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.
+
+definition mk_tuple ≝ λqin,cin,qout,cout,mv.
+ qin @ cin :: 〈comma,false〉:: qout @ cout :: 〈comma,false〉 :: [mv].
+
(* 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 ∧
- |qin| = n ∧ |qout| = n (* ∧ |mv| = ? *) ∧
- t = qin@〈comma,false〉::qout@〈comma,false〉::mv.
+ λn,t.∃qin,cin,qout,cout,mv.
+ no_marks qin ∧ no_marks qout ∧
+ only_bits qin ∧ only_bits qout ∧
+ bit_or_null cin = true ∧ bit_or_null cout = true ∧ bit_or_null mv = true ∧
+ (cout = null → mv = null) ∧
+ |qin| = n ∧ |qout| = n ∧
+ t = mk_tuple qin 〈cin,false〉 qout 〈cout,false〉 〈mv,false〉.
-inductive table_TM : nat → list STape → Prop ≝
-| ttm_nil : ∀n.table_TM n []
-| ttm_cons : ∀n,t1,T.tuple_TM n t1 → table_TM n T → table_TM n (t1@〈bar,false〉::T).
+inductive table_TM (n:nat) : list STape → Prop ≝
+| ttm_nil : table_TM n []
+| ttm_cons : ∀t1,T.tuple_TM n t1 → table_TM n T → table_TM n (t1@〈bar,false〉::T).
-lemma no_grids_in_table: ∀n.∀l.table_TM n l → no_grids l.
-#n #l #t elim t
- [normalize #n #x #H destruct
- |#m #t1 #t2 * #qin * #qout * #mv * * * * * *
- #Hmarks #Hqin #Hqout #Hmv #_ #_ #Heq #Ht2 #Hind
- whd >Heq #x #membx
- cases (memb_append … membx) -membx #membx
- [cases (memb_append … membx) -membx #membx
- [@bit_not_grid @Hqin //
- |cases (orb_true_l … membx) -membx #membx
- [>(\P membx) //
- |cases (memb_append … membx) -membx #membx
- [@bit_not_grid @Hqout //
- |cases (orb_true_l … membx) -membx #membx
- [>(\P membx) //
- |@bit_not_grid @Hmv //
- ]
- ]
- ]
- ]
- |cases (orb_true_l … membx) -membx #membx
- [>(\P membx) //
- |@Hind //
- ]
+inductive match_in_table (qin:list STape) (cin: STape)
+ (qout:list STape) (cout:STape) (mv:STape)
+: list STape → Prop ≝
+| mit_hd :
+ ∀tb.
+ match_in_table qin cin qout cout mv
+ (mk_tuple qin cin qout cout mv @〈bar,false〉::tb)
+| mit_tl :
+ ∀qin0,cin0,qout0,cout0,mv0,tb.
+ match_in_table qin cin qout cout mv tb →
+ match_in_table qin cin qout cout mv
+ (mk_tuple qin0 cin0 qout0 cout0 mv0@〈bar,false〉::tb).
+
+axiom append_l1_injective :
+ ∀A.∀l1,l2,l3,l4:list A. |l1| = |l2| → l1@l3 = l2@l4 → l1 = l2.
+axiom append_l2_injective :
+ ∀A.∀l1,l2,l3,l4:list A. |l1| = |l2| → l1@l3 = l2@l4 → l3 = l4.
+axiom cons_injective_l : ∀A.∀a1,a2:A.∀l1,l2.a1::l1 = a2::l2 → a1 = a2.
+axiom cons_injective_r : ∀A.∀a1,a2:A.∀l1,l2.a1::l1 = a2::l2 → l1 = l2.
+axiom tuple_len : ∀n,t.tuple_TM n t → |t| = 2*n+5.
+axiom append_eq_tech1 :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → |l1| < |l3| → ∃la:list A.l1@a::la = l3.
+axiom append_eq_tech2 :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l4 = false → ∃la:list A.l3 = l1@a::la.
+(*axiom list_decompose_cases :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → ∃la,lb:list A.l3 = la@a::lb ∨ l4 = la@a::lb.
+axiom list_decompose_l :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l4 = false →
+ ∃la,lb.l2 = la@lb ∧ l3 = l1@a::la.
+axiom list_decompose_r :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l3 = false →
+ ∃la,lb.l1 = la@lb ∧ l4 = lb@a::l2.
+axiom list_decompose_memb :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → |l1| < |l3| → memb A a l3 = true.*)
+
+lemma generic_match_to_match_in_table :
+ ∀n,T.table_TM n T →
+ ∀qin,cin,qout,cout,mv.|qin| = n → |qout| = n →
+ only_bits qin → only_bits qout →
+ bit_or_null (\fst cin) = true → bit_or_null (\fst cout) = true →
+ bit_or_null (\fst mv) = true →
+ ∀t1,t2.
+ T = (t1@qin@cin::〈comma,false〉::qout@cout::〈comma,false〉::[mv])@t2 →
+ match_in_table qin cin qout cout mv T.
+#n #T #Htable #qin #cin #qout #cout #mv #Hlenqin #Hlenqout
+#Hqinbits #Hqoutbits #Hcin #Hcout #Hmv
+elim Htable
+[ #t1 #t2 <associative_append cases (t1@qin) normalize
+ [ #Hfalse destruct (Hfalse) | #c0 #t0 #Hfalse destruct (Hfalse) ]
+| #tuple #T0 * #qin0 * #cin0 * #qout0 * #cout0 * #mv0
+ * * * * * * * * * *
+ #Hqin0marks #Hqout0marks #Hqin0bits #Hqout0bits #Hcin0 #Hcout0 #Hmv0 #Hcout0mv0
+ #Hlenqin0 #Hlenqout0 #Htuple #Htable0 #IH #t1 #t2 #HT
+ cases t1 in HT;
+ [ >Htuple normalize in ⊢ (??%%→?);
+ >associative_append >associative_append #HT
+ cut (qin0 = qin ∧ (〈cin0,false〉 = cin ∧ (qout0 = qout ∧
+ (〈cout0,false〉 = cout ∧ (〈mv0,false〉 = mv ∧ 〈bar,false〉::T0 = t2)))))
+ [ lapply (append_l1_injective … HT) [ >Hlenqin @Hlenqin0 ]
+ #Hqin % [ @Hqin ] -Hqin
+ lapply (append_l2_injective … HT) [ >Hlenqin @Hlenqin0 ] -HT #HT
+ lapply (cons_injective_l ????? HT) #Hcin % [ @Hcin ] -Hcin
+ lapply (cons_injective_r ????? HT) -HT #HT
+ lapply (cons_injective_r ????? HT) -HT
+ >associative_append >associative_append #HT
+ lapply (append_l1_injective … HT) [ >Hlenqout @Hlenqout0 ]
+ #Hqout % [ @Hqout ] -Hqout
+ lapply (append_l2_injective … HT) [ >Hlenqout @Hlenqout0 ] -HT normalize #HT
+ lapply (cons_injective_l ????? HT) #Hcout % [ @Hcout ] -Hcout
+ lapply (cons_injective_r ????? HT) -HT #HT
+ lapply (cons_injective_r ????? HT) -HT #HT
+ lapply (cons_injective_l ????? HT) #Hmv % [ @Hmv ] -Hmv
+ @(cons_injective_r ????? HT) ]
+ -HT * #Hqin * #Hcin * #Hqout * #Hcout * #Hmv #HT0
+ >(?:qin0@(〈cin0,false〉::〈comma,false〉::qout0@[〈cout0,false〉;〈comma,false〉;〈mv0,false〉])@〈bar,false〉::T0
+ = mk_tuple qin cin qout cout mv@〈bar,false〉::T0)
+ [|>Hqin >Hqout >Hcin >Hcout >Hmv normalize >associative_append >associative_append
+ normalize >associative_append % ]
+ %
+ | #c0 #cs0 #HT cut (∃cs1.c0::cs0 = tuple@〈bar,false〉::cs1)
+ [ cases (append_eq_tech1 ?????? HT ?)
+ [ -HT #ta #Hta cases (append_eq_tech2 … Hta ?)
+ [ -Hta #tb #Htb %{tb} @Htb
+ | @daemon ]
+ | @le_S_S >length_append >(plus_n_O (|tuple|)) >commutative_plus @le_plus
+ [ @le_O_n
+ | >Htuple normalize >length_append >length_append @le_plus [ >Hlenqin >Hlenqin0 % ]
+ @le_S_S @le_S_S >length_append >length_append @le_plus [ >Hlenqout >Hlenqout0 % ] %] ]
]
+ * #cs1 #Hcs1 >Hcs1 in HT; >associative_append >associative_append #HT
+ lapply (append_l2_injective … HT) // -HT #HT
+ lapply (cons_injective_r ????? HT) -HT
+ <associative_append #HT >Htuple %2 @(IH ?? HT)
]
+]
+qed.
+
+lemma no_grids_in_tuple : ∀n,l.tuple_TM n l → no_grids l.
+#n #l * #qin * #cin * #qout * #cout * #mv * * * * * * * * * *
+#_ #_ #Hqin #Hqout #Hcin #Hcout #Hmv #_ #_ #_ #Hl >Hl
+#c #Hc normalize in Hc; cases (memb_append … Hc) -Hc #Hc
+[ @bit_not_grid @(Hqin … Hc)
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈cin,false〉 = true) in Hc; >(\P Hc) @bit_or_null_not_grid //
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
+| cases (memb_append …Hc) -Hc #Hc
+[ @bit_not_grid @(Hqout … Hc)
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈cout,false〉 = true) in Hc; >(\P Hc) @bit_or_null_not_grid //
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
+| >(memb_single … Hc) @bit_or_null_not_grid @Hmv
+]]]]]]
+qed.
+
+lemma no_marks_in_tuple : ∀n,l.tuple_TM n l → no_marks l.
+#n #l * #qin * #cin * #qout * #cout * #mv * * * * * * * * * *
+#Hqin #Hqout #_ #_ #_ #_ #_ #_ #_ #_ #Hl >Hl
+#c #Hc normalize in Hc; cases (memb_append … Hc) -Hc #Hc
+[ @(Hqin … Hc)
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈cin,false〉 = true) in Hc; >(\P Hc) %
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
+| cases (memb_append … Hc) -Hc #Hc
+[ @(Hqout … Hc)
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈cout,false〉 = true) in Hc; >(\P Hc) %
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
+| >(memb_single … Hc) %
+]]]]]]
+qed.
+
+lemma no_grids_in_table: ∀n.∀l.table_TM n l → no_grids l.
+#n #l #t elim t
+ [normalize #c #H destruct
+ |#t1 #t2 #Ht1 #Ht2 #IH lapply (no_grids_in_tuple … Ht1) -Ht1 #Ht1 #x #Hx
+ cases (memb_append … Hx) -Hx #Hx
+ [ @(Ht1 … Hx)
+ | cases (orb_true_l … Hx) -Hx #Hx
+ [ >(\P Hx) %
+ | @(IH … Hx) ] ] ]
qed.
lemma no_marks_in_table: ∀n.∀l.table_TM n l → no_marks l.
#n #l #t elim t
- [normalize #n #x #H destruct
- |#m #t1 #t2 * #qin * #qout * #mv * * * * * *
- #Hmarks #_ #_ #_ #_ #_ #_ #Ht2 #Hind
- #x #Hx cases (memb_append … Hx) -Hx #Hx
- [@Hmarks //
- |cases (orb_true_l … Hx) -Hx #Hx
- [>(\P Hx) //
- |@Hind //
- ]
- ]
- ]
+ [normalize #c #H destruct
+ |#t1 #t2 #Ht1 #Ht2 #IH lapply (no_marks_in_tuple … Ht1) -Ht1 #Ht1 #x #Hx
+ cases (memb_append … Hx) -Hx #Hx
+ [ @(Ht1 … Hx)
+ | cases (orb_true_l … Hx) -Hx #Hx
+ [ >(\P Hx) %
+ | @(IH … Hx) ] ] ]
qed.
-
+axiom last_of_table: ∀n,l,b.¬ table_TM n (l@[〈bar,b〉]).
+
(*
l0 x* a l1 x0* a0 l2 ------> l0 x a* l1 x0 a0* l2
^ ^
]]]]
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) →
∨
(* non facciamo match e marchiamo la prossima tupla *)
((〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
- ∃c2,l5,l6,l7.l4 = l5@〈bar,false〉::〈c2,false〉::l6@〈comma,false〉::l7 ∧
+ ∃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,true〉::l3@〈comma,false〉::
- l5@〈bar,false〉::〈c2,true〉::l6@〈comma,false〉::l7))
+ (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
#A #l #a cases l normalize /2/
qed.
+axiom tech_split2 : ∀A,l1,l2,l3,l4,x.
+ memb A x l1 = false → memb ? x l3 = false →
+ l1@x::l2 = l3@x::l4 → l1 = l3 ∧ l2 = l4.
+
+axiom injective_append : ∀A,l.injective … (λx.append A x l).
+
lemma sem_match_tuple_step:
accRealize ? match_tuple_step (inr … (inl … (inr … 0)))
R_match_tuple_step_true R_match_tuple_step_false.
|#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
cases (Htapee … Htaped ???) -Htaped -Htapee
[* #rs3 * * (* we proceed by cases on rs4 *)
- [(* rs4 is empty *)
- * #d * #b * * * #Heq1 #Hnobars
- whd in ⊢ ((???%)→?); #Htemp destruct (Htemp)
- #Htapee *
- [* #tapef * whd in ⊢ (%→?); >Htapee -Htapee #Htapef
- cases (Htapef … (refl …)) -Htapef #_ #Htapef >Htapef -Htapef
- whd in ⊢ (%→?); #H lapply (H … ???? (refl …)) #Htapeout
- %2 %
- [% [@Hnoteq |@daemon]
- |>Htapeout %
- ]
- |* #tapef * whd in ⊢ (%→?); >Htapee -Htapee #Htapef
- cases (Htapef … (refl …)) whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)
- ]
+ [(* 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 >Hcut <associative_append >Hcut
+ <associative_append #Htable @(absurd … Htable)
+ @last_of_table
|(* rs4 not empty *)
* #d2 #b2 #rs3' * #d * #b * * * #Heq1 #Hnobars
- cut (is_grid d2 = false) [@daemon (* no grids in table *)] #Hd2
+ 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〉 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 *
- [* #tapef * whd in ⊢ (%→?); #Htapef
+ [(* 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 * whd in ⊢ (%→?);
+ * #tapeg >Htapef -Htapef *
+ (* move_l *)
+ whd in ⊢ (%→?);
#H lapply (H … (refl …)) whd in ⊢ (???%→?); -H #Htapeg
- >Htapeg -Htapeg whd in ⊢ (%→?); #Htapeout
+ >Htapeg -Htapeg
+ (* init_current *)
+ whd in ⊢ (%→?); #Htapeout
%1 cases (some_option_hd ? (reverse ? (reverse ? la)) 〈c',true〉)
* #c00 #b00 #Hoption
lapply
[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
- |@daemon
+ |>reverse_append >reverse_cons >reverse_reverse
+ >reverse_append >reverse_reverse
+ >reverse_cons >reverse_append >reverse_reverse
+ >reverse_append >reverse_cons >reverse_reverse
+ >reverse_reverse
+ #Htapeout % [@Hnoteq]
+ @(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@〈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)
+ [|>associative_append normalize
+ >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)))
+ [|>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 %
+ ]
+ ]
]
]
]
cases (Htapef … (refl …)) -Htapef
whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)
]
- |@daemon (* no marks in table *)
+ |(* no marks in table *)
+ #x #membx @(no_marks_in_table … Htable)
+ @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
|(* no grids in table *)
- #x #Hx cases (memb_append … Hx)
- [-Hx #Hx @bit_not_grid @Hl3 cases la in H3; normalize
- [#H3 destruct (H3) @Hx | #y #tl #H3 destruct (H3)
- @memb_append_l2 @memb_cons @Hx ]
- |-Hx #Hx @(no_grids_in_table … Htable)
- @memb_append_l2 @memb_cons @memb_cons @memb_append_l2 @Hx
- ]
- |whd in ⊢ (??%?); >(bit_not_grid … Hd') >(bit_not_bar … Hd') %
+ #x #membx @(no_grids_in_table … Htable)
+ @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_or_null_not_grid … Hd') >(bit_or_null_not_bar … Hd') %
]
]
- |(* no marks in l3 *)
- @daemon
- |(* no marks in l2@[〈bar,false〉] *) @daemon
+ |#x #membx @(no_marks_in_table … Htable)
+ @memb_append_l2 @memb_cons @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 %
]
]
qed.
-
\ No newline at end of file
+
+(*
+ MATCH TUPLE
+
+ scrolls through the tuples in the transition table until one matching the
+ current configuration is found
+*)
+
+definition match_tuple ≝ whileTM ? match_tuple_step (inr … (inl … (inr … 0))).
+
+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) →
+ (* 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).
projects / helm.git / blobdiff