include "basics/vectors.ma".
(* include "basics/relations.ma". *)
+(*
record tape (sig:FinSet): Type[0] ≝
-{ left : list sig;
- right: list sig
+{ left : list (option sig);
+ right: list (option sig)
}.
+*)
+
+inductive tape (sig:FinSet) : Type[0] ≝
+| niltape : tape sig
+| leftof : sig → list sig → tape sig
+| rightof : sig → list sig → tape sig
+| midtape : list sig → sig → list sig → tape sig.
+
+definition left ≝
+ λsig.λt:tape sig.match t with
+ [ niltape ⇒ []
+ | leftof _ _ ⇒ []
+ | rightof s l ⇒ s::l
+ | midtape l _ _ ⇒ l ].
+
+definition right ≝
+ λsig.λt:tape sig.match t with
+ [ niltape ⇒ []
+ | leftof s r ⇒ s::r
+ | rightof _ _ ⇒ []
+ | midtape _ _ r ⇒ r ].
+
+
+definition current ≝
+ λsig.λt:tape sig.match t with
+ [ midtape _ c _ ⇒ Some ? c
+ | _ ⇒ None ? ].
inductive move : Type[0] ≝
| L : move
| R : move
+| N : move
.
(* We do not distinuish an input tape *)
ctape: tape sig
}.
-definition option_hd ≝ λA.λl:list A.
+(* definition option_hd ≝ λA.λl:list (option A).
match l with
[nil ⇒ None ?
- |cons a _ ⇒ Some ? a
+ |cons a _ ⇒ a
].
+ *)
-definition tape_move ≝ λsig.λt: tape sig.λm:option (sig × move).
- match m with
+(*definition tape_write ≝ λsig.λt:tape sig.λs:sig.
+ <left ? t) s (right ? t).
[ None ⇒ t
- | Some m1 ⇒
- match \snd m1 with
- [ R ⇒ mk_tape sig ((\fst m1)::(left ? t)) (tail ? (right ? t))
- | L ⇒ mk_tape sig (tail ? (left ? t)) ((\fst m1)::(right ? t))
- ]
- ].
+ | Some s' ⇒ midtape ? (left ? t) s' (right ? t) ].*)
+
+definition tape_move_left ≝ λsig:FinSet.λlt:list sig.λc:sig.λrt:list sig.
+ match lt with
+ [ nil ⇒ leftof sig c rt
+ | cons c0 lt0 ⇒ midtape sig lt0 c0 (c::rt) ].
+
+definition tape_move_right ≝ λsig:FinSet.λlt:list sig.λc:sig.λrt:list sig.
+ match rt with
+ [ nil ⇒ rightof sig c lt
+ | cons c0 rt0 ⇒ midtape sig (c::lt) c0 rt0 ].
+definition tape_move ≝ λsig.λt: tape sig.λm:option (sig × move).
+ match m with
+ [ None ⇒ t
+ | Some m' ⇒
+ let 〈s,m1〉 ≝ m' in
+ match m1 with
+ [ R ⇒ tape_move_right ? (left ? t) s (right ? t)
+ | L ⇒ tape_move_left ? (left ? t) s (right ? t)
+ | N ⇒ midtape ? (left ? t) s (right ? t)
+ ] ].
+(*
+ (None,[]) → □
+ (None,a::[]) → □
+ (None,a::b::rs) → None::b::rs
+ (Some a,[]) → [Some a]
+ (Some a,b::rs) → Some a::rs
+ *)
+(*
+definition option_cons ≝ λA.λa:option A.λl.
+ match a with
+ [ None ⇒ match l with
+ [ nil ⇒ []
+ | cons _ _ ⇒ a::l ]
+ | Some _ ⇒ a::l ].
+
+(* definition tape_update := λsig.λt: tape sig.λs:option sig.
+ let newright ≝
+ match right ? t with
+ [ nil ⇒ match s with
+ [ None ⇒ []
+ | Some a ⇒ [Some ? a] ]
+ | cons b rs ⇒ match s with
+ [ None ⇒ match rs with
+ [ nil ⇒ []
+ | cons _ _ ⇒ None ?::rs ]
+ | Some a ⇒ Some ? a::rs ] ]
+ in mk_tape ? (left ? t) newright. *)
+
+definition tape_move ≝ λsig.λt:tape sig.λm:option sig × move.
+ let 〈s,m1〉 ≝ m in match m1 with
+ [ R ⇒ mk_tape sig (option_cons ? s (left ? t)) (tail ? (right ? t))
+ | L ⇒ mk_tape sig (tail ? (left ? t))
+ (option_cons ? (option_hd ? (left ? t))
+ (option_cons ? s (tail ? (right ? t))))
+ | N ⇒ mk_tape sig (left ? t) (option_cons ? s (tail ? (right ? t)))
+ ].
+*)
+
definition step ≝ λsig.λM:TM sig.λc:config sig (states sig M).
- let current_char ≝ option_hd ? (right ? (ctape ?? c)) in
+ let current_char ≝ current ? (ctape ?? c) in
let 〈news,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
mk_config ?? news (tape_move sig (ctape ?? c) mv).
halt ? M2 (cstate ?? c0) = false →
step sig (seq sig M1 M2) (lift_confR sig (states ? M1) (states ? M2) c0) =
lift_confR sig (states ? M1) (states ? M2) (step sig M2 c0).
-#sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s * #lt
-#rs #Hhalt
-whd in ⊢ (???(????%));whd in ⊢ (???%);
-lapply (refl ? (trans ?? 〈s,option_hd sig rs〉))
-cases (trans ?? 〈s,option_hd sig rs〉) in ⊢ (???% → %);
-#s0 #m0 #Heq whd in ⊢ (???%);
-whd in ⊢ (??(???%)?); whd in ⊢ (??%?);
->(trans_liftR … Heq)
-[% | //]
+#sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
+ lapply (refl ? (trans ?? 〈s,current sig t〉))
+ cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
+ #s0 #m0 cases t
+ [ #Heq #Hhalt
+ | 2,3: #s1 #l1 #Heq #Hhalt
+ |#ls #s1 #rs #Heq #Hhalt ]
+ whd in ⊢ (???(????%)); >Heq
+ whd in ⊢ (???%);
+ whd in ⊢ (??(???%)?); whd in ⊢ (??%?);
+ >(trans_liftR … Heq) //
qed.
lemma step_lift_confL : ∀sig,M1,M2,c0.
halt ? M1 (cstate ?? c0) = false →
step sig (seq sig M1 M2) (lift_confL sig (states ? M1) (states ? M2) c0) =
lift_confL sig ?? (step sig M1 c0).
-#sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s * #lt
-#rs #Hhalt
-whd in ⊢ (???(????%));whd in ⊢ (???%);
-lapply (refl ? (trans ?? 〈s,option_hd sig rs〉))
-cases (trans ?? 〈s,option_hd sig rs〉) in ⊢ (???% → %);
-#s0 #m0 #Heq whd in ⊢ (???%);
-whd in ⊢ (??(???%)?); whd in ⊢ (??%?);
->(trans_liftL … Heq)
-[% | //]
+#sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
+ lapply (refl ? (trans ?? 〈s,current sig t〉))
+ cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
+ #s0 #m0 cases t
+ [ #Heq #Hhalt
+ | 2,3: #s1 #l1 #Heq #Hhalt
+ |#ls #s1 #rs #Heq #Hhalt ]
+ whd in ⊢ (???(????%)); >Heq
+ whd in ⊢ (???%);
+ whd in ⊢ (??(???%)?); whd in ⊢ (??%?);
+ >(trans_liftL … Heq) //
qed.
lemma loop_lift : ∀A,B,k,lift,f,g,h,hlift,c1,c2.
\ / GNU General Public License Version 2
V_____________________________________________________________*)
-include "turing/mono.ma".
include "basics/star.ma".
+include "turing/mono.ma".
definition while_trans ≝ λsig. λM : TM sig. λq:states sig M. λp.
let 〈s,a〉 ≝ p in
(start sig M)
(λs.halt sig M s ∧ ¬ s==qacc).
-axiom daemon : ∀X:Prop.X.
+(* axiom daemon : ∀X:Prop.X. *)
lemma while_trans_false : ∀sig,M,q,p.
\fst p ≠ q → trans sig (whileTM sig M q) p = trans sig M p.
]
]
qed.
-(*
+
+(* inductive move_states : Type[0] ≝
+| start : move_states
+| q1 : move_states
+| q2 : move_states
+| q3 : move_states
+| qacc : move_states
+| qfail : move_states.
+
+definition
+*)
+
+definition mystates : FinSet → FinSet ≝ λalpha:FinSet.FinProd (initN 5) alpha.
+
+definition move_char ≝
+ λalpha:FinSet.λsep:alpha.
+ mk_TM alpha (mystates alpha)
+ (λp.let 〈q,a〉 ≝ p in
+ let 〈q',b〉 ≝ q in
+ match a with
+ [ None ⇒ 〈〈4,sep〉,None ?〉
+ | Some a' ⇒
+ match q' with
+ [ O ⇒ (* qinit *)
+ match a' == sep with
+ [ true ⇒ 〈〈4,sep〉,None ?〉
+ | false ⇒ 〈〈1,a'〉,Some ? 〈a',L〉〉 ]
+ | S q' ⇒ match q' with
+ [ O ⇒ (* q1 *)
+ 〈〈2,a'〉,Some ? 〈b,R〉〉
+ | S q' ⇒ match q' with
+ [ O ⇒ (* q2 *)
+ 〈〈3,sep〉,Some ? 〈b,R〉〉
+ | S q' ⇒ match q' with
+ [ O ⇒ (* qacc *)
+ 〈〈3,sep〉,None ?〉
+ | S q' ⇒ (* qfail *)
+ 〈〈4,sep〉,None ?〉 ] ] ] ] ])
+ 〈0,sep〉
+ (λq.let 〈q',a〉 ≝ q in q' == 3 ∨ q' == 4).
+
+definition mk_tape :
+ ∀sig:FinSet.list sig → option sig → list sig → tape sig ≝
+ λsig,lt,c,rt.match c with
+ [ Some c' ⇒ midtape sig lt c' rt
+ | None ⇒ match lt with
+ [ nil ⇒ match rt with
+ [ nil ⇒ niltape ?
+ | cons r0 rs0 ⇒ leftof ? r0 rs0 ]
+ | cons l0 ls0 ⇒ rightof ? l0 ls0 ] ].
+
+lemma cmove_q0_q1 :
+ ∀alpha:FinSet.∀sep,a,ls,a0,rs.
+ a0 == sep = false →
+ step alpha (move_char alpha sep)
+ (mk_config ?? 〈0,a〉 (mk_tape … ls (Some ? a0) rs)) =
+ mk_config alpha (states ? (move_char alpha sep)) 〈1,a0〉
+ (tape_move_left alpha ls a0 rs).
+#alpha #sep #a *
+[ #a0 #rs #Ha0 whd in ⊢ (??%?);
+ normalize in match (trans ???); >Ha0 %
+| #a1 #ls #a0 #rs #Ha0 whd in ⊢ (??%?);
+ normalize in match (trans ???); >Ha0 %
+]
+qed.
+
+lemma cmove_q1_q2 :
+ ∀alpha:FinSet.∀sep,a,ls,a0,rs.
+ step alpha (move_char alpha sep)
+ (mk_config ?? 〈1,a〉 (mk_tape … ls (Some ? a0) rs)) =
+ mk_config alpha (states ? (move_char alpha sep)) 〈2,a0〉
+ (tape_move_right alpha ls a rs).
+#alpha #sep #a #ls #a0 * //
+qed.
+
+lemma cmove_q2_q3 :
+ ∀alpha:FinSet.∀sep,a,ls,a0,rs.
+ step alpha (move_char alpha sep)
+ (mk_config ?? 〈2,a〉 (mk_tape … ls (Some ? a0) rs)) =
+ mk_config alpha (states ? (move_char alpha sep)) 〈3,sep〉
+ (tape_move_right alpha ls a rs).
+#alpha #sep #a #ls #a0 * //
+qed.
+
+definition option_hd ≝
+ λA.λl:list A. match l with
+ [ nil ⇒ None ?
+ | cons a _ ⇒ Some ? a ].
+
+definition Rmove_char_true ≝
+ λalpha,sep,t1,t2.
+ ∀a,b,ls,rs. b ≠ sep →
+ t1 = midtape alpha (a::ls) b rs →
+ t2 = mk_tape alpha (a::b::ls) (option_hd ? rs) (tail ? rs).
+
+definition Rmove_char_false ≝
+ λalpha,sep,t1,t2.
+ (current alpha t1 = None alpha → t2 = t1) ∧
+ (current alpha t1 = Some alpha sep → t2 = t1).
+
+lemma loop_S_true :
+ ∀A,n,f,p,a. p a = true →
+ loop A (S n) f p a = Some ? a. /2/
+qed.
+
+lemma loop_S_false :
+ ∀A,n,f,p,a. p a = false →
+ loop A (S n) f p a = loop A n f p (f a).
+normalize #A #n #f #p #a #Hpa >Hpa %
+qed.
+
+notation < "𝐅" non associative with precedence 90
+ for @{'bigF}.
+notation < "𝐃" non associative with precedence 90
+ for @{'bigD}.
+
+interpretation "FinSet" 'bigF = (mk_FinSet ???).
+interpretation "DeqSet" 'bigD = (mk_DeqSet ???).
+
+lemma trans_init_sep:
+ ∀alpha,sep,x.
+ trans ? (move_char alpha sep) 〈〈0,x〉,Some ? sep〉 = 〈〈4,sep〉,None ?〉.
+#alpha #sep #x normalize >(\b ?) //
+qed.
+
+lemma trans_init_not_sep:
+ ∀alpha,sep,x,y.y == sep = false →
+ trans ? (move_char alpha sep) 〈〈0,x〉,Some ? y〉 = 〈〈1,y〉,Some ? 〈y,L〉〉.
+#alpha #sep #x #y #H1 normalize >H1 //
+qed.
+
+lemma sem_move_char :
+ ∀alpha,sep.
+ accRealize alpha (move_char alpha sep)
+ 〈3,sep〉 (Rmove_char_true alpha sep) (Rmove_char_false alpha sep).
+#alpha #sep *
+[@(ex_intro ?? 2)
+ @(ex_intro … (mk_config ?? 〈4,sep〉 (niltape ?)))
+ % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 whd % #_ % ]
+|#l0 #lt0 @(ex_intro ?? 2)
+ @(ex_intro … (mk_config ?? 〈4,sep〉 (leftof ? l0 lt0)))
+ % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 whd % #_ % ]
+|#r0 #rt0 @(ex_intro ?? 2)
+ @(ex_intro … (mk_config ?? 〈4,sep〉 (rightof ? r0 rt0)))
+ % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 whd % #_ % ]
+| #lt #c #rt cases (true_or_false (c == sep)) #Hc
+ [ @(ex_intro ?? 2)
+ @(ex_intro ?? (mk_config ?? 〈4,sep〉 (midtape ? lt c rt)))
+ %
+ [%
+ [ >(\P Hc) >loop_S_false //
+ >loop_S_true
+ [ @eq_f whd in ⊢ (??%?); >trans_init_sep %
+ |>(\P Hc) whd in ⊢(??(???(???%))?);
+ >trans_init_sep % ]
+ | #Hfalse destruct
+ ]
+ |#_ % #_ % ]
+ | @(ex_intro ?? 4)
+ cases lt
+ [ @ex_intro
+ [|%
+ [ %
+ [ >loop_S_false //
+ >cmove_q0_q1 //
+ | normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
+ ]
+ | normalize in ⊢ (%→?); #_ %
+ [ normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
+ | normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
+ @False_ind @(absurd ?? (\Pf Hc)) %
+ ]
+ ]
+ ]
+ | #l0 #lt @ex_intro
+ [| %
+ [ %
+ [ >loop_S_false //
+ >cmove_q0_q1 //
+ | #_ #a #b #ls #rs #Hb #Htape
+ destruct (Htape)
+ >cmove_q1_q2
+ >cmove_q2_q3
+ cases rs normalize //
+ ]
+ | normalize in ⊢ (% → ?); * #Hfalse
+ @False_ind /2/
+ ]
+ ]
+ ]
+ ]
+]
+qed.
+
+definition R_while_cmove :
+ λalpha,sep,t1,t2.
+ ∀a,b,ls,rs. b ≠ sep → memb ? sep rs = false →
+ t1 = midtape alpha (a::ls) b (rs@sep::rs') →
+ t2 = midtape alpha (a::rev ? rs@b::ls) sep rs'.
+
+(* (*
(* We do not distinuish an input tape *)
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