definition STape ≝ FinProd … FSUnialpha FinBool.
definition only_bits ≝ λl.
- ∀c.memb STape c l = true → is_bit (\fst c) = true.
+ ∀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.
qed.
definition mk_tuple ≝ λqin,cin,qout,cout,mv.
- qin @ cin :: 〈comma,false〉:: qout @ cout :: 〈comma,false〉 :: [mv].
-
+ 〈bar,false〉 :: 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,cin,qout,cout,mv.
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).
+| ttm_cons : ∀t1,T.tuple_TM n t1 → table_TM n T → table_TM n (t1@T).
-inductive match_in_table (qin:list STape) (cin: STape)
+inductive match_in_table (n:nat) (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)
+ tuple_TM n (mk_tuple qin cin qout cout mv) →
+ match_in_table n qin cin qout cout mv
+ (mk_tuple qin cin qout cout mv @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).
+ tuple_TM n (mk_tuple qin0 cin0 qout0 cout0 mv0) →
+ match_in_table n qin cin qout cout mv tb →
+ match_in_table n qin cin qout cout mv
+ (mk_tuple qin0 cin0 qout0 cout0 mv0@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 tuple_len : ∀n,t.tuple_TM n t → |t| = 2*n+6.
axiom append_eq_tech1 :
- ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → |l1| < |l3| → ∃la:list A.l1@a::la = l3.
+ ∀A,l1,l2,l3,l4.l1@l2 = l3@l4 → |l1| < |l3| → ∃la:list A.l1@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.
+ ∃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 :
+(*axiom list_decompose_memb :
∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → |l1| < |l3| → memb A a l3 = true.*)
+lemma table_invert_r : ∀n,t,T.
+ tuple_TM n t → table_TM n (t@T) → table_TM n T.
+#n #t #T #Htuple #Htable inversion Htable
+[ cases Htuple #qin * #cin * #qout * #cout * #mv * #_ #Ht >Ht
+ normalize #Hfalse destruct (Hfalse)
+| #t0 #T0 #Htuple0 #Htable0 #_ #Heq
+ lapply (append_l2_injective ?????? Heq)
+ [ >(tuple_len … Htuple) >(tuple_len … Htuple0) % ]
+ -Heq #Heq destruct (Heq) // ]
+qed.
+
+lemma match_in_table_to_tuple :
+ ∀n,T,qin,cin,qout,cout,mv.
+ match_in_table n qin cin qout cout mv T → table_TM n T →
+ tuple_TM n (mk_tuple qin cin qout cout mv).
+#n #T #qin #cin #qout #cout #mv #Hmatch elim Hmatch
+[ //
+| #qin0 #cin0 #qout0 #cout0 #mv0 #tb #Htuple #Hmatch #IH #Htable
+ @IH @(table_invert_r ???? Htable) @Htuple
+]
+qed.
+
+lemma match_in_table_append :
+ ∀n,T,qin,cin,qout,cout,mv,t.
+ tuple_TM n t →
+ match_in_table n qin cin qout cout mv (t@T) →
+ t = mk_tuple qin cin qout cout mv ∨ match_in_table n qin cin qout cout mv T.
+#n #T #qin #cin #qout #cout #mv #t #Ht #Hmatch inversion Hmatch
+[ #T0 #H #H1 % >(append_l1_injective … H1) //
+ >(tuple_len … Ht) >(tuple_len … H) %
+| #qin0 #cin0 #qout0 #cout0 #mv0 #T0 #H #H1 #_ #H2 %2
+ >(append_l2_injective … H2) // >(tuple_len … Ht) >(tuple_len … H) %
+]
+qed.
+
+lemma generic_match_to_match_in_table_tech :
+ ∀n,t,T0,T1,T2.tuple_TM n t → table_TM n (T1@〈bar,false〉::T2) →
+ t@T0 = T1@〈bar,false〉::T2 → T1 = [] ∨ ∃T3.T1 = t@T3.
+#n #t #T0 #T1 #T2 #Ht cases T1
+[ #_ #_ % %
+| normalize #c #T1c #Htable #Heq %2
+ cases Ht in Heq; #qin * #cin * #qout * #cout * #mv **********
+ #Hqin1 #Hqout1 #Hqin2 #Hqout2 #Hcin #Hcout #Hmv #Hcoutmv #Hqinlen #Hqoutlen
+ #Heqt >Heqt whd in ⊢ (??%%→?); #Ht lapply (cons_injective_r ????? Ht)
+ #Ht' cases (list_decompose_r STape … (sym_eq … Ht') ?)
+ [ #la * #lb * #HT1c #HT0 %{lb} >HT1c @(eq_f2 ??? (append ?) (c::la)) //
+ >HT0 in Ht'; >HT1c >associative_append in ⊢ (???%→?); #Ht'
+ <(append_l1_injective_r … Ht') // <(cons_injective_l ????? Ht) %
+ |@(noteq_to_eqnot ? true) @(not_to_not … not_eq_true_false) #Hbar @sym_eq
+ cases (memb_append … Hbar) -Hbar #Hbar
+ [@(Hqin2 … Hbar)
+ |cases (orb_true_l … Hbar) -Hbar
+ [#Hbar lapply (\P Hbar) -Hbar #Hbar destruct (Hbar) @Hcin
+ |whd in ⊢ ((??%?)→?); #Hbar cases (memb_append … Hbar) -Hbar #Hbar
+ [@(Hqout2 … Hbar)
+ |cases (orb_true_l … Hbar) -Hbar
+ [#Hbar lapply (\P Hbar) -Hbar #Hbar destruct (Hbar) @Hcout
+ |#Hbar cases (orb_true_l … Hbar) -Hbar
+ [whd in ⊢ ((??%?)→?); #Hbar @Hbar
+ |#Hbar lapply (memb_single … Hbar) -Hbar #Hbar destruct (Hbar) @Hmv
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+qed.
+
lemma generic_match_to_match_in_table :
∀n,T.table_TM n T →
∀qin,cin,qout,cout,mv.|qin| = n → |qout| = n →
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.
+ T = (t1@〈bar,false〉::qin@cin::〈comma,false〉::qout@cout::〈comma,false〉::[mv])@t2 →
+ match_in_table n 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
+[ * [ #t2 normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
+ | #c0 #t1 #t2 normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
+| #tuple #T0 #H1 #Htable0#IH #t1 #t2 #HT cases H1 #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 ⊢ (??%%→?);
+ #Hlenqin0 #Hlenqout0 #Htuple
+ lapply (generic_match_to_match_in_table_tech n ? T0 t1
+ (qin@cin::〈comma,false〉::qout@[cout;〈comma,false〉;mv]@t2) H1) #Htmp
+ >Htuple in H1; #H1
+ lapply (ttm_cons … T0 H1 Htable0) <Htuple in ⊢ (%→?); >HT
+ >associative_append normalize >associative_append normalize
+ >associative_append #Htable cases (Htmp Htable ?)
+ [ #Ht1 >Htuple in HT; >Ht1 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 ]
+ (〈cout0,false〉 = cout ∧ (〈mv0,false〉 = mv ∧ T0 = t2)))))
+ [ lapply (cons_injective_r ????? HT) -HT #HT
+ 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_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)
- ]
+ >(?:〈bar,false〉::qin0@(〈cin0,false〉::〈comma,false〉::qout0@
+ [〈cout0,false〉;〈comma,false〉;〈mv0,false〉])@T0 = tuple@T0)
+ [ >Htuple >Hqin >Hqout >Hcin >Hcout >Hmv % //
+ | >Htuple normalize >associative_append normalize >associative_append
+ normalize >associative_append % ]
+ | * #T3 #HT3 >HT3 in HT; >associative_append; >associative_append #HT
+ lapply (append_l2_injective … HT) // -HT #HT %2 //
+ @(IH T3 t2) >HT >associative_append %
+ |>HT >associative_append normalize >associative_append normalize
+ >associative_append % ]
]
qed.
+(*
+lemma table_invert_l : ∀n,T0,qin,cin,qout,cout,mv.
+ table_TM n (mk_tuple qin cin qout cout mv@〈bar,false〉::T0) →
+ tuple_TM n (mk_tuple qin cin qout cout mv).
+#n #T #qin #cin #qout #cout #mv #HT inversion HT
+[ change with (append ???) in ⊢ (??(??%?)?→?);cases qin [ #Hfalse | #t0 #ts0 #Hfalse] normalize in Hfalse; destruct (Hfalse)
+| #t0 #T0 #Ht0 #HT0 #_
+
+
+lemma table_invert_r : ∀n,T0,qin,cin,qout,cout,mv.
+ table n (mk_tuple qin cin qout cout mv@〈bar,false〉::T0) → table n T0.
+*)
+
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
+#c #Hc cases (orb_true_l … Hc) -Hc #Hc
+[ >(\P 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 //
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
+#c #Hc cases (orb_true_l … Hc) -Hc #Hc
+[ >(\P 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) %
|#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) ] ] ]
+ | @(IH … Hx) ] ]
qed.
lemma no_marks_in_table: ∀n.∀l.table_TM n l → no_marks l.
|#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) ] ] ]
+ | @(IH … Hx) ] ]
qed.
axiom last_of_table: ∀n,l,b.¬ table_TM n (l@[〈bar,b〉]).
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.
∨
(no_bars rs1 ∧ t2 = midtape ? (reverse ? rs1@c::ls) 〈grid,false〉 rs2).
+axiom daemon :∀P:Prop.P.
+
axiom tech_split :
∀A:DeqSet.∀f,l.
(∀x.memb A x l = true → f x = false) ∨
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〉 ?)
]
qed.
*)
+
definition init_current ≝
seq ? (adv_to_mark_l ? (is_marked ?))
(seq ? (clear_mark ?)
(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:%]
- #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 ?)
+|#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 % [%]
+ [* #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
[@(not_to_not …H1) normalize #H destruct %
|#x #tl @not_to_not normalize #H destruct //
]
- ] #Hnoteq %2
+ ] #Hnoteq
cut (bit_or_null d' = true)
[cases la in H3;
[normalize in ⊢ (%→?); #H destruct //
* #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
>Htapeg -Htapeg
(* init_current *)
whd in ⊢ (%→?); #Htapeout
- %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))
- c' (reverse ? la) false ls bar (〈d2,true〉::rs3'@〈grid,false〉::rs) c00 b00 ?????) -Htapeout
- [whd in ⊢ (??(??%??)?); @eq_f3 [2:%|3: %]
- >associative_append
- generalize in match (〈c',true〉::reverse ? la@〈grid,false〉::ls); #l
- whd in ⊢ (???(???%)); >associative_append >associative_append %
- |>reverse_cons @Hoption
- |cases la in H2;
- [normalize in ⊢ (%→?); #Htemp destruct (Htemp)
- @injective_notb @notgridc
- |#x #tl normalize in ⊢ (%→?); #Htemp destruct (Htemp)
- @bit_or_null_not_grid @(Hl1bars 〈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 |@Hl1bars]
- |#Hl1' #x #Hx @bit_or_null_not_grid @Hl1'
- @memb_append_l1 @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
+ 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
+ [(* by Hoption, H2 *) @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
>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)))
+ >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 >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 % ]
+ <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
- >(Htapeout … (refl …)) %
- [ %
- [ @Hnoteq
- | whd #x #Hx @Hnobars @memb_append_l2 @memb_cons //
- ]
- | %
- ]
+ 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 @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.
-
(*
MATCH TUPLE
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.
+
+(* possible variante ?
+definition weakR_match_tuple ≝ λt1,t2.
+ (∀ls,cur,rs,b. t1 = midtape STape ls 〈grid,b〉 rs → t2 = t1) ∧
+ (∀c,l1,c1,l2,l3,ls0,rs0,n.
+ t1 = midtape STape (〈grid,false〉::ls0) 〈bit c,true〉 rs
+ (l1@〈grid,false〉::l2@〈bit c1,true〉::l3@〈grid,false〉::rs0) →
+ only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
+ table_TM (S n) (l2@〈c1,false〉::l3) →
+ (* facciamo match *)
+ (∃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〉::ls0) 〈grid,false〉
+ (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〉 ∧
+ ∀l4,newc,mv,l5.
+ 〈c1,false〉::l3 ≠ l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)).
+*)
+
+definition R_match_tuple0 ≝ λt1,t2.
+ ∀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@〈bar,false〉::〈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) (l2@〈bar,false〉::〈c1,false〉::l3) →
+ (* facciamo match *)
+ (∃l4,newc,mv,l5.
+ 〈bar,false〉::〈c1,false〉::l3 = l4@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5 ∧
+ t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
+ (l2@l4@〈bar,false〉::〈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〉 ∧
+ ∀l4,newc,mv,l5.
+ 〈bar,false〉::〈c1,false〉::l3 ≠ l4@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)).
+
+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 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)
+[ #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)
+ ]
+| (* 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 %
+ [ (* 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
+ >Hl3 in Htable; >append_cons #Htable
+ >(?: l2@〈bar,false〉::〈c1,true〉::l3@〈grid,false〉::rs0
+ = (l2@[〈bar,false〉])@〈c1,true〉::la@〈comma,false〉::(lb@〈comma,false〉::mv::
+ lc)@〈grid,false〉::rs0) in Hrs;
+ [| >associative_append normalize >Hl3
+ >associative_append normalize % ] #Hrs
+ cases (Htc ????????? Hl1bitnull Hl1marks ?? Hlabitnull Hl1len ? Htable Hls Hcur Hrs)
+ [5: <Hl1len @Hlalen
+ |4: whd in ⊢ (??%?); >Hc1 %
+ |3: whd in ⊢ (??%?); >Hc %
+ |-Htc *
+ [ (* 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 %
+ ]
+ | (* 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 >Heqlblc
+ >associative_append normalize //
+ | @Hl1len
+ | @Hl1marks
+ | @Hl1bitnull
+ | (*???*) @daemon
+ | @Hc
+ | >associative_append normalize
+ >associative_append normalize
+ >associative_append %
+ |-IH *
+ [ (* 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 %
+ | >Houtc @eq_f >associative_append normalize
+ >associative_append normalize >associative_append
+ normalize >associative_append %
+ ]
+ | (* the while fails finding a tuple: there are no matches in the whole table *)
+ * #Houtc #Hdiff1 %2 %
+ [ @Houtc
+ | #l50 #newc #mv0 #l51 >Heqlblc
+ @daemon
+ ]
+ ]
+ ]
+ ]
+ | (* 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
+ (* 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 → only_bits_or_nulls l1 → is_bit c1 = true → n = |l1| →
- table_TM (S n) (〈c1,false〉::l2) →
+ 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〉::〈c1,true〉::l2@〈grid,false〉::rs) →
+ (l1@〈grid,false〉::〈bar,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 ∧
+ 〈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@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l4@〈grid,false〉::rs))
+ (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.
- 〈c1,false〉::l2 ≠ l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::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
projects / helm.git / blobdiff