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.
* // 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_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,false〉].
+ 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).
+
+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.
+
+
+lemma generic_match_to_match_in_table :
+ ∀n,T.table_TM n T →
+ ∀qin,cin,qout,cout,mv.|qin| = n → |qout| = n →
+ ∀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
+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 #Hqinbits #Hqoutbits #Hcin0 #Hcout0 #Hmv0 #Hcout0mv0
+ #Hlenqin0 #Hlenqout0 #Htuple #Htable0 #IH #t1 #t2 #HT
+ cases t1 in HT;
+ [ >Htuple normalize in ⊢ (??%%→?);
+ >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
+ destruct (HT)
+
lemma no_grids_in_table: ∀n.∀l.table_TM n l → no_grids l.
#n #l #t elim t
qed.
definition init_current_on_match ≝
- seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
- (seq ? (move_l ?)
- (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
- (seq ? (move_r ?) (mark ?)))).
+ (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,l3,d,rs. no_grids l1 → no_grids l2 → is_grid c = false → is_grid (\fst d) = false →
- t1 = midtape STape (l1@〈grid,false〉::l2@〈c,false〉::〈grid,false〉::l3) d rs →
- t2 = midtape STape (〈grid,false〉::l3) 〈c,true〉
- ((reverse ? (l1@〈grid,false〉::l2)@d::rs)).
+ ∀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_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
- (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)
+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 #l3 #d #rs #Hl1 #Hl2 #Hc #Hd #Hintape
-cases HR -HR #ta * whd in ⊢ (%→?); #Hta cases (Hta … Hintape) -Hta -Hintape
- [ * >Hd #Hfalse normalize in Hfalse; destruct (Hfalse) ]
-* #_ #Hta lapply (Hta l1 〈grid,false〉 ? (refl ??) (refl …) Hl1) -Hta #Hta
-* #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta) -Htb -Hta
-generalize in match Hl2; cases l2
- [#Hl2 whd in ⊢ ((???(??%%%))→?); #Htb
- * #tc * whd in ⊢ (%→?); #Htc cases (Htc … Htb) -Htb
+#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 …) Hl2)
+ * #_ #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 >reverse_append %
- |#d #tl #Htl whd in ⊢ ((???(??%%%))→?); #Htb
- * #tc * whd in ⊢ (%→?); #Htc cases (Htc … Htb) -Htc
+ >Houtc %
+ |#d #tl #Htl whd in ⊢ ((???(??%%%))→?); #Hta
+ * #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Htb
[* >(Htl … (memb_hd …)) #Htemp destruct (Htemp)]
- * #Hd >append_cons #Htc lapply (Htc … (refl ??) (refl …) ?)
+ * #Hd >append_cons #Htb lapply (Htb … (refl ??) (refl …) ?)
[#x #membx cases (memb_append … membx) -membx #membx
- [@Htl @memb_cons @membx | >(memb_single … membx) @Hc]] #Htc
- * #td * whd in ⊢ (%→?); #Htd lapply (Htd … Htc) -Htc -Htd
- >reverse_append >associative_append whd in ⊢ ((???(??%%%))→?); #Htd
- whd in ⊢ (%→?); #Houtc lapply (Houtc … Htd) -Houtc #Houtc
- >Houtc >reverse_append >reverse_cons >reverse_cons
- >associative_append >associative_append >associative_append %
+ [@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.
(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 ∧
+ 〈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))
+ (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).
+ 〈c1,false〉::l2 ≠ l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4).