definition no_nulls ≝
λl:list STape.∀x.memb ? x l = true → is_null (\fst x) = false.
+definition current_of_alpha ≝ λc:STape.
+ match \fst c with [ null ⇒ None ? | _ ⇒ Some ? c ].
+
+definition legal_tape ≝ λls,c,rs.
+ let t ≝ mk_tape STape ls (current_of_alpha c) rs in
+ left ? t = ls ∧ right ? t = rs ∧ current ? t = current_of_alpha c.
+
+lemma legal_tape_cases :
+ ∀ls,c,rs.legal_tape ls c rs →
+ \fst c ≠ null ∨ (\fst c = null ∧ (ls = [] ∨ rs = [])).
+#ls #c #rs cases c #c0 #bc0 cases c0
+[ #c1 normalize #_ % % #Hfalse destruct (Hfalse)
+| cases ls
+ [ #_ %2 % // % %
+ | #l0 #ls0 cases rs
+ [ #_ %2 % // %2 %
+ | #r0 #rs0 normalize * * #_ #Hrs destruct (Hrs) ]
+ ]
+|*: #_ % % #Hfalse destruct (Hfalse) ]
+qed.
+
+axiom legal_tape_conditions :
+ ∀ls,c,rs.(\fst c ≠ null ∨ ls = [] ∨ rs = []) → legal_tape ls c rs.
+(*#ls #c #rs *
+[ *
+ [ >(eq_pair_fst_snd ?? c) cases (\fst c)
+ [ #c0 #Hc % % %
+ | * #Hfalse @False_ind /2/
+ |*: #Hc % % %
+ ]
+ | cases ls [ * #Hfalse @False_ind /2/ ]
+ #l0 #ls0
+
+ #Hc
+*)
+
definition R_move_tape_r_abstract ≝ λt1,t2.
∀rs,n,table,curc,curconfig,ls.
bit_or_null curc = true → only_bits_or_nulls curconfig → table_TM n (reverse ? table) →
t1 = midtape STape (table@〈grid,false〉::〈curc,false〉::curconfig@〈grid,false〉::ls)
〈grid,false〉 rs →
no_nulls rs → no_marks rs →
+ legal_tape ls 〈curc,false〉 rs →
∀t1'.t1' = lift_tape ls 〈curc,false〉 rs →
∃ls1,rs1,newc.
(t2 = midtape STape ls1 〈grid,false〉 (reverse ? curconfig@〈newc,false〉::
〈grid,false〉::reverse ? table@〈grid,false〉::rs1) ∧
lift_tape ls1 〈newc,false〉 rs1 =
- tape_move_right STape ls 〈curc,false〉 rs).
+ tape_move_right STape ls 〈curc,false〉 rs ∧ legal_tape ls1 〈newc,false〉 rs1).
lemma lift_tape_not_null :
∀ls,c,rs. is_null (\fst c) = false →
∀t1,t2.R_move_tape_r t1 t2 → R_move_tape_r_abstract t1 t2.
#t1 #t2 whd in ⊢(%→?); #Hconcrete
#rs #n #table #curc #curconfig #ls #Hcurc #Hcurconfig #Htable #Ht1
-#Hrsnonulls #Hrsnomarks #t1' #Ht1'
+#Hrsnonulls #Hrsnomarks #Htape #t1' #Ht1'
cases (Hconcrete … Htable Ht1) //
[ * #Hrs #Ht2
@(ex_intro ?? (〈curc,false〉::ls)) @(ex_intro ?? [])
@(ex_intro ?? null) %
- [ >Ht2 %
- | >Hrs % ]
+ [ %
+ [ >Ht2 %
+ | >Hrs % ]
+ | % % % ]
| * * #r0 #br0 * #rs0 * #Hrs
cut (br0 = false) [@(Hrsnomarks 〈r0,br0〉) >Hrs @memb_hd]
#Hbr0 >Hbr0 in Hrs; #Hrs #Ht2
@(ex_intro ?? (〈curc,false〉::ls)) @(ex_intro ?? rs0)
@(ex_intro ?? r0) %
- [ >Ht2 //
- | >Hrs >lift_tape_not_null
- [ %
- | @(Hrsnonulls 〈r0,false〉) >Hrs @memb_hd ] ]
+ [ %
+ [ >Ht2 //
+ | >Hrs >lift_tape_not_null
+ [ %
+ | @(Hrsnonulls 〈r0,false〉) >Hrs @memb_hd ] ]
+ | @legal_tape_conditions % % % #Hr0 >Hr0 in Hrs; #Hrs
+ lapply (Hrsnonulls 〈null,false〉 ?)
+ [ >Hrs @memb_hd | normalize #H destruct (H) ]
+ ]
+]
qed.
definition R_move_tape_l_abstract ≝ λt1,t2.
t1 = midtape STape (table@〈grid,false〉::〈curc,false〉::curconfig@〈grid,false〉::ls)
〈grid,false〉 rs →
no_nulls ls → no_marks ls →
+ legal_tape ls 〈curc,false〉 rs →
∀t1'.t1' = lift_tape ls 〈curc,false〉 rs →
∃ls1,rs1,newc.
(t2 = midtape STape ls1 〈grid,false〉 (reverse ? curconfig@〈newc,false〉::
〈grid,false〉::reverse ? table@〈grid,false〉::rs1) ∧
lift_tape ls1 〈newc,false〉 rs1 =
- tape_move_left STape ls 〈curc,false〉 rs).
+ tape_move_left STape ls 〈curc,false〉 rs ∧ legal_tape ls1 〈newc,false〉 rs1).
lemma mtl_concrete_to_abstract :
∀t1,t2.R_move_tape_l t1 t2 → R_move_tape_l_abstract t1 t2.
#t1 #t2 whd in ⊢(%→?); #Hconcrete
#rs #n #table #curc #curconfig #ls #Hcurc #Hcurconfig #Htable #Ht1
-#Hlsnonulls #Hlsnomarks #t1' #Ht1'
+#Hlsnonulls #Hlsnomarks #Htape #t1' #Ht1'
cases (Hconcrete … Htable Ht1) //
[ * #Hls #Ht2
@(ex_intro ?? [])
@(ex_intro ?? (〈curc,false〉::rs))
@(ex_intro ?? null) %
- [ >Ht2 %
- | >Hls % ]
+ [ %
+ [ >Ht2 %
+ | >Hls % ]
+ | % % % ]
| * * #l0 #bl0 * #ls0 * #Hls
cut (bl0 = false) [@(Hlsnomarks 〈l0,bl0〉) >Hls @memb_hd]
#Hbl0 >Hbl0 in Hls; #Hls #Ht2
@(ex_intro ?? ls0)
@(ex_intro ?? (〈curc,false〉::rs))
@(ex_intro ?? l0) %
- [ >Ht2 %
- | >Hls >lift_tape_not_null
- [ %
- | @(Hlsnonulls 〈l0,false〉) >Hls @memb_hd ] ]
+ [ %
+ [ >Ht2 %
+ | >Hls >lift_tape_not_null
+ [ %
+ | @(Hlsnonulls 〈l0,false〉) >Hls @memb_hd ] ]
+ | @legal_tape_conditions % % % #Hl0 >Hl0 in Hls; #Hls
+ lapply (Hlsnonulls 〈null,false〉 ?)
+ [ >Hls @memb_hd | normalize #H destruct (H) ]
+ ]
qed.
lemma Realize_to_Realize :
t1 = midtape STape (table1@〈grid,false〉::〈curc,false〉::curconfig@〈grid,false〉::ls)
〈c,false〉 (table2@〈grid,false〉::rs) →
no_nulls ls → no_nulls rs → no_marks ls → no_marks rs →
+ legal_tape ls 〈curc,false〉 rs →
∀t1'.t1' = lift_tape ls 〈curc,false〉 rs →
∃ls1,rs1,newc.
+ legal_tape ls1 〈newc,false〉 rs1 ∧
(t2 = midtape STape ls1 〈grid,false〉 (reverse ? curconfig@〈newc,false〉::
〈grid,false〉::reverse ? table1@〈c,false〉::table2@〈grid,false〉::rs1) ∧
((c = bit false ∧ lift_tape ls1 〈newc,false〉 rs1 = tape_move_left STape ls 〈curc,false〉 rs) ∨
#k * #outc * #Hloop #HR
@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
#rs #n #table1 #c #table2 #curc #curconfig #ls
-#Hcurc #Hc #Hcurconfig #Htable #Hintape #Hls #Hrs #Hls1 #Hrs1 #t1' #Ht1'
+#Hcurc #Hc #Hcurconfig #Htable #Hintape #Hls #Hrs #Hls1 #Hrs1 #Htape #t1' #Ht1'
generalize in match HR; -HR *
[ * #ta * whd in ⊢ (%→?); #Hta cases (Hta 〈c,false〉 ?)
[| >Hintape % ] -Hta #Hceq #Hta lapply (\P Hceq) -Hceq #Hceq destruct (Hta Hceq)
[ @daemon ] -Htb >append_cons <associative_append #Htb
whd in ⊢ (%→?); #Houtc lapply (Houtc … Htb … Ht1') //
[ >reverse_append >reverse_append >reverse_reverse @Htable |]
- -Houtc -Htb * #ls1 * #rs1 * #newc * #Houtc #Hnewtape
+ -Houtc -Htb * #ls1 * #rs1 * #newc * * #Houtc #Hnewtape #Hnewtapelegal
@(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? newc) %
- [ >Houtc >reverse_append >reverse_append >reverse_reverse
- >associative_append >associative_append %
- | % % % // ]
+ [ //
+ | %
+ [ >Houtc >reverse_append >reverse_append >reverse_reverse
+ >associative_append >associative_append %
+ | % % % // ]
+ ]
| * #ta * whd in ⊢ (%→?); #Hta cases (Hta 〈c,false〉 ?)
[| >Hintape % ] -Hta #Hcneq cut (c ≠ bit false)
[ lapply (\Pf Hcneq) @not_to_not #Heq >Heq % ] -Hcneq #Hcneq #Hta destruct (Hta)
[ @daemon ] -Htc >append_cons <associative_append #Htc
whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc … Ht1') //
[ >reverse_append >reverse_append >reverse_reverse @Htable |]
- -Houtc -Htc * #ls1 * #rs1 * #newc * #Houtc #Hnewtape
+ -Houtc -Htc * #ls1 * #rs1 * #newc * * #Houtc #Hnewtape #Hnewtapelegal
@(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? newc) %
- [ >Houtc >reverse_append >reverse_append >reverse_reverse
- >associative_append >associative_append %
- | % %2 % // ]
+ [ //
+ | %
+ [ >Houtc >reverse_append >reverse_append >reverse_reverse
+ >associative_append >associative_append %
+ | % %2 % // ]
+ ]
| * #tb * whd in ⊢ (%→?); #Htb cases (Htb 〈c,false〉 ?)
[| >Hintape % ] -Htb #Hcneq' cut (c ≠ bit true)
[ lapply (\Pf Hcneq') @not_to_not #Heq >Heq % ] -Hcneq' #Hcneq' #Htb destruct (Htb)
[ * >(bit_or_null_not_grid … Hcurc) #Hfalse destruct (Hfalse) ]
* #_ #Houtc lapply (Houtc … (refl ??) (refl ??) ?) [@daemon] -Houtc #Houtc
@(ex_intro ?? ls) @(ex_intro ?? rs) @(ex_intro ?? curc) %
- [ @Houtc
- | %2 % // % // % //
- generalize in match Hcneq; generalize in match Hcneq';
- cases c in Hc; normalize //
- [ * #_ normalize [ #Hfalse @False_ind cases Hfalse /2/ | #_ #Hfalse @False_ind cases Hfalse /2/ ]
- |*: #Hfalse destruct (Hfalse) ]
+ [ //
+ | %
+ [ @Houtc
+ | %2 % // % // % //
+ generalize in match Hcneq; generalize in match Hcneq';
+ cases c in Hc; normalize //
+ [ * #_ normalize [ #Hfalse @False_ind cases Hfalse /2/ | #_ #Hfalse @False_ind cases Hfalse /2/ ]
+ |*: #Hfalse destruct (Hfalse) ]
+ ]
]
]
]
definition map_move ≝
λc,mv.match c with [ null ⇒ None ? | _ ⇒ Some ? 〈c,false,move_of_unialpha mv〉 ].
-definition current_of_alpha ≝ λc:STape.
- match \fst c with [ null ⇒ None ? | _ ⇒ Some ? c ].
-
-definition legal_tape ≝ λls,c,rs.
- let t ≝ mk_tape STape ls (current_of_alpha c) rs in
- left ? t = ls ∧ right ? t = rs ∧ current ? t = current_of_alpha c.
-
-lemma legal_tape_cases :
- ∀ls,c,rs.legal_tape ls c rs →
- \fst c ≠ null ∨ (\fst c = null ∧ ls = []) ∨ (\fst c = null ∧ rs = []).
-#ls #c #rs cases c #c0 #bc0 cases c0
-[ #c1 normalize #_ % % % #Hfalse destruct (Hfalse)
-| cases ls
- [ #_ % %2 % %
- | #l0 #ls0 cases rs
- [ #_ %2 % %
- | #r0 #rs0 normalize * * #Hls #Hrs destruct (Hrs) ]
- ]
-|*: #_ % % % #Hfalse destruct (Hfalse) ]
-qed.
-
+(* - aggiungere a legal_tape le condizioni
+ only_bits ls, rs; bit_or_null c
+ - ci vuole un lemma che dimostri
+ bit_or_null c1 = true bit_or_null mv = true
+ mv ≠ null → c1 ≠ null
+ dal fatto che c1 e mv sono contenuti nella table
+ *)
definition R_exec_move ≝ λt1,t2.
∀n,curconfig,ls,rs,c0,c1,s0,s1,table1,newconfig,mv,table2.
table_TM n (table1@〈comma,false〉::〈s1,false〉::newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2) →
no_marks curconfig → only_bits (curconfig@[〈s0,false〉]) → only_bits (〈s1,false〉::newconfig) →
+ |curconfig| = |newconfig| →
no_nulls ls → no_nulls rs → no_marks ls → no_marks rs →
legal_tape ls 〈c0,false〉 rs →
t1 = midtape STape (〈c0,false〉::curconfig@〈s0,false〉::〈grid,false〉::ls) 〈grid,false〉
(〈s1,false〉::newconfig@〈c2,false〉::〈grid,false〉::
table1@〈comma,false〉::〈s1,false〉::newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs1) ∧
lift_tape ls1 〈c2,false〉 rs1 =
- tape_move STape t1' (map_move c1 mv).
+ tape_move STape t1' (map_move c1 mv) ∧ legal_tape ls1 〈c2,false〉 rs1.
(* move the following 2 lemmata to mono.ma *)
lemma tape_move_left_eq :
//
qed.
+lemma lift_tape_not_null :
+ ∀ls,c,bc,rs.c ≠ null → lift_tape ls 〈c,bc〉 rs = midtape ? ls 〈c,bc〉 rs.
+#ls #c #bc #rs cases c //
+#Hfalse @False_ind /2/
+qed.
+
+lemma merge_char_not_null :
+ ∀c1,c2.c1 ≠ null → merge_char c1 c2 ≠ null.
+#c1 #c2 @not_to_not cases c2
+[ #c1' normalize #Hfalse destruct (Hfalse)
+| normalize //
+| *: normalize #Hfalse destruct (Hfalse)
+]
+qed.
+
lemma sem_exec_move : Realize ? exec_move R_exec_move.
#intape
cases (sem_seq … sem_init_copy
#k * #outc * #Hloop #HR
@(ex_intro ?? k) @(ex_intro ?? outc) % [ @Hloop ] -Hloop
#n #curconfig #ls #rs #c0 #c1 #s0 #s1 #table1 #newconfig #mv #table2
-#Htable #Hcurconfig1 #Hcurconfig2 #Hnewconfig #Hls #Hrs #Hls1 #Hrs1 #Htape #Hintape #t1' #Ht1'
+#Htable #Hcurconfig1 #Hcurconfig2 #Hnewconfig #Hlen #Hls #Hrs #Hls1 #Hrs1 #Htape #Hintape #t1' #Ht1'
cases HR -HR #ta * whd in ⊢ (%→?); #Hta
lapply (Hta (〈c0,false〉::curconfig) table1 s0 ls s1
(newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs) … Hintape) -Hta
[ (*Hcurconfig2*) @daemon
| (*Htable*) @daemon
-| (*FIXME*) @daemon
-| (*FIXME*) @daemon
+| (*bit_or_null c0 = true *) @daemon
+| (*Hcurconfig1*) @daemon
| #Hta * #tb * whd in ⊢ (%→?); #Htb
lapply (Htb (〈grid,false〉::ls) s0 s1 c0 c1 (〈mv,false〉::table2@〈grid,false〉::rs) newconfig (〈comma,false〉::reverse ? table1) curconfig Hta ????????) -Htb
- [9:|*:(* rivedere le precondizioni *) @daemon ]
+ [9:|*:(* bit_or_null c0,c1; |curconfig| = |newconfig|*) @daemon ]
#Htb * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc whd in ⊢(???(??%%%)→?);#Htc
whd in ⊢ (%→?); #Houtc whd in Htc:(???%); whd in Htc:(???(??%%%));
lapply (Houtc rs n
(〈comma,false〉::〈c1,false〉::reverse ? newconfig@〈s1,false〉::〈comma,false〉::reverse ? table1)
mv table2 (merge_char c0 c1) (reverse ? newconfig@[〈s1,false〉]) ls ?????
- Hls Hrs Hls1 Hrs1 ??)
- [3: >Htc @(eq_f3 … (midtape ?))
+ Hls Hrs Hls1 Hrs1 ???)
+ [3: cases (true_or_false (c0 == null)) #Hc0
+ [ >(\P Hc0) in Htape; #Htape cases (legal_tape_cases … Htape)
+ [ * #Hfalse @False_ind /2/
+ | * #_ *
+ [ #Hlsnil @legal_tape_conditions % %2 //
+ | #Hrsnil @legal_tape_conditions %2 //
+ ]
+ ]
+ | @legal_tape_conditions % % @merge_char_not_null @(\Pf Hc0) ]
+ |4:>Htc @(eq_f3 … (midtape ?))
[ @eq_f @eq_f >associative_append >associative_append %
| %
| % ]
>associative_append >associative_append >associative_append
>associative_append >associative_append
@Htable
- | (* only bits or nulls c1,c2 → only bits or nulls (merge c1 c2) *) @daemon
- | (* add to hyps? *) @daemon
+ | (* Hnewconfig *) @daemon
+ | (* bit_or_null mv = true *) @daemon
| (* bit_or_null c1,c2 → bit_or_null (merge_char c1 c2) *) @daemon
- | -Houtc * #ls1 * #rs1 * #newc * #Houtc *
+ | -Houtc * #ls1 * #rs1 * #newc * #Hnewtapelegal * #Houtc *
[ *
[ * #Hmv #Htapemove
@(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? newc)
%
- [ >Houtc -Houtc >reverse_append
- >reverse_reverse >reverse_single @eq_f
- >reverse_cons >reverse_cons >reverse_append >reverse_cons
- >reverse_cons >reverse_reverse >reverse_reverse
- >associative_append >associative_append
- >associative_append >associative_append
- >associative_append >associative_append %
- | >Hmv >Ht1' >Htapemove
- (* mv = bit false -→ c1 = bit ?
- whd in Htapemove:(???%); whd in ⊢ (???%);
- whd in match (lift_tape ???) in ⊢ (???%);
- >Htapemove *)
- cut (∃c1'.c1 = bit c1') [ @daemon ] * #c1' #Hc1
- >Hc1 >tape_move_left_eq cases c0
- [ #c0' %
- | cases ls
- [ cases rs
- [ %
- | #r0 #rs0 % ]
- | #l0 #ls0 cases rs
- [ %
- | #r0 #rs0
-
- ls #... null # ... # rs
- ls #... c # ... # rs
-
- ∃t.left ? t = ls, right ? t = rs, current t = [[ c ]]
-
- % ]
-
-
- @eq_f3
- [
-
- whd in match (merge_char ??);
- whd in match (map_move ??);
-
+ [ %
+ [ >Houtc -Houtc >reverse_append
+ >reverse_reverse >reverse_single @eq_f
+ >reverse_cons >reverse_cons >reverse_append >reverse_cons
+ >reverse_cons >reverse_reverse >reverse_reverse
+ >associative_append >associative_append
+ >associative_append >associative_append
+ >associative_append >associative_append %
+ | >Hmv >Ht1' >Htapemove
+ (* mv = bit false -→ c1 = bit ? *)
+ cut (∃c1'.c1 = bit c1') [ @daemon ] * #c1' #Hc1
+ >Hc1 >tape_move_left_eq cases (legal_tape_cases … Htape)
+ [ #Hc0 >lift_tape_not_null /2/
+ | * #Hc0 *
+ [ #Hls >Hc0 >Hls whd in ⊢ (??%%); cases rs;
+ [%| #r0 #rs0 %]
+ | #Hrs >Hc0 >Hrs cases ls; [%| #l0 #ls0 %]
+ ]
+ ]
+ ]
+ | //
]
- |
-
-
-
- >Heqcurconfig
->append_cons in Hintape; >associative_append
-cases (Hta
-
- cases (Hta … Hintape) -Hta
-[ * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
-* #_ #Hta lapply (Hta ? 〈comma,true〉 … (refl ??) (refl ??) ?)
-[ @daemon ] -Hta #Hta
-* #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta)
-
-definition move_of_unialpha ≝
- λc.match c with
- [ bit x ⇒ match x with [ true ⇒ R | false ⇒ L ]
- | _ ⇒ N ].
+ | * #Hmv #Htapemove
+ @(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? newc) %
+ [ %
+ [ >Houtc -Houtc >reverse_append
+ >reverse_reverse >reverse_single @eq_f
+ >reverse_cons >reverse_cons >reverse_append >reverse_cons
+ >reverse_cons >reverse_reverse >reverse_reverse
+ >associative_append >associative_append
+ >associative_append >associative_append
+ >associative_append >associative_append %
+ |>Hmv >Ht1' >Htapemove
+ cut (∃c1'.c1 = bit c1') [ @daemon ] * #c1' #Hc1
+ >Hc1 >tape_move_right_eq cases (legal_tape_cases … Htape)
+ [ #Hc0 >lift_tape_not_null /2/
+ | * #Hc0 *
+ [ #Hls >Hc0 >Hls whd in ⊢ (??%%); cases rs;
+ [%| #r0 #rs0 %]
+ | #Hrs >Hc0 >Hrs cases ls; [%| #l0 #ls0 %]
+ ]
+ ]
+ ]
+ | //
+ ]
+ ]
+ | * * * #Hmv #Hlseq #Hrseq #Hnewc
+ @(ex_intro ?? ls1) @(ex_intro ?? rs1) @(ex_intro ?? newc) %
+ [ %
+ [ >Houtc -Houtc >reverse_append
+ >reverse_reverse >reverse_single @eq_f
+ >reverse_cons >reverse_cons >reverse_append >reverse_cons
+ >reverse_cons >reverse_reverse >reverse_reverse
+ >associative_append >associative_append
+ >associative_append >associative_append
+ >associative_append >associative_append %
+ |>Hmv >Ht1' cases c1 in Hnewc;
+ [ #c1' whd in ⊢ (??%?→?);#Hnewc <Hnewc
+ >Hlseq >Hrseq cases Htape * #Hleft #Hright #_
+ whd in ⊢ (??%%); >Hleft >Hright %
+ | whd in ⊢ (??%?→?); #Hnewc >Hnewc >Hlseq >Hrseq %
+ |*: whd in ⊢ (??%?→?);#Hnewc <Hnewc
+ >Hlseq >Hrseq cases Htape * #Hleft #Hright #_
+ whd in ⊢ (??%%); >Hleft >Hright % ]
+ ]
+ | //
+ ]
+ ]
+]
+qed.
definition R_uni_step ≝ λt1,t2.
∀n,table,c,c1,ls,rs,curs,curc,news,newc,mv.
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