definition STape ≝ FinProd … FSUnialpha FinBool.
-definition only_bits ≝ λl.
- ∀c.memb STape c l = true → is_bit (\fst c) = true.
+definition only_bits_or_nulls ≝ λl.
+ ∀c.memb STape c l = true → bit_or_null (\fst c) = true.
definition no_grids ≝ λl.
∀c.memb STape c l = true → is_grid (\fst c) = false.
* // normalize #H destruct
qed.
+lemma bit_or_null_not_grid: ∀d. bit_or_null d = true → is_grid d = false.
+* // normalize #H destruct
+qed.
+
lemma bit_not_bar: ∀d. is_bit d = true → is_bar d = false.
* // normalize #H destruct
qed.
+lemma bit_or_null_not_bar: ∀d. bit_or_null d = true → is_bar d = false.
+* // normalize #H destruct
+qed.
+
(* by definition, a tuple is not marked *)
definition tuple_TM : nat → list STape → Prop ≝
λn,t.∃qin,qout,mv.
no_marks t ∧
- only_bits qin ∧ only_bits qout ∧ only_bits mv ∧
+ only_bits_or_nulls qin ∧ only_bits_or_nulls qout ∧ bit_or_null mv = true ∧
|qin| = n ∧ |qout| = n (* ∧ |mv| = ? *) ∧
- t = qin@〈comma,false〉::qout@〈comma,false〉::mv.
+ t = qin@〈comma,false〉::qout@〈comma,false〉::[〈mv,false〉].
inductive table_TM : nat → list STape → Prop ≝
| ttm_nil : ∀n.table_TM n []
whd >Heq #x #membx
cases (memb_append … membx) -membx #membx
[cases (memb_append … membx) -membx #membx
- [@bit_not_grid @Hqin //
+ [@bit_or_null_not_grid @Hqin //
|cases (orb_true_l … membx) -membx #membx
[>(\P membx) //
|cases (memb_append … membx) -membx #membx
- [@bit_not_grid @Hqout //
+ [@bit_or_null_not_grid @Hqout //
|cases (orb_true_l … membx) -membx #membx
[>(\P membx) //
- |@bit_not_grid @Hmv //
+ |@bit_or_null_not_grid >(memb_single … membx) @Hmv
]
]
]
]]]]
qed.
+definition init_current_on_match ≝
+ (seq ? (move_l ?)
+ (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
+ (seq ? (move_r ?) (mark ?)))).
+
+definition R_init_current_on_match ≝ λt1,t2.
+ ∀l1,l2,c,rs. no_grids l1 → is_grid c = false →
+ t1 = midtape STape (l1@〈c,false〉::〈grid,false〉::l2) 〈grid,false〉 rs →
+ t2 = midtape STape (〈grid,false〉::l2) 〈c,true〉 ((reverse ? l1)@〈grid,false〉::rs).
+
+lemma sem_init_current_on_match :
+ Realize ? init_current_on_match R_init_current_on_match.
+#intape
+cases (sem_seq ????? (sem_move_l ?)
+ (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
+ (sem_seq ????? (sem_move_r ?) (sem_mark ?))) intape)
+#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
+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
+ >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 …) ?)
+ [#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
+ >reverse_append >associative_append whd in ⊢ ((???(??%%%))→?); #Htc
+ whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc #Houtc
+ >Houtc >reverse_cons >associative_append %
+ ]
+qed.
+
+(*
+definition init_current_gen ≝
+ seq ? (adv_to_mark_l ? (is_marked ?))
+ (seq ? (clear_mark ?)
+ (seq ? (move_l ?)
+ (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
+ (seq ? (move_r ?) (mark ?))))).
+
+definition R_init_current_gen ≝ λt1,t2.
+ ∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 →
+ Some ? 〈c0,b0〉 = option_hd ? (reverse ? (〈c,true〉::l2)) →
+ t1 = midtape STape (l1@〈c,true〉::l2@〈grid,b〉::l3) 〈c1,false〉 rs →
+ t2 = midtape STape (〈grid,b〉::l3) 〈c0,true〉
+ ((tail ? (reverse ? (l1@〈c,false〉::l2))@〈c1,false〉::rs)).
+
+lemma sem_init_current_gen : Realize ? init_current_gen R_init_current_gen.
+#intape
+cases (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
+ (sem_seq ????? (sem_clear_mark ?)
+ (sem_seq ????? (sem_move_l ?)
+ (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
+ (sem_seq ????? (sem_move_r ?) (sem_mark ?))))) intape)
+#k * #outc * #Hloop #HR
+@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
+#l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hintape
+cases HR -HR #ta * whd in ⊢ (%→?); #Hta cases (Hta … Hintape) -Hta -Hintape
+ [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
+* #_ #Hta lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%] -Hta #Hta
+* #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta) -Htb -Hta #Htb
+* #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb
+generalize in match Hc; generalize in match Hl2; cases l2
+ [#_ whd in ⊢ ((???%)→?); #Htemp destruct (Htemp)
+ whd in ⊢ ((???(??%%%))→?); #Htc
+ * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd
+ [2: * whd in ⊢ (??%?→?); #Htemp destruct (Htemp) ]
+ * #_ #Htd >Htd in Htc; -Htd #Htd
+ * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Htd
+ >reverse_append >reverse_cons
+ whd in ⊢ ((???(??%%%))→?); #Hte
+ whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
+ >Houtc %
+ |#d #tl #Htl #Hc0 whd in ⊢ ((???(??%%%))→?); #Htc
+ * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd
+ [* >(Htl … (memb_hd …)) whd in ⊢ (??%?→?); #Htemp destruct (Htemp)]
+ * #Hd #Htd lapply (Htd … (refl ??) (refl ??) ?)
+ [#x #membx @Htl @memb_cons @membx] -Htd #Htd
+ * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Htd
+ >reverse_append >reverse_cons >reverse_cons
+ >reverse_cons in Hc0; >reverse_cons cases (reverse ? tl)
+ [normalize in ⊢ (%→?); #Hc0 destruct (Hc0) #Hte
+ whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
+ >Houtc %
+ |* #c2 #b2 #tl2 normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
+ whd in ⊢ ((???(??%%%))→?); #Hte
+ whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
+ >Houtc >associative_append >associative_append >associative_append %
+ ]
+ ]
+qed.
+*)
definition init_current ≝
seq ? (adv_to_mark_l ? (is_marked ?))
(seq ? (clear_mark ?)
definition R_match_tuple_step_true ≝ λt1,t2.
∀ls,c,l1,l2,c1,l3,l4,rs,n.
- is_bit c = true → only_bits l1 → no_marks l1 (* → no_grids l2 *) → is_bit c1 = true →
- only_bits l3 → n = |l1| → |l1| = |l3| →
+ 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) →
|#x #tl @not_to_not normalize #H destruct //
]
] #Hnoteq %2
- cut (is_bit d' = true)
+ cut (bit_or_null d' = true)
[cases la in H3;
[normalize in ⊢ (%→?); #H destruct //
|#x #tl #H @(Hl3 〈d',false〉)
] #Hd'
>Htapec in Hor; -Htapec *
[* #taped * whd in ⊢ (%→?); #H @False_ind
- cases (H … (refl …)) >(bit_not_grid ? Hd') #Htemp destruct (Htemp)
+ 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
@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 (∀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〉)
- @daemon (* no grids in table *)] #Hd2
+ [@(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
whd in ⊢ (%→?); #Htapeout
%1 cases (some_option_hd ? (reverse ? (reverse ? la)) 〈c',true〉)
* #c00 #b00 #Hoption
- (* cut
- ((reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? (l2@[〈bar,false〉])@(〈grid,false〉::reverse ? lb))
- = *)
lapply
(Htapeout (reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? (l2@[〈bar,false〉])@(〈grid,false〉::reverse ? lb))
c' (reverse ? la) false ls bar (〈d2,true〉::rs3'@〈grid,false〉::rs) c00 b00 ?????) -Htapeout
[normalize in ⊢ (%→?); #Htemp destruct (Htemp)
@injective_notb @notgridc
|#x #tl normalize in ⊢ (%→?); #Htemp destruct (Htemp)
- @bit_not_grid @(Hl1bars 〈c',false〉) @memb_append_l2 @memb_hd
+ @bit_or_null_not_grid @(Hl1bars 〈c',false〉) @memb_append_l2 @memb_hd
]
- |cut (only_bits (la@(〈c',false〉::lb)))
+ |cut (only_bits_or_nulls (la@(〈c',false〉::lb)))
[<H2 whd #c0 #Hmemb cases (orb_true_l … Hmemb)
[#eqc0 >(\P eqc0) @Hc |@Hl1bars]
- |#Hl1' #x #Hx @bit_not_grid @Hl1'
+ |#Hl1' #x #Hx @bit_or_null_not_grid @Hl1'
@memb_append_l1 @daemon
]
|@daemon
(* cut
(〈c1,false〉::l3@〈comma,false〉::l4= la@〈d',false〉::rs3@〈bar,false〉::〈d2,b2〉::rs3')
[@daemon] #Hcut *)
- cut (l4=rs32@〈bar,false〉::〈d2,b2〉::rs3')
+ cut (l4=rs32@〈bar,false〉::〈d2,false〉::rs3')
[ >Hrs3 in Heq1; @daemon ] #Hl4
@(ex_intro … rs32) @(ex_intro … rs3') %
- [(* by showing b2 = false: @Hl4 *) @daemon
+ [@Hl4
|>Htapeout @eq_f2
[@daemon
|(*>Hrs3 *)>append_cons
>associative_append normalize
% ]
@eq_f2 [|%]
- @(injective_append … (〈d2,b2〉::rs3'))
+ @(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)))
[|>associative_append >associative_append
@memb_append_l2 @memb_cons
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_not_grid … Hd') >(bit_not_bar … Hd') %
+ |whd in ⊢ (??%?); >(bit_or_null_not_grid … Hd') >(bit_or_null_not_bar … Hd') %
]
]
|#x #membx @(no_marks_in_table … Htable)
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,true〉::l2) →
+ 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) →
(* facciamo match *)
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