nop ?) tc_true.
+
+definition R_match_tuple_step_true_new ≝ λt1,t2.
+ ∃ls,cur,rs.t1 = midtape STape ls cur rs \wedge
+ \fst cur ≠ grid ∧
+ (∀ls0,c,l1,l2,c1,l3,l4,rs0,n.
+ only_bits_or_nulls l1 → no_marks l1 (* → no_grids l2 *) →
+ bit_or_null c = true → bit_or_null c1 = true →
+ only_bits_or_nulls l3 → S n = |l1| → |l1| = |l3| →
+ table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) →
+ ls = 〈grid,false〉::ls0 → cur = 〈c,true〉 →
+ rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0 →
+ (* facciamo match *)
+ (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
+ t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
+ (l2@〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs0))
+ ∨
+ (* non facciamo match e marchiamo la prossima tupla *)
+ (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
+ ∃c2,l5,l6.l4 = l5@〈bar,false〉::〈c2,false〉::l6 ∧
+ (* condizioni su l5 l6 l7 *)
+ t2 = midtape STape (〈grid,false〉::ls0) 〈c,true〉
+ (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::
+ l5@〈bar,false〉::〈c2,true〉::l6@〈grid,false〉::rs0))
+ ∨
+ (* non facciamo match e non c'è una prossima tupla:
+ non specifichiamo condizioni sul nastro di output, perché
+ non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
+ (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)).
+(* universal version
definition R_match_tuple_step_true ≝ λt1,t2.
∀ls,cur,rs.t1 = midtape STape ls cur rs →
\fst cur ≠ grid ∧
non specifichiamo condizioni sul nastro di output, perché
non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
(〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)).
-
+*)
+
+definition R_match_tuple_step_true ≝ λt1,t2.
+ ∃ls,cur,rs.t1 = midtape STape ls cur rs \wedge
+ \fst cur ≠ grid ∧
+ (∀ls0,c,l1,l2,c1,l3,l4,rs0,n.
+ only_bits_or_nulls l1 → no_marks l1 (* → no_grids l2 *) →
+ bit_or_null c = true → bit_or_null c1 = true →
+ only_bits_or_nulls l3 → S n = |l1| → |l1| = |l3| →
+ table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) →
+ ls = 〈grid,false〉::ls0 → cur = 〈c,true〉 →
+ rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0 →
+ (* facciamo match *)
+ (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
+ t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
+ (l2@〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs0))
+ ∨
+ (* non facciamo match e marchiamo la prossima tupla *)
+ (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
+ ∃c2,l5,l6.l4 = l5@〈bar,false〉::〈c2,false〉::l6 ∧
+ (* condizioni su l5 l6 l7 *)
+ t2 = midtape STape (〈grid,false〉::ls0) 〈c,true〉
+ (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::
+ l5@〈bar,false〉::〈c2,true〉::l6@〈grid,false〉::rs0))
+ ∨
+ (* non facciamo match e non c'è una prossima tupla:
+ non specifichiamo condizioni sul nastro di output, perché
+ non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
+ (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)).
+
definition R_match_tuple_step_false ≝ λt1,t2.
∀ls,c,rs.t1 = midtape STape ls c rs → is_grid (\fst c) = true ∧ t2 = t1.
(sem_mark ?) (sem_seq … (sem_move_l …) (sem_init_current …))))))
(sem_nop ?) …)
[(* is_grid: termination case *)
- 2:#t1 #t2 #t3 whd in ⊢ (%→?); #H #H1 whd #ls #c #rs #Ht1
- cases (H c ?) [2: >Ht1 %] #Hgrid #Heq %
- [@injective_notb @Hgrid | <Heq @H1]
+ 2:#t1 #t2 #t3 whd in ⊢ (%→?); * #Hc #H #H1 whd #ls #c #rs #Ht1 %
+ [lapply(Hc c ?) [>Ht1 %] #Hgrid @injective_notb @Hgrid |>H1 @H]
|#tapea #tapeout #tapeb whd in ⊢ (%→?); #Hcur
* #tapec * whd in ⊢ (%→?); #Hcompare #Hor
- #ls #cur #rs #Htapea >Htapea in Hcur; #Hcur cases (Hcur ? (refl ??))
- -Hcur #Hcur #Htapeb %
- [ % #Hfalse >Hfalse in Hcur; normalize #Hfalse1 destruct (Hfalse1)]
- #ls0 #c #l1 #l2 #c1 #l3 #l4 #rs0 #n #Hl1bitnull #Hl1marks #Hc #Hc1 #Hl3 #eqn
- #eqlen #Htable #Hls #Hcur #Hrs -Htapea >Hls in Htapeb; >Hcur >Hrs #Htapeb
+ cases Hcur * #c * -Hcur #Hcur #Hgrid #Htapeb cases (current_to_midtape … Hcur)
+ #ls * #rs #Htapea @(ex_intro … ls) @(ex_intro … c) @(ex_intro … rs) %
+ [%[@Htapea | cases (true_or_false (\fst c == grid))
+ [#eqc @False_ind >(\P eqc) in Hgrid; normalize #H destruct |#eqc @(\Pf eqc)]]]
+ #ls0 #cur #l1 #l2 #c1 #l3 #l4 #rs0 #n #Hl1bitnull #Hl1marks #Hc #Hc1 #Hl3 #eqn
+ #eqlen #Htable #Hls -Hcur #Hcur #Hrs >Htapea in Htapeb; >Hls >Hcur >Hrs #Htapeb
cases (Hcompare … Htapeb) -Hcompare -Htapeb * #_ #_ #Hcompare
- cases (Hcompare c c1 l1 l3 l2 (l4@〈grid,false〉::rs0) eqlen Hl1bitnull Hl3 Hl1marks … (refl …) Hc ?)
+ cases (Hcompare cur c1 l1 l3 l2 (l4@〈grid,false〉::rs0) eqlen Hl1bitnull Hl3 Hl1marks … (refl …) Hc ?)
-Hcompare
[* #Htemp destruct (Htemp) #Htapec %1 % % [%]
>Htapec in Hor; -Htapec *
- [2: * #t3 * whd in ⊢ (%→?); #H @False_ind
- cases (H … (refl …)) whd in ⊢ ((??%?)→?); #H destruct (H)
- |* #taped * whd in ⊢ (%→?); #Htaped cases (Htaped ? (refl …)) -Htaped *
+ [2: * #t3 * whd in ⊢ (%→?); * #H #_ @False_ind
+ lapply (H … (refl …)) whd in ⊢ ((??%?)→?); #H destruct (H)
+ |* #taped * whd in ⊢ (%→?); * #_
#Htaped whd in ⊢ (%→?); #Htapeout >Htapeout >Htaped
%
]
|* #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #Htapec
- cut (〈c,false〉::l1 ≠ 〈c1,false〉::l3)
+ cut (〈cur,false〉::l1 ≠ 〈c1,false〉::l3)
[>H2 >H3 elim la
- [@(not_to_not …H1) normalize #H destruct %
+ [@(not_to_not …H1) normalize #H destruct (H) %
|#x #tl @not_to_not normalize #H destruct //
]
] #Hnoteq
]
] #Hd'
>Htapec in Hor; -Htapec *
- [* #taped * whd in ⊢ (%→?); #H @False_ind
- cases (H … (refl …)) >(bit_or_null_not_grid ? Hd') #Htemp destruct (Htemp)
- |* #taped * whd in ⊢ (%→?); #H cases (H … (refl …)) -H #_
+ [* #taped * whd in ⊢ (%→?); * * #c0 * normalize in ⊢ (%→?);
+ #Hdes destruct (Hdes) >(bit_or_null_not_grid ? Hd') #Htemp destruct (Htemp)
+ |* #taped * whd in ⊢ (%→?); * #_ (* * #_ #H cases (H … (refl …)) -H #_ *)
#Htaped * #tapee * whd in ⊢ (%→?); #Htapee
<(associative_append ? lc (〈comma,false〉::l4)) in Htaped; #Htaped
cases (Htapee … Htaped ???) -Htaped -Htapee
>Hb2 in Heq1; #Heq1 -Hb2 -b2
whd in ⊢ ((???%)→?); #Htemp destruct (Htemp) #Htapee >Htapee -Htapee *
[(* we know current is not grid *)
- * #tapef * whd in ⊢ (%→?); #Htapef
- cases (Htapef … (refl …)) >Hd2 #Htemp destruct (Htemp)
- |* #tapef * whd in ⊢ (%→?); #Htapef
- cases (Htapef … (refl …)) #_ -Htapef #Htapef
+ * #tapef * whd in ⊢ (%→?); * * #c0 *
+ normalize in ⊢ (%→?); #Hdes destruct (Hdes) >Hd2
+ #Htemp destruct (Htemp)
+ |* #tapef * whd in ⊢ (%→?); * #_ #Htapef
* #tapeg >Htapef -Htapef *
(* move_l *)
- whd in ⊢ (%→?);
- #H lapply (H … (refl …)) whd in ⊢ (???%→?); -H #Htapeg
+ whd in ⊢ (%→?); * #_ #H lapply (H … (refl …)) whd in ⊢ (???%→?); -H #Htapeg
>Htapeg -Htapeg
(* init_current *)
whd in ⊢ (%→?); #Htapeout
]
]
|* #Hnobars #Htapee >Htapee -Htapee *
- [whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
- cases (Htapef … (refl …)) -Htapef #_ #Htapef >Htapef -Htapef
- whd in ⊢ (%→?); #Htapeout %2 %
+ [whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); * #_
+ #Htapef >Htapef -Htapef
+ whd in ⊢ (%→?); * #Htapeout #_ %2 %
[% [//] whd #x #Hx @Hnobars @memb_append_l2 @memb_cons //
| >(Htapeout … (refl …)) % ]
- |whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
- cases (Htapef … (refl …)) -Htapef
- whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)
+ |whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?);
+ * #Hc0 lapply(Hc0 … (refl … )) normalize in ⊢ (%→?); #Htemp destruct (Htemp)
]
|(* no marks in table *)
#x #membx @(no_marks_in_table … Htable)
#intape #k #outc #Hloop
lapply (sem_while … sem_match_tuple_step intape k outc Hloop) [%] -Hloop
* #ta * #Hstar @(star_ind_l ??????? Hstar)
-[ #tb whd in ⊢ (%→?); #Hleft
+[ whd in ⊢ (%→?); #Hleft
#ls #cur #rs #Htb cases (Hleft … Htb) #Hgrid #Houtc %
[ #_ @Houtc
| #c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs
]
| (* 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
+ #tc #td whd in ⊢ (%→?); #Htc
#Hstar1 #IH whd in ⊢ (%→?); #Hright lapply (IH Hright) -IH whd in ⊢ (%→?); #IH
#ls #cur #rs #Htb %
[ (* cur can't be true because we assume at least one iteration *)
- #Hcur cases (Htc … Htb) * #Hfalse @False_ind @Hfalse @(is_grid_true … Hcur)
+ #Hcur cases Htc #ls' * #c' * #rs' * * >Htb #Hdes destruct (Hdes)
+ #Hfalse @False_ind @(absurd … (is_grid_true … Hcur) Hfalse)
| (* 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
+ #Hl1len #Htable
+ cases Htc #ls' * #c' * #rs' * * >Htb #Hdes destruct (Hdes)
+ -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)
| (* 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
+ [ #_ @daemon (* long normalize *) (*
+ normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse)
+ *)
+ |2,3: #x #xs #_ @daemon (* long normalize *) (*
+ normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse) *) ]
+ #ls1 #cur1 #rs1 #Htc @daemon (* long normalize *) (*
+ normalize in ⊢ (??%?→?); #Hcur1
cases (IH … Htc) -IH #IH #_ %2 %
[ destruct (Hcur1) >IH [ >Htc % | % ]
| #l4 #newc #mv0 #l5
(* no_bars except the first one, where the tuple does not match ⇒
no match *)
@daemon
- ]
+ ] *)
]
]
qed.
+(* termination *)
+lemma WF_mts_niltape:
+ WF ? (inv ? R_match_tuple_step_true) (niltape (FinProd FSUnialpha FinBool)).
+@wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * * #H destruct
+qed.
+
+lemma WF_mts_rightof:
+ ∀a,ls. WF ? (inv ? R_match_tuple_step_true) (rightof (FinProd FSUnialpha FinBool) a ls).
+#a #ls @wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * * #H destruct
+qed.
+
+lemma WF_mts_leftof:
+ ∀a,ls. WF ? (inv ? R_match_tuple_step_true) (leftof (FinProd FSUnialpha FinBool) a ls).
+#a #ls @wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * * #H destruct
+qed.
+
+lemma WF_cst_midtape_grid:
+ ∀ls,b,rs. WF ? (inv ? R_match_tuple_step_true)
+ (midtape (FinProd … FSUnialpha FinBool) ls 〈grid,b〉 rs).
+#ls #b #rs @wf #t1 whd in ⊢ (%→?); * #ls' * #c' * #rs' * * #H destruct
+* #Hfalse @False_ind @Hfalse %
+qed.
+
+definition Pre_match_tuple ≝ λt.
+ ∃ls,cur,rs. t = midtape STape ls cur rs ∧
+ (is_grid (\fst cur) = true ∨
+ (∃ls0,c,l1,l2,c1,l3,l4,rs0,n.
+ only_bits_or_nulls l1 ∧ no_marks l1 ∧
+ bit_or_null c = true ∧ bit_or_null c1 = true ∧
+ only_bits_or_nulls l3 ∧ S n = |l1| ∧|l1| = |l3| ∧
+ table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) ∧
+ ls = 〈grid,false〉::ls0 ∧ cur = 〈c,true〉 ∧
+ rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0)).
+
+lemma acc_Realize_to_acc_GRealize: ∀sig,M.∀q:states sig M.∀P,R1,R2.
+ M ⊨ [q:R1,R2] → accGRealize sig M q P R1 R2.
+#alpha #M #q #Pre #R1 #R2 #HR #t #HPre
+cases (HR t) -HR #k * #outc * * #Hloop #HRtrue #HRfalse
+@(ex_intro ?? k) @(ex_intro ?? outc) %
+ [ % [@Hloop] @HRtrue | @HRfalse]
+qed.
+
+
+lemma terminate_match_tuple:
+ ∀t. Pre_match_tuple t → Terminate ? match_tuple t.
+#t #HPre
+@(terminate_while_guarded ???
+ Pre_match_tuple …
+ (acc_Realize_to_acc_GRealize ??? Pre_match_tuple … sem_match_tuple_step)
+ … HPre) [%]
+ [-HPre -t #t1 #t2 #HPre cases HPre #ls * * #curl #curr * #rs * #Ht1 *
+ [(* absurd case *)
+ #Hgrid * #ls1 * #cur1 * #rs1 * * >Ht1 #Hdes destruct (Hdes)
+ #Habs @False_ind @(absurd ?? Habs) @(is_grid_true … Hgrid)
+ |* #ls0 * #c * #l1 * #l2 * #c1 * #l3 * #l4 * #rs0 * #n
+ * * * * * * * * * *
+ #Hl1 #Hmarksl1 #Hc #Hc1 #Hl3 #lenl1 #eqlen #Htable #Hls #Hcur #Hrs
+ * #ls1 * #cur1 * #rs1 * * >Ht1 #Hdes destruct (Hdes) #Hdes #H
+ lapply (H … Hl1 Hmarksl1 Hc Hc1 Hl3 lenl1 eqlen Htable Hls Hcur Hrs)
+ -H *
+ [* [ * #Hdes #Ht2 >Ht2
+ @ex_intro [2:@ex_intro [2: @ex_intro [2: % [%]|]|]|]
+ %1 %
+ |* #test * #c2 * #l5 * #l6 * #Hl4 #Ht2
+ cut (∃l7,l8. l6 = l7@〈comma,false 〉::l8 ∧ |l7| = |l1|) [@daemon]
+ * #l7 * #l8 * #Hl6 #eqlen1
+ @ex_intro [2:@ex_intro [2: @ex_intro [2: % [@Ht2]|]|]|] %2
+ @(ex_intro … ls0) @(ex_intro … c) @(ex_intro … l1)
+ @(ex_intro … (l2@〈c1,false〉::l3@〈comma,false〉::l5@[〈bar,false〉]))
+ @(ex_intro … c2) @(ex_intro … l7) @(ex_intro … l8)
+ @(ex_intro … rs0) @(ex_intro … n)
+ % [2: >Hl6 >associative_append >associative_append @eq_f @eq_f @eq_f
+ @eq_f >associative_append @eq_f @eq_f >associative_append % ]
+ % [2: %] % [2: %] % [2:@daemon] % [2: @sym_eq @eqlen1]
+ % [2: @lenl1] % [2: #x #memx @daemon]
+ % [2: @daemon] % [2: @Hc] % [2: @Hmarksl1] @Hl1
+ ]
+ |* * #_ #_ #H cases (current_to_midtape … H) #ls * #rs #Ht1
+ >Ht1 @ex_intro [2:@ex_intro [2: @ex_intro [2: % [%]|]|]|] %1 %
+ ]
+ ]
+ |cases HPre -HPre #ls * * #curl #curr * #rs * #Ht *
+ [#Hgrid >Ht >(is_grid_true … Hgrid) @WF_cst_midtape_grid
+ |* #ls0 * #c * #l1 * #l2 * #c1 * #l3 * #l4
+ cut (∃len. |l4| = len) [/2/] * #lenl4
+ lapply l4 lapply l3 lapply c1 lapply l2 lapply l1 lapply c lapply ls0 lapply Ht
+ lapply curr lapply curl lapply ls lapply rs lapply t -l4 -l3 -l2 -l1 -c1 -curr -curl -ls -t
+ -c -ls0 -rs
+ (* by induction on the length of l4 *)
+ @(nat_elim1 lenl4)
+ #len #Hind #t #rs #ls #cl #cr #Ht #ls0 #c #l1 #l2 #c1 #l3 #l4 #Hlen
+ * #rs0 * #n * * * * * * * * * *
+ #Hl1 #Hmarksl1 #Hc #Hc1 #Hl3 #lenl1 #eqlen #Htable #Hls #Hcur #Hrs
+ % #t1 >Ht whd in ⊢ (%→?); * #ls1 * #cur * #rs1 * * #Hdes destruct (Hdes)
+ #Hgrid #H lapply (H … Hl1 Hmarksl1 Hc Hc1 Hl3 lenl1 eqlen Htable Hls Hcur Hrs)
+ -H *
+ [* [ * #Hdes destruct (Hdes) #Ht1 >Ht1 @WF_cst_midtape_grid
+ | * #_ * #c2 * #l5 * #l6 * #Hl4 #Ht1
+ cut (∃l7,l8. l6 = l7@〈comma,false 〉::l8 ∧ |l7| = |l1|) [@daemon]
+ * #l7 * #l8 * #Hl6 #eqlen1
+ @(Hind … Ht1 ls0 c l1 (l2@〈c1,false〉::l3@〈comma,false〉::l5@[〈bar,false〉]) c2 l7 l8 … (refl …))
+ [<Hlen >Hl4 >Hl6 >length_append normalize in match (length … (cons …));
+ >length_append normalize in match (length … (cons …)); <plus_n_Sm
+ @le_S_S @daemon
+ |@(ex_intro … rs0) @(ex_intro … n) %
+ [2: >Hl6 >associative_append >associative_append @eq_f @eq_f @eq_f
+ @eq_f >associative_append @eq_f @eq_f >associative_append % ]
+ % [2: %] % [2: %] % [2:@daemon] % [2: @sym_eq @eqlen1]
+ % [2: @lenl1] % [2: #x #memx @daemon]
+ % [2: @daemon] % [2: @Hc] % [2: @Hmarksl1] @Hl1
+ ]
+ ]
+ |* * #_ #_ #H cases (current_to_midtape … H) #ls * #rs #Ht1
+ >Ht1 //
+ ]
+ ]
+qed.
+
definition R_match_tuple ≝ λt1,t2.
∀ls,c,l1,c1,l2,rs,n.
is_bit c = true → is_bit c1 = 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.
+lemma sem_match_tuple0 : GRealize ? match_tuple Pre_match_tuple R_match_tuple0.
+@WRealize_to_GRealize [@terminate_match_tuple | @wsem_match_tuple]
+qed.
-lemma sem_match_tuple : Realize ? match_tuple R_match_tuple.
-generalize in match sem_match_tuple0; @Realize_to_Realize
+lemma sem_match_tuple : GRealize ? match_tuple Pre_match_tuple R_match_tuple.
+generalize in match sem_match_tuple0; @GRealize_to_GRealize
#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
projects / helm.git / blobdiff