include "turing/mono.ma".
+definition single_finalTM ≝
+ λsig.λM:TM sig.seq ? M (nop ?).
+
+lemma sem_single_final :
+ ∀sig,M,R.Realize ? M R → Realize ? (single_finalTM sig M) R.
+#sig #M #R #HR #intape
+cases (sem_seq ????? HR (sem_nop …) intape)
+#k * #outc * #Hloop * #ta * #Hta whd in ⊢ (%→?); #Houtc
+@(ex_intro ?? k) @(ex_intro ?? outc) % [ @Hloop | >Houtc // ]
+qed.
+
definition if_trans ≝ λsig. λM1,M2,M3 : TM sig. λq:states sig M1.
λp. let 〈s,a〉 ≝ p in
match s with
[ inl s2 ⇒ halt sig thenM s2
| inr s3 ⇒ halt sig elseM s3 ]]).
-axiom daemon : ∀X:Prop.X.
+axiom daemon : ∀X:Prop.X.
+axiom tdaemon : ∀X:Type[0].X.
-theorem sem_if: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,acc.
+axiom sem_if: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,acc.
accRealize sig M1 acc Rtrue Rfalse → Realize sig M2 R2 → Realize sig M3 R3 →
Realize sig (ifTM sig M1 M2 M3 acc) (λt1,t2. (Rtrue ∘ R2) t1 t2 ∨ (Rfalse ∘ R3) t1 t2).
+(*
#sig #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #acc #HaccR #HR2 #HR3 #t
cases (HaccR t) #k1 * #outc1 * * #Hloop1 #HMtrue #HMfalse
cases (true_or_false (cstate ?? outc1 == acc)) #Hacc
[cases (HR2 (ctape sig ? outc1)) #k2 * #outc2 * #Hloop2 #HM2
@(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confL … outc2)))
%
- [@(loop_split ???????????
+ [@(loop_merge ???????????
(loop_lift ???
(lift_confL sig (states ? M1) (FinSum (states ? M2) (states ? M3)))
(step sig M1) (step sig (ifTM sig M1 M2 M3 acc))
generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
>(trans_liftL_true sig M1 M2 ??)
[ whd in ⊢ (??%?); whd in ⊢ (???%);
- @config_eq whd in ⊢ (???%); //
- | @(loop_Some ?????? Hloop10) ]
- ||4:cases outc1 #s1 #t1 %
+ @daemon
+(* @config_eq whd in ⊢ (???%); // *)
+ | @(loop_Some ?????? Hloop10)
+ | @tdaemon
+ | skip ]
+ ]
+ |
+ |4:cases outc1 #s1 #t1 %
|5:
@(loop_liftR … Hloop2)
% //
]
qed.
-(* We do not distinuish an input tape *)
-
-record TM (sig:FinSet): Type[1] ≝
-{ states : FinSet;
- trans : states × (option sig) → states × (option (sig × move));
- start: states;
- halt : states → bool
-}.
-
-record config (sig:FinSet) (M:TM sig): Type[0] ≝
-{ cstate : states sig M;
- ctape: tape sig
-}.
-
-definition option_hd ≝ λA.λl:list A.
- match l with
- [nil ⇒ None ?
- |cons a _ ⇒ Some ? a
- ].
-
-definition tape_move ≝ λsig.λt: tape sig.λm:option (sig × move).
- match m with
- [ 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))
- ]
- ].
-
-definition step ≝ λsig.λM:TM sig.λc:config sig M.
- let current_char ≝ option_hd ? (right ? (ctape ?? c)) in
- let 〈news,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
- mk_config ?? news (tape_move sig (ctape ?? c) mv).
-
-let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
- match n with
- [ O ⇒ None ?
- | S m ⇒ if p a then (Some ? a) else loop A m f p (f a)
- ].
-
-lemma loop_incr : ∀A,f,p,k1,k2,a1,a2.
- loop A k1 f p a1 = Some ? a2 →
- loop A (k2+k1) f p a1 = Some ? a2.
-#A #f #p #k1 #k2 #a1 #a2 generalize in match a1; elim k1
-[normalize #a0 #Hfalse destruct
-|#k1' #IH #a0 <plus_n_Sm whd in ⊢ (??%? → ??%?);
- cases (true_or_false (p a0)) #Hpa0 >Hpa0 whd in ⊢ (??%? → ??%?); // @IH
-]
-qed.
-
-lemma loop_split : ∀A,f,p,q.(∀b. p b = false → q b = false) →
- ∀k1,k2,a1,a2,a3,a4.
- loop A k1 f p a1 = Some ? a2 →
- f a2 = a3 → q a2 = false →
- loop A k2 f q a3 = Some ? a4 →
- loop A (k1+k2) f q a1 = Some ? a4.
-#Sig #f #p #q #Hpq #k1 elim k1
- [normalize #k2 #a1 #a2 #a3 #a4 #H destruct
- |#k1' #Hind #k2 #a1 #a2 #a3 #a4 normalize in ⊢ (%→?);
- cases (true_or_false (p a1)) #pa1 >pa1 normalize in ⊢ (%→?);
- [#eqa1a2 destruct #eqa2a3 #Hqa2 #H
- whd in ⊢ (??(??%???)?); >plus_n_Sm @loop_incr
- whd in ⊢ (??%?); >Hqa2 >eqa2a3 @H
- |normalize >(Hpq … pa1) normalize
- #H1 #H2 #H3 @(Hind … H2) //
- ]
- ]
-qed.
-
-(*
-lemma loop_split : ∀A,f,p,q.(∀b. p b = false → q b = false) →
- ∀k1,k2,a1,a2,a3.
- loop A k1 f p a1 = Some ? a2 →
- loop A k2 f q a2 = Some ? a3 →
- loop A (k1+k2) f q a1 = Some ? a3.
-#Sig #f #p #q #Hpq #k1 elim k1
- [normalize #k2 #a1 #a2 #a3 #H destruct
- |#k1' #Hind #k2 #a1 #a2 #a3 normalize in ⊢ (%→?→?);
- cases (true_or_false (p a1)) #pa1 >pa1 normalize in ⊢ (%→?);
- [#eqa1a2 destruct #H @loop_incr //
- |normalize >(Hpq … pa1) normalize
- #H1 #H2 @(Hind … H2) //
- ]
- ]
-qed.
-*)
-
-definition initc ≝ λsig.λM:TM sig.λt.
- mk_config sig M (start sig M) t.
-
-definition Realize ≝ λsig.λM:TM sig.λR:relation (tape sig).
-∀t.∃i.∃outc.
- loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) (initc sig M t) = Some ? outc ∧
- R t (ctape ?? outc).
-
-(* Compositions *)
-
-definition seq_trans ≝ λsig. λM1,M2 : TM sig.
-λp. let 〈s,a〉 ≝ p in
- match s with
- [ inl s1 ⇒
- if halt sig M1 s1 then 〈inr … (start sig M2), None ?〉
- else
- let 〈news1,m〉 ≝ trans sig M1 〈s1,a〉 in
- 〈inl … news1,m〉
- | inr s2 ⇒
- let 〈news2,m〉 ≝ trans sig M2 〈s2,a〉 in
- 〈inr … news2,m〉
- ].
-
-definition seq ≝ λsig. λM1,M2 : TM sig.
- mk_TM sig
- (FinSum (states sig M1) (states sig M2))
- (seq_trans sig M1 M2)
- (inl … (start sig M1))
- (λs.match s with
- [ inl _ ⇒ false | inr s2 ⇒ halt sig M2 s2]).
-
-definition Rcomp ≝ λA.λR1,R2:relation A.λa1,a2.
- ∃am.R1 a1 am ∧ R2 am a2.
-
-(*
-definition injectRl ≝ λsig.λM1.λM2.λR.
- λc1,c2. ∃c11,c12.
- inl … (cstate sig M1 c11) = cstate sig (seq sig M1 M2) c1 ∧
- inl … (cstate sig M1 c12) = cstate sig (seq sig M1 M2) c2 ∧
- ctape sig M1 c11 = ctape sig (seq sig M1 M2) c1 ∧
- ctape sig M1 c12 = ctape sig (seq sig M1 M2) c2 ∧
- R c11 c12.
-
-definition injectRr ≝ λsig.λM1.λM2.λR.
- λc1,c2. ∃c21,c22.
- inr … (cstate sig M2 c21) = cstate sig (seq sig M1 M2) c1 ∧
- inr … (cstate sig M2 c22) = cstate sig (seq sig M1 M2) c2 ∧
- ctape sig M2 c21 = ctape sig (seq sig M1 M2) c1 ∧
- ctape sig M2 c22 = ctape sig (seq sig M1 M2) c2 ∧
- R c21 c22.
-
-definition Rlink ≝ λsig.λM1,M2.λc1,c2.
- ctape sig (seq sig M1 M2) c1 = ctape sig (seq sig M1 M2) c2 ∧
- cstate sig (seq sig M1 M2) c1 = inl … (halt sig M1) ∧
- cstate sig (seq sig M1 M2) c2 = inr … (start sig M2). *)
-
-interpretation "relation composition" 'compose R1 R2 = (Rcomp ? R1 R2).
-
-definition lift_confL ≝
- λsig,M1,M2,c.match c with
- [ mk_config s t ⇒ mk_config ? (seq sig M1 M2) (inl … s) t ].
-definition lift_confR ≝
- λsig,M1,M2,c.match c with
- [ mk_config s t ⇒ mk_config ? (seq sig M1 M2) (inr … s) t ].
-
-definition halt_liftL ≝
- λsig.λM1,M2:TM sig.λs:FinSum (states ? M1) (states ? M2).
- match s with
- [ inl s1 ⇒ halt sig M1 s1
- | inr _ ⇒ true ]. (* should be vacuous in all cases we use halt_liftL *)
-
-definition halt_liftR ≝
- λsig.λM1,M2:TM sig.λs:FinSum (states ? M1) (states ? M2).
- match s with
- [ inl _ ⇒ false
- | inr s2 ⇒ halt sig M2 s2 ].
-
-lemma p_halt_liftL : ∀sig,M1,M2,c.
- halt sig M1 (cstate … c) =
- halt_liftL sig M1 M2 (cstate … (lift_confL … c)).
-#sig #M1 #M2 #c cases c #s #t %
-qed.
-
-lemma trans_liftL : ∀sig,M1,M2,s,a,news,move.
- halt ? M1 s = false →
- trans sig M1 〈s,a〉 = 〈news,move〉 →
- trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inl … news,move〉.
-#sig (*#M1*) * #Q1 #T1 #init1 #halt1 #M2 #s #a #news #move
-#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
-qed.
-
-lemma config_eq :
- ∀sig,M,c1,c2.
- cstate sig M c1 = cstate sig M c2 →
- ctape sig M c1 = ctape sig M c2 → c1 = c2.
-#sig #M1 * #s1 #t1 * #s2 #t2 //
-qed.
-
-lemma step_lift_confL : ∀sig,M1,M2,c0.
- halt ? M1 (cstate ?? c0) = false →
- step sig (seq sig M1 M2) (lift_confL sig M1 M2 c0) =
- lift_confL sig M1 M2 (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)
-[% | //]
-qed.
-
-lemma loop_liftL : ∀sig,k,M1,M2,c1,c2.
- loop ? k (step sig M1) (λc.halt sig M1 (cstate ?? c)) c1 = Some ? c2 →
- loop ? k (step sig (seq sig M1 M2))
- (λc.halt_liftL sig M1 M2 (cstate ?? c)) (lift_confL … c1) =
- Some ? (lift_confL … c2).
-#sig #k #M1 #M2 #c1 #c2 generalize in match c1;
-elim k
-[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
-|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
- lapply (refl ? (halt ?? (cstate sig M1 c0)))
- cases (halt ?? (cstate sig M1 c0)) in ⊢ (???% → ?); #Hc0 >Hc0
- [ >(?: halt_liftL ??? (cstate sig (seq ? M1 M2) (lift_confL … c0)) = true)
- [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2) %
- | // ]
- | >(?: halt_liftL ??? (cstate sig (seq ? M1 M2) (lift_confL … c0)) = false)
- [whd in ⊢ (??%? → ??%?); #Hc2 <(IH ? Hc2) @eq_f
- @step_lift_confL //
- | // ]
-qed.
-
-STOP!
-
-lemma loop_liftR : ∀sig,k,M1,M2,c1,c2.
- loop ? k (step sig M2) (λc.halt sig M2 (cstate ?? c)) c1 = Some ? c2 →
- loop ? k (step sig (seq sig M1 M2))
- (λc.halt sig (seq sig M1 M2) (cstate ?? c)) (lift_confR … c1) =
- Some ? (lift_confR … c2).
-#sig #k #M1 #M2 #c1 #c2
-elim k
-[normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
-|#k0 #IH whd in ⊢ (??%? → ??%?);
- lapply (refl ? (halt ?? (cstate sig M2 c1)))
- cases (halt ?? (cstate sig M2 c1)) in ⊢ (???% → ?); #Hc0 >Hc0
- [ >(?: halt ?? (cstate sig (seq ? M1 M2) (lift_confR … c1)) = true)
- [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2)
- | (* ... *) ]
- | >(?: halt ?? (cstate sig (seq ? M1 M2) (lift_confR … c1)) = false)
- [whd in ⊢ (??%? → ??%?); #Hc2 <IH
- [@eq_f (* @step_lift_confR // *)
- |
- | // ]
-qed. *)
-
-lemma loop_Some :
- ∀A,k,f,p,a,b.loop A k f p a = Some ? b → p b = true.
-#A #k #f #p #a #b elim k
-[normalize #Hfalse destruct
-|#k0 #IH whd in ⊢ (??%? → ?); cases (p a)
- [ normalize #H1 destruct
-
-lemma trans_liftL_true : ∀sig,M1,M2,s,a.
- halt ? M1 s = true →
- trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inr … (start ? M2),None ?〉.
-#sig #M1 #M2 #s #a
-#Hhalt whd in ⊢ (??%?); >Hhalt %
-qed.
-
-lemma eq_ctape_lift_conf_L : ∀sig,M1,M2,outc.
- ctape sig (seq sig M1 M2) (lift_confL … outc) = ctape … outc.
-#sig #M1 #M2 #outc cases outc #s #t %
-qed.
-
-lemma eq_ctape_lift_conf_R : ∀sig,M1,M2,outc.
- ctape sig (seq sig M1 M2) (lift_confR … outc) = ctape … outc.
-#sig #M1 #M2 #outc cases outc #s #t %
-qed.
-
-theorem sem_seq: ∀sig,M1,M2,R1,R2.
- Realize sig M1 R1 → Realize sig M2 R2 →
- Realize sig (seq sig M1 M2) (R1 ∘ R2).
-#sig #M1 #M2 #R1 #R2 #HR1 #HR2 #t
-cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
-cases (HR2 (ctape sig M1 outc1)) #k2 * #outc2 * #Hloop2 #HM2
-@(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
-%
-[@(loop_split ??????????? (loop_liftL … Hloop1))
- [* *
- [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
- | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
- ||4:cases outc1 #s1 #t1 %
- |5:@(loop_liftR … Hloop2)
- |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
- generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
- >(trans_liftL_true sig M1 M2 ??)
- [ whd in ⊢ (??%?); whd in ⊢ (???%);
- @config_eq //
- | @(loop_Some ?????? Hloop10) ]
- ]
-| @(ex_intro … (ctape ? (seq sig M1 M2) (lift_confL … outc1)))
- % //
-]
-qed.
-
-(* boolean machines: machines with two distinguished halting states *)
-
-
-
-(* old stuff *)
-definition empty_tapes ≝ λsig.λn.
-mk_Vector ? n (make_list (tape sig) (mk_tape sig [] []) n) ?.
-elim n // normalize //
-qed.
-
-definition init ≝ λsig.λM:TM sig.λi:(list sig).
- mk_config ??
- (start sig M)
- (vec_cons ? (mk_tape sig [] i) ? (empty_tapes sig (tapes_no sig M)))
- [ ].
-
-definition stop ≝ λsig.λM:TM sig.λc:config sig M.
- halt sig M (state sig M c).
-
-let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
- match n with
- [ O ⇒ None ?
- | S m ⇒ if p a then (Some ? a) else loop A m f p (f a)
- ].
-
-(* Compute ? M f states that f is computed by M *)
-definition Compute ≝ λsig.λM:TM sig.λf:(list sig) → (list sig).
-∀l.∃i.∃c.
- loop ? i (step sig M) (stop sig M) (init sig M l) = Some ? c ∧
- out ?? c = f l.
-
-(* for decision problems, we accept a string if on termination
-output is not empty *)
-
-definition ComputeB ≝ λsig.λM:TM sig.λf:(list sig) → bool.
-∀l.∃i.∃c.
- loop ? i (step sig M) (stop sig M) (init sig M l) = Some ? c ∧
- (isnilb ? (out ?? c) = false).
-
-(* alternative approach.
-We define the notion of computation. The notion must be constructive,
-since we want to define functions over it, like lenght and size
-
-Perche' serve Type[2] se sposto a e b a destra? *)
-
-inductive cmove (A:Type[0]) (f:A→A) (p:A →bool) (a,b:A): Type[0] ≝
- mk_move: p a = false → b = f a → cmove A f p a b.
-
-inductive cstar (A:Type[0]) (M:A→A→Type[0]) : A →A → Type[0] ≝
-| empty : ∀a. cstar A M a a
-| more : ∀a,b,c. M a b → cstar A M b c → cstar A M a c.
-
-definition computation ≝ λsig.λM:TM sig.
- cstar ? (cmove ? (step sig M) (stop sig M)).
-
-definition Compute_expl ≝ λsig.λM:TM sig.λf:(list sig) → (list sig).
- ∀l.∃c.computation sig M (init sig M l) c →
- (stop sig M c = true) ∧ out ?? c = f l.
-
-definition ComputeB_expl ≝ λsig.λM:TM sig.λf:(list sig) → bool.
- ∀l.∃c.computation sig M (init sig M l) c →
- (stop sig M c = true) ∧ (isnilb ? (out ?? c) = false).
+*)
\ No newline at end of file
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