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) →
- no_nulls ls → no_nulls rs → no_marks ls → no_marks rs →
+ no_marks curconfig → only_bits (curconfig@[〈s0,false〉]) →
+ only_bits (〈s1,false〉::newconfig) → bit_or_null c1 = true →
+ |curconfig| = |newconfig| →
legal_tape ls 〈c0,false〉 rs →
t1 = midtape STape (〈c0,false〉::curconfig@〈s0,false〉::〈grid,false〉::ls) 〈grid,false〉
(table1@〈comma,true〉::〈s1,false〉::newconfig@〈c1,false〉::〈comma,false〉::〈mv,false〉::table2@〈grid,false〉::rs) →
(〈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 merge_char_null : ∀c.merge_char null c = c.
+* //
+qed.
+
+lemma merge_char_cases : ∀c1,c2.merge_char c1 c2 = c1 ∨ merge_char c1 c2 = c2.
+#c1 *
+[ #c1' %2 %
+| % %
+| *: %2 % ]
+qed.
+
+(* lemma merge_char_c_bit :
+ ∀c1,c2.is_bit c2 = true → merge_char c1 c2 = c2.
+#c1 *
+[ #c2' #_ %
+|*: normalize #Hfalse destruct (Hfalse) ]
+qed.
+
+lemma merge_char_c_bit :
+ ∀c1,c2.is_null c2 = true → merge_char c1 c2 = c1.
+#c1 *
+[ #c2' #_ %
+|*: 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 #Hc1 #Hlen #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 ?))
+ mv table2 (merge_char c0 c1) (reverse ? newconfig@[〈s1,false〉]) ls ????????)
+ [3: cases Htape -Htape * * #Hnomarks #Hbits #Hc0 #Hlsrs % [ % [ %
+ [ #x #Hx cases (orb_true_l … Hx) #Hx'
+ [ >(\P Hx') %
+ | @Hnomarks @memb_cons // ]
+ | @Hbits ]
+ | cases (merge_char_cases c0 c1) #Hmerge >Hmerge // ]
+ | cases (true_or_false (c0 == null)) #Hc0'
+ [ cases Hlsrs -Hlsrs
+ [ *
+ [ >(\P Hc0') * #Hfalse @False_ind /2/
+ | #Hlsnil % %2 // ]
+ | #Hrsnil %2 // ]
+ | % % @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
- | (* bit_or_null c1,c2 → bit_or_null (merge_char c1 c2) *) @daemon
- | -Houtc * #ls1 * #rs1 * #newc * #Houtc *
+ | (* well formedness of table *) @daemon
+ | (* Hnewconfig *) @daemon
+ | (* bit_or_null mv = true (well formedness of table) *) @daemon
+ | -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 ? *)
+ cut (∃c1'.c1 = bit c1') [ @daemon ] * #c1' #Hc1
+ >Hc1 >tape_move_left_eq >(legal_tape_left … Htape)
+ >(legal_tape_right … Htape) %
+ ]
+ | //
+ ]
+ | * #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 >(legal_tape_left … Htape)
+ >(legal_tape_right … Htape) %
+ ]
+ | //
+ ]
+ ]
+ | * * * #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
>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 ??);
-
+ |>Hmv >Ht1' cases c1 in Hnewc;
+ [ #c1' whd in ⊢ (??%?→?);#Hnewc <Hnewc
+ >Hlseq >Hrseq whd in ⊢ (??%%);
+ >(legal_tape_left … Htape) >(legal_tape_right … Htape) %
+ | whd in ⊢ (??%?→?); #Hnewc >Hnewc >Hlseq >Hrseq %
+ |*: whd in ⊢ (??%?→?);#Hnewc <Hnewc
+ >Hlseq >Hrseq whd in ⊢ (??%%);
+ >(legal_tape_left … Htape) >(legal_tape_right … Htape) %
+ ]
]
- |
-
-
+ | //
+ ]
+ ]
+ ]
+]
+qed.
+
+(*
+if is_false(current) (* current state is not final *)
+ then init_match;
+ match_tuple;
+ if is_marked(current) = false (* match ok *)
+ then
+ exec_move
+ move_r;
+ else sink;
+ else nop;
+*)
+
+definition uni_step ≝
+ ifTM ? (test_char STape (λc.\fst c == bit false))
+ (single_finalTM ? (seq ? init_match
+ (seq ? match_tuple
+ (ifTM ? (test_char ? (λc.¬is_marked ? c))
+ (seq ? exec_move (move_r …))
+ (nop ?) (* sink *)
+ tc_true))))
+ (nop ?)
+ tc_true.
+
+definition R_uni_step_true ≝ λt1,t2.
+ ∀n,t0,table,s0,s1,c0,c1,ls,rs,curconfig,newconfig,mv.
+ table_TM (S n) (〈t0,false〉::table) →
+ match_in_table (〈s0,false〉::curconfig@[〈c0,false〉])
+ (〈s1,false〉::newconfig@[〈c1,false〉]) mv (〈t0,false〉::table) →
+ legal_tape ls 〈c0,false〉 rs →
+ t1 = midtape STape (〈grid,false〉::ls) 〈s0,false〉
+ (curconfig@〈c0,false〉::〈grid,false〉::〈t0,false〉::table@〈grid,false〉::rs) →
+ ∀t1'.t1' = lift_tape ls 〈c0,false〉 rs →
+ s0 = bit false ∧
+ ∃ls1,rs1,c2.
+ (t2 = midtape STape (〈grid,false〉::ls1) 〈s1,false〉
+ (newconfig@〈c2,false〉::〈grid,false〉::〈t0,false〉::table@〈grid,false〉::rs1) ∧
+ lift_tape ls1 〈c2,false〉 rs1 =
+ tape_move STape t1' (map_move c1 mv) ∧ legal_tape ls1 〈c2,false〉 rs1).
+
+definition R_uni_step_false ≝ λt1,t2.
+ ∀b. current STape t1 = Some ? 〈bit b,false〉 → b = true ∧ t2 = t1.
- >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 ].
-
-definition R_uni_step ≝ λt1,t2.
- ∀n,table,c,c1,ls,rs,curs,curc,news,newc,mv.
- table_TM n table →
- match_in_table (〈c,false〉::curs@[〈curc,false〉])
- (〈c1,false〉::news@[〈newc,false〉]) mv table →
- t1 = midtape STape (〈grid,false〉::ls) 〈c,false〉
- (curs@〈curc,false〉::〈grid,false〉::table@〈grid,false〉::rs) →
- ∀t1',ls1,rs1.t1' = lift_tape ls 〈curc,false〉 rs →
- (t2 = midtape STape (〈grid,false〉::ls1) 〈c1,false〉
- (news@〈newc,false〉::〈grid,false〉::table@〈grid,false〉::rs1) ∧
- lift_tape ls1 〈newc,false〉 rs1 =
- tape_move STape t1' (Some ? 〈〈newc,false〉,move_of_unialpha mv〉)).
-
-definition no_nulls ≝
- λl:list STape.∀x.memb ? x l = true → is_null (\fst x) = false.
-
+axiom sem_match_tuple : Realize ? match_tuple R_match_tuple.
+
+lemma sem_uni_step :
+ accRealize ? uni_step (inr … (inl … (inr … 0)))
+ R_uni_step_true R_uni_step_false.
+@(acc_sem_if_app … (sem_test_char ? (λc:STape.\fst c == bit false))
+ (sem_seq … sem_init_match
+ (sem_seq … sem_match_tuple
+ (sem_if … (sem_test_char ? (λc.¬is_marked ? c))
+ (sem_seq … sem_exec_move (sem_move_r …))
+ (sem_nop …))))
+ (sem_nop …)
+ …)
+[ #intape #outtape
+ #ta whd in ⊢ (%→?); #Hta #HR
+ #n #t0 #table #s0 #s1 #c0 #c1 #ls #rs #curconfig #newconfig #mv
+ #Htable #Hmatch #Htape #Hintape #t1' #Ht1'
+ >Hintape in Hta; #Hta cases (Hta ? (refl ??)) -Hta
+ #Hs0 lapply (\P Hs0) -Hs0 #Hs0 #Hta % //
+ cases HR -HR
+ #tb * whd in ⊢ (%→?); #Htb
+ lapply (Htb (〈grid,false〉::ls) (curconfig@[〈c0,false〉]) (table@〈grid,false〉::rs) s0 t0 ???)
+ [ >Hta >associative_append %
+ | (* da decidere se aggiungere un'assunzione o utilizzare Hmatch *) @daemon
+ | (* idem *) @daemon
+ | -Hta -Htb #Htb *
+ #tc * whd in ⊢ (%→?); #Htc cases (Htc … Htable … Htb) -Htb -Htc
+ [| * #Hcurrent #Hfalse @False_ind
+ (* absurd by Hmatch *) @daemon
+ | >Hs0 %
+ | (* da decidere se aggiungere un'assunzione o utilizzare Hmatch *) @daemon
+ | (* Htable *) @daemon
+ | (* Htable, Hmatch → |config| = n *) @daemon ]
+ * #table1 * #newc * #mv1 * #table2 * #Htableeq #Htc *
+ [ * #td * whd in ⊢ (%→?); >Htc -Htc #Htd
+ cases (Htd ? (refl ??)) #_ -Htd
+ cut (newc = 〈s1,false〉::newconfig@[〈c1,false〉]) [@daemon] #Hnewc
+ >Hnewc #Htd
+ cut (mv1 = 〈\fst mv1,false〉)
+ [ >(eq_pair_fst_snd … mv1) @eq_f (*Htable, Htableeq*) @daemon ] #Hmv1
+ * #te * whd in ⊢ (%→?); #Hte
+ cut (td = midtape STape (〈c0,false〉::reverse STape curconfig@〈s0,false〉::〈grid,false〉::ls)
+ 〈grid,false〉
+ ((table1@〈s0,false〉::curconfig@[〈c0,false〉])@〈comma,true〉::〈s1,false〉::
+ newconfig@〈c1,false〉::〈comma,false〉::〈\fst mv1,false〉::table2@〈grid,false〉::rs))
+ [ >Htd @eq_f3 //
+ [ >reverse_append >reverse_single %
+ | >associative_append >associative_append normalize
+ >associative_append >associative_append <Hmv1 %
+ ]
+ ]
+ -Htd #Htd lapply (Hte … (S n) … Htd … Ht1') -Htd -Hte
+ [ //
+ | (*|curconfig| = |newconfig|*) @daemon
+ | (* Htable → bit_or_null c1 = true *) @daemon
+ | (* only_bits (〈s1,false〉::newconfig) *) @daemon
+ | (* only_bits (curconfig@[〈s0,false〉]) *) @daemon
+ | (* no_marks (reverse ? curconfig) *) @daemon
+ | <Hmv1 >Hnewc in Htableeq;
+ >associative_append >associative_append normalize
+ >associative_append >associative_append
+ #Htableeq <Htableeq // ]
+ * #ls1 * #rs1 * #c2 * * #Hte #Hliftte #Hlegalte
+ whd in ⊢ (%→?); #Houttape lapply (Houttape … Hte) -Houttape #Houttape
+ whd in Houttape:(???%); whd in Houttape:(???(??%%%));
+ @ex_intro [| @(ex_intro ?? rs1) @ex_intro [| % [ %
+ [ >Houttape @eq_f @eq_f @eq_f @eq_f
+ change with ((〈t0,false〉::table)@?) in ⊢ (???%);
+ >Htableeq >associative_append >associative_append
+ >associative_append normalize >associative_append
+ >associative_append normalize >Hnewc <Hmv1
+ >associative_append normalize >associative_append %
+ | >(?: mv = \fst mv1) [| (*Hmatch, Htableeq*) @daemon ]
+ @Hliftte
+ ]
+ | //
+ ]
+ ]
+ ]
+ | * #td * whd in ⊢ (%→%→?); >Htc #Htd
+ cases (Htd ? (refl ??)) normalize in ⊢ (%→?);
+ #Hfalse destruct (Hfalse)
+ ]
+ ]
+| #t1 #t2 #t3 whd in ⊢ (%→%→?); #Ht1 #Ht2
+ #b #Hb cases (Ht1 ? Hb) #Hb' #Ht3 >Ht2 % //
+ cases b in Hb'; normalize #H1 //
+]
+qed.
projects / helm.git / blobdiff