definition mark_next_tuple ≝
seq ? (adv_to_mark_r ? bar_or_grid)
(ifTM ? (test_char ? (λc:STape.is_bar (\fst c)))
- (move_right_and_mark ?) (nop ?) 1).
+ (move_right_and_mark ?) (nop ?) tc_true).
definition R_mark_next_tuple ≝
λt1,t2.
Realize ? mark_next_tuple R_mark_next_tuple.
#intape
lapply (sem_seq ? (adv_to_mark_r ? bar_or_grid)
- (ifTM ? (test_char ? (λc:STape.is_bar (\fst c))) (move_right_and_mark ?) (nop ?) 1) ????)
+ (ifTM ? (test_char ? (λc:STape.is_bar (\fst c))) (move_right_and_mark ?) (nop ?) tc_true) ????)
[@sem_if [5: // |6: @sem_move_right_and_mark |7: // |*:skip]
| //
|||#Hif cases (Hif intape) -Hif
| -Hta #Hta cases Hright
[ * #tb * whd in ⊢ (%→?); #Hcurrent
@False_ind cases (Hcurrent 〈grid,false〉 ?)
- [ normalize #Hfalse destruct (Hfalse)
+ [ normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
| >Hta % ]
| * #tb * whd in ⊢ (%→?); #Hcurrent
cases (Hcurrent 〈grid,false〉 ?)
(nop ?) tc_true.
definition R_match_tuple_step_true ≝ λt1,t2.
- ∀ls,c,l1,l2,c1,l3,l4,rs,n.
- bit_or_null c = true → only_bits_or_nulls l1 → no_marks l1 (* → no_grids l2 *) → bit_or_null c1 = true →
- only_bits_or_nulls l3 → n = |l1| → |l1| = |l3| →
- table_TM (S n) (l2@〈bar,false〉::〈c1,false〉::l3@〈comma,false〉::l4) →
- t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉
- (l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs) →
- (* facciamo match *)
- (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
- t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
- (l2@〈bar,false〉::〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs))
- ∨
- (* 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〉::ls) 〈c,true〉
- (l1@〈grid,false〉::l2@〈bar,false〉::〈c1,false〉::l3@〈comma,false〉::
- l5@〈bar,false〉::〈c2,true〉::l6@〈grid,false〉::rs))
- ∨
- (* 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〉)).
+ ∀ls,cur,rs.t1 = midtape STape ls cur rs →
+ \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.
axiom injective_append : ∀A,l.injective … (λx.append A x l).
lemma sem_match_tuple_step:
- accRealize ? match_tuple_step (inr … (inl … (inr … 0)))
+ accRealize ? match_tuple_step (inr … (inl … (inr … start_nop)))
R_match_tuple_step_true R_match_tuple_step_false.
@(acc_sem_if_app … (sem_test_char ? (λc:STape.¬ is_grid (\fst c))) …
(sem_seq … sem_compare
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]
-|#tapea #tapeout #tapeb whd in ⊢ (%→?); #Htapea
- * #tapec * #Hcompare #Hor
- #ls #c #l1 #l2 #c1 #l3 #l4 #rs #n #Hc #Hl1bars #Hl1marks #Hc1 #Hl3 #eqn
- #eqlen #Htable #Htapea1 cases (Htapea 〈c,true〉 ?) >Htapea1 [2:%]
+|#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 (Hcompare … Htapeb) -Hcompare -Htapeb * #_ #_ #Hcompare
+ cases (Hcompare c 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 *
+ #Htaped whd in ⊢ (%→?); #Htapeout >Htapeout >Htaped
+ %
+ ]
+ |* #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #Htapec
+ cut (〈c,false〉::l1 ≠ 〈c1,false〉::l3)
+ [>H2 >H3 elim la
+ [@(not_to_not …H1) normalize #H destruct %
+ |#x #tl @not_to_not normalize #H destruct //
+ ]
+ ] #Hnoteq
+ cut (bit_or_null d' = true)
+ [cases la in H3;
+ [normalize in ⊢ (%→?); #H destruct //
+ |#x #tl #H @(Hl3 〈d',false〉)
+ normalize in H; destruct @memb_append_l2 @memb_hd
+ ]
+ ] #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 #_
+ #Htaped * #tapee * whd in ⊢ (%→?); #Htapee
+ <(associative_append ? lc (〈comma,false〉::l4)) in Htaped; #Htaped
+ cases (Htapee … Htaped ???) -Htaped -Htapee
+ [* #rs3 * * (* we proceed by cases on rs4 *)
+ [(* rs4 is empty : the case is absurd since the tape
+ cannot end with a bar *)
+ * #d * #b * * * #Heq1 @False_ind
+ cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
+ >Hcut in Htable; >H3 >associative_append
+ normalize >Heq1 <associative_append >Hcut
+ <associative_append #Htable @(absurd … Htable)
+ @last_of_table
+ |(* rs4 not empty *)
+ * #d2 #b2 #rs3' * #d * #b * * * #Heq1 #Hnobars
+ cut (memb STape 〈d2,b2〉 (l2@〈c1,false〉::l3@〈comma,false〉::l4) = true)
+ [@memb_append_l2
+ cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
+ >Hcut >H3 >associative_append @memb_append_l2
+ @memb_cons >Heq1 @memb_append_l2 @memb_cons @memb_hd] #d2intable
+ cut (is_grid d2 = false)
+ [@(no_grids_in_table … Htable … 〈d2,b2〉 d2intable)] #Hd2
+ cut (b2 = false)
+ [@(no_marks_in_table … Htable … 〈d2,b2〉 d2intable)] #Hb2
+ >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
+ * #tapeg >Htapef -Htapef *
+ (* move_l *)
+ whd in ⊢ (%→?);
+ #H lapply (H … (refl …)) whd in ⊢ (???%→?); -H #Htapeg
+ >Htapeg -Htapeg
+ (* init_current *)
+ whd in ⊢ (%→?); #Htapeout
+ cases (some_option_hd ? (reverse ? (reverse ? la)) 〈c',true〉)
+ * #c00 #b00 #Hoption
+ lapply
+ (Htapeout (reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? l2@(〈grid,false〉::reverse ? lb))
+ c' (reverse ? la) false ls0 bar (〈d2,true〉::rs3'@〈grid,false〉::rs0) c00 b00 ?????) -Htapeout
+ [whd in ⊢ (??(??%??)?); @eq_f3 [2:%|3: %]
+ >associative_append
+ generalize in match (〈c',true〉::reverse ? la@〈grid,false〉::ls0); #l
+ whd in ⊢ (???(???%)); >associative_append >associative_append %
+ |>reverse_cons @Hoption
+ |cases la in H2;
+ [normalize in ⊢ (%→?); #Htemp destruct (Htemp)
+ @bit_or_null_not_grid @Hc
+ |#x #tl normalize in ⊢ (%→?); #Htemp destruct (Htemp)
+ @bit_or_null_not_grid @(Hl1bitnull 〈c',false〉) @memb_append_l2 @memb_hd
+ ]
+ |cut (only_bits_or_nulls (la@(〈c',false〉::lb)))
+ [<H2 whd #c0 #Hmemb cases (orb_true_l … Hmemb)
+ [#eqc0 >(\P eqc0) @Hc |@Hl1bitnull]
+ |#Hl1' #x #Hx @bit_or_null_not_grid @Hl1'
+ @memb_append_l1 @daemon
+ ]
+ |@daemon] #Htapeout % %2 % //
+ @(ex_intro … d2)
+ cut (∃rs32.rs3 = lc@〈comma,false〉::rs32)
+ [ (*cases (tech_split STape (λc.c == 〈bar,false〉) l4)
+ [
+ | * #l41 * * #cbar #bfalse * #l42 * * #Hbar #Hl4 #Hl41
+ @(ex_intro ?? l41) >Hl4 in Heq1; #Heq1
+
+ cut (sublist … lc l3)
+ [ #x #Hx cases la in H3;
+ [ normalize #H3 destruct (H3) @Hx
+ | #p #la' normalize #Hla' destruct (Hla')
+ @memb_append_l2 @memb_cons @Hx ] ] #Hsublist*)
+ @daemon]
+ * #rs32 #Hrs3
+ (* cut
+ (〈c1,false〉::l3@〈comma,false〉::l4= la@〈d',false〉::rs3@〈bar,false〉::〈d2,b2〉::rs3')
+ [@daemon] #Hcut *)
+ cut (l4=rs32@〈bar,false〉::〈d2,false〉::rs3')
+ [ >Hrs3 in Heq1; @daemon ] #Hl4
+ @(ex_intro … rs32) @(ex_intro … rs3') % [@Hl4]
+ >Htapeout @eq_f2
+ [@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)
+ [|>associative_append normalize
+ >associative_append normalize
+ >associative_append normalize
+ >associative_append normalize
+ % ]
+ >reverse_append >reverse_append >reverse_cons
+ >reverse_reverse >reverse_cons >reverse_reverse
+ >reverse_append >reverse_append >reverse_cons
+ >reverse_reverse >reverse_reverse >reverse_reverse
+ >(?:(la@[〈c',false〉])@((((lb@[〈grid,false〉])@l2)@la)@[〈d',false〉])@rs3
+ =((la@〈c',false〉::lb)@([〈grid,false〉]@l2@la@[〈d',false〉]@rs3)))
+ [|>associative_append >associative_append
+ >associative_append >associative_append >associative_append
+ >associative_append % ]
+ <H2 normalize in ⊢ (??%?); >Hrs3
+ >associative_append >associative_append normalize
+ >associative_append >associative_append
+ @eq_f @eq_f @eq_f
+ >(?:la@(〈d',false〉::lc@〈comma,false〉::rs32)@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs0 =
+ (la@〈d',false〉::lc)@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs0 )
+ [| >associative_append normalize >associative_append % ]
+ <H3 %
+ ]
+ ]
+ ]
+ |* #Hnobars #Htapee >Htapee -Htapee *
+ [whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
+ cases (Htapef … (refl …)) -Htapef #_ #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)
+ ]
+ |(* no marks in table *)
+ #x #membx @(no_marks_in_table … Htable)
+ @memb_append_l2
+ cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
+ >H3 >associative_append @memb_append_l2 @memb_cons @membx
+ |(* no grids in table *)
+ #x #membx @(no_grids_in_table … Htable)
+ @memb_append_l2
+ cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
+ >H3 >associative_append @memb_append_l2 @memb_cons @membx
+ |whd in ⊢ (??%?); >(bit_or_null_not_grid … Hd') >(bit_or_null_not_bar … Hd') %
+ ]
+ ]
+ |#x #membx @(no_marks_in_table … Htable)
+ @memb_append_l2 @memb_cons @memb_append_l1 @membx
+ |#x #membx @(no_marks_in_table … Htable)
+ @memb_append_l1 @membx
+ |%
+ ]
+ ]
+qed.
+
+ cases (Htapea 〈c,true〉 ?) >Htapea1 [2:%]
#notgridc -Htapea -Htapea1 -tapea #Htapeb
cases (Hcompare … Htapeb) -Hcompare -Htapeb * #_ #_ #Hcompare
- cases (Hcompare c c1 l1 l3 (l2@[〈bar,false〉]) (l4@〈grid,false〉::rs) eqlen Hl1bars Hl3 Hl1marks … (refl …) Hc ?)
+ cases (Hcompare c c1 l1 l3 l2 (l4@〈grid,false〉::rs) eqlen Hl1bars 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 *
- #Htaped whd in ⊢ (%→?); #Htapeout >Htapeout >Htaped >associative_append
+ #Htaped whd in ⊢ (%→?); #Htapeout >Htapeout >Htaped
%
]
|* #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #Htapec
* #d * #b * * * #Heq1 @False_ind
cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
>Hcut in Htable; >H3 >associative_append
- normalize >Heq1 >Hcut <associative_append >Hcut
+ normalize >Heq1 <associative_append >Hcut
<associative_append #Htable @(absurd … Htable)
- @last_of_table
+ @last_of_table
|(* rs4 not empty *)
* #d2 #b2 #rs3' * #d * #b * * * #Heq1 #Hnobars
- cut (memb STape 〈d2,b2〉 (l2@〈bar,false〉::〈c1,false〉::l3@〈comma,false〉::l4) = true)
- [@memb_append_l2 @memb_cons
+ cut (memb STape 〈d2,b2〉 (l2@〈c1,false〉::l3@〈comma,false〉::l4) = true)
+ [@memb_append_l2
cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
>Hcut >H3 >associative_append @memb_append_l2
@memb_cons >Heq1 @memb_append_l2 @memb_cons @memb_hd] #d2intable
%1 cases (some_option_hd ? (reverse ? (reverse ? la)) 〈c',true〉)
* #c00 #b00 #Hoption
lapply
- (Htapeout (reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? (l2@[〈bar,false〉])@(〈grid,false〉::reverse ? lb))
+ (Htapeout (reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? l2@(〈grid,false〉::reverse ? lb))
c' (reverse ? la) false ls bar (〈d2,true〉::rs3'@〈grid,false〉::rs) c00 b00 ?????) -Htapeout
[whd in ⊢ (??(??%??)?); @eq_f3 [2:%|3: %]
>associative_append
|>Htapeout @eq_f2
[@daemon
|(*>Hrs3 *)>append_cons
- > (?:l1@〈grid,false〉::l2@〈bar,false〉::〈c1,false〉::l3@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs
- = (l1@〈grid,false〉::l2@〈bar,false〉::〈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
+ = (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@[〈bar,false〉])@〈d2,true〉::rs3'@〈grid,false〉::rs)
[|>associative_append normalize
>associative_append normalize
>associative_append normalize
% ]
@eq_f2 [|%]
@(injective_append … (〈d2,false〉::rs3'))
- >(?:(la@[〈c',false〉])@((((lb@[〈grid,false〉])@l2@[〈bar,false〉])@la)@[〈d',false〉])@rs3
- =((la@〈c',false〉::lb)@([〈grid,false〉]@l2@[〈bar,false〉]@la@[〈d',false〉]@rs3)))
+ >(?:(la@[〈c',false〉])@((((lb@[〈grid,false〉])@l2)@la)@[〈d',false〉])@rs3
+ =((la@〈c',false〉::lb)@([〈grid,false〉]@l2@la@[〈d',false〉]@rs3)))
[|>associative_append >associative_append
>associative_append >associative_append >associative_append
- >associative_append >associative_append % ]
+ >associative_append % ]
<H2 normalize (* <Hrs3 *)
>associative_append >associative_append >associative_append
@eq_f normalize @eq_f >associative_append
- >associative_append @eq_f normalize @eq_f
- >(append_cons ? 〈d',false〉) >associative_append
- <Heq1 >Hl4 <associative_append <append_cons
- <H3
- >associative_append normalize
- >associative_append normalize %
+ >associative_append >associative_append @eq_f normalize
+ >associative_append
+ change with ((〈c1,false〉::l3)@?) in ⊢ (???%);
+ >H3 >associative_append @eq_f @eq_f
+ >Hrs3 >associative_append normalize >associative_append %
]
]
]
]
|(* no marks in table *)
#x #membx @(no_marks_in_table … Htable)
- @memb_append_l2 @memb_cons
+ @memb_append_l2
cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
>H3 >associative_append @memb_append_l2 @memb_cons @membx
|(* no grids in table *)
#x #membx @(no_grids_in_table … Htable)
- @memb_append_l2 @memb_cons
+ @memb_append_l2
cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
>H3 >associative_append @memb_append_l2 @memb_cons @membx
|whd in ⊢ (??%?); >(bit_or_null_not_grid … Hd') >(bit_or_null_not_bar … Hd') %
]
]
|#x #membx @(no_marks_in_table … Htable)
- @memb_append_l2 @memb_cons @memb_cons @memb_append_l1 @membx
+ @memb_append_l2 @memb_cons @memb_append_l1 @membx
|#x #membx @(no_marks_in_table … Htable)
- cases (memb_append … membx) -membx #membx
- [@memb_append_l1 @membx | @memb_append_l2 >(memb_single … membx) @memb_hd]
- |>associative_append %
+ @memb_append_l1 @membx
+ |%
]
]
qed.
current configuration is found
*)
-definition match_tuple ≝ whileTM ? match_tuple_step (inr … (inl … (inr … 0))).
+definition match_tuple ≝ whileTM ? match_tuple_step (inr … (inl … (inr … start_nop))).
+
+lemma is_grid_true : ∀c.is_grid c = true → c = grid.
+* 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_or_nulls 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) →
+ ∀ls,cur,rs.
+ t1 = midtape STape ls cur rs →
+ (is_grid (\fst cur) = true → t2 = t1) ∧
+ (∀c,l1,c1,l2,l3,ls0,rs0,n.
+ ls = 〈grid,false〉::ls0 →
+ cur = 〈c,true〉 →
+ rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈grid,false〉::rs0 →
+ is_bit c = true → is_bit c1 = true →
+ only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
+ table_TM (S n) (〈c1,false〉::l3) →
(* facciamo match *)
- (∃l3,newc,mv,l4.
- 〈c1,false〉::l2 = l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4 ∧
+ (∃l4,newc,mv,l5.
+ 〈c1,false〉::l3 = l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5 ∧
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))
+ (l2@l4@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l5@
+ 〈grid,false〉::rs0))
∨
(* 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).
+ ∀l4,newc,mv,l5.
+ 〈c1,false〉::l3 ≠ l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)).
+
+lemma wsem_match_tuple : WRealize ? match_tuple R_match_tuple.
+#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
+ #ls #cur #rs #Htb cases (Hleft … Htb) #Hgrid #Houtc %
+ [ #_ @Houtc
+ | #c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs
+ >Hcur in Hgrid; #Hgrid >(is_grid_true … Hgrid) normalize in ⊢ (%→?);
+ #Hc destruct (Hc)
+ ]
+| #tb #tc #td whd in ⊢ (%→?); #Htc
+ #Hstar1 #IH whd in ⊢ (%→?); #Hright lapply (IH Hright) -IH whd in ⊢ (%→?); #IH
+ #ls #cur #rs #Htb %
+ [ #Hcur
+ #l1 #c1 #l2 #rs #n #Hc #Hc1 #Hl1bitnull #Hl1marks #Hl1len #Htable
+ cut (∃la,lb,mv,lc.l2 = la@〈comma,false〉::lb@〈comma,false〉::mv::lc ∧
+ S n = |la| ∧ only_bits_or_nulls la)
+ [@daemon] * #la * #lb * #mv * #lc * * #Hl2 #Hlalen #Hlabitnull
+ >Hl2 in Htable;
+ >(?: 〈c1,true〉::(la@〈comma,false〉::lb@〈comma,false〉::mv::lc)@〈grid,false〉::rs =
+ 〈c1,true〉::la@〈comma,false〉::(lb@〈comma,false〉::mv::lc)@〈grid,false〉::rs)
+ [#Htable #Htb cases (Htc ??? [] ???????????? Htable Htb) //
+ [5: <Hl1len @Hlalen
+ |4: whd in ⊢ (??%?); >Hc1 %
+ |3: whd in ⊢ (??%?); >Hc %
+ | * #Heq destruct (Heq) #Htc
+
+
+ %
+ %{[]} %{lb} %{mv} %{lc} % [%]
+
+
+
+ =midtape STape (〈grid,false〉::ls) 〈c,true〉
+ (l1@〈grid,false〉::〈c1,true〉::l2@〈grid,false〉::rs)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ #Houtc % %
+ [ generalize in match Hl1bits; -Hl1bits cases l1' in Hl1;
+ [ normalize #Hl1 #c1 destruct (Hl1) %
+ | * #c' #b' #l0 #Heq normalize in Heq; destruct (Heq)
+ #Hl1bits lapply (Hl1bits 〈c',false〉 ?) [ @memb_hd ]
+ >Hc #Hfalse destruct ]
+ | @Houtc ]
+| #tb #tc #td whd in ⊢ (%→?→(?→%)→%→?); #Htc #Hstar1 #Hind #Htd
+ lapply (Hind Htd) -Hind #Hind
projects / helm.git / commitdiff