include "basics/star.ma".
include "turing/mono.ma".
+(* The following machine implements a while-loop over a body machine $M$.
+We just need to extend $M$ adding a single transition leading back from a
+distinguished final state $q$ to the initial state. *)
+
definition while_trans ≝ λsig. λM : TM sig. λq:states sig M. λp.
let 〈s,a〉 ≝ p in
- if s == q then 〈start ? M, None ?〉
+ if s == q then 〈start ? M, None ?,N〉
else trans ? M p.
definition whileTM ≝ λsig. λM : TM sig. λqacc: states ? M.
(while_trans sig M qacc)
(start sig M)
(λs.halt sig M s ∧ ¬ s==qacc).
-
-(* 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.
-axiom tech1: ∀A.∀R1,R2:relation A.
+lemma tech1: ∀A.∀R1,R2:relation A.
∀a,b. (R1 ∘ ((star ? R1) ∘ R2)) a b → ((star ? R1) ∘ R2) a b.
-
+#A #R1 #R2 #a #b #H lapply (sub_assoc_l ?????? H) @sub_comp_l -a -b
+#a #b * #c * /2/
+qed.
+
lemma halt_while_acc :
∀sig,M,acc.halt sig (whileTM sig M acc) acc = false.
-#sig #M #acc normalize >(\b ?) //
-cases (halt sig M acc) %
+#sig #M #acc normalize >(\b ?) // cases (halt sig M acc) %
+qed.
+
+lemma halt_while_not_acc :
+ ∀sig,M,acc,s.s == acc = false → halt sig (whileTM sig M acc) s = halt sig M s.
+#sig #M #acc #s #neqs normalize >neqs cases (halt sig M s) %
qed.
lemma step_while_acc :
#sig #M #acc * #s #t #Hs normalize >(\b Hs) %
qed.
-lemma loop_p_true :
- ∀A,k,f,p,a.p a = true → loop A (S k) f p a = Some ? a.
-#A #k #f #p #a #Ha normalize >Ha %
-qed.
-
theorem sem_while: ∀sig,M,acc,Rtrue,Rfalse.
halt sig M acc = true →
- accRealize sig M acc Rtrue Rfalse →
- WRealize sig (whileTM sig M acc) ((star ? Rtrue) ∘ Rfalse).
+ M ⊨ [acc: Rtrue,Rfalse] →
+ whileTM sig M acc ⊫ (star ? Rtrue) ∘ Rfalse.
#sig #M #acc #Rtrue #Rfalse #Hacctrue #HaccR #t #i
generalize in match t;
@(nat_elim1 … i) #m #Hind #intape #outc #Hloop
]
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.
- left ? t1 ≠ [] → current alpha t1 ≠ None alpha →
- 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 #H2 @False_ind @(absurd ?? H2) %]
-|#l0 #lt0 @(ex_intro ?? 2)
- @(ex_intro … (mk_config ?? 〈4,sep〉 (leftof ? l0 lt0)))
- % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 #H2 @False_ind @(absurd ?? H2) %]
-|#r0 #rt0 @(ex_intro ?? 2)
- @(ex_intro … (mk_config ?? 〈4,sep〉 (rightof ? r0 rt0)))
- % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 #H2 #H3 @False_ind @(absurd ?? H3) %]
-| #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
+theorem terminate_while: ∀sig,M,acc,Rtrue,Rfalse,t.
+ halt sig M acc = true →
+ M ⊨ [acc: Rtrue,Rfalse] →
+ WF ? (inv … Rtrue) t → whileTM sig M acc ↓ t.
+#sig #M #acc #Rtrue #Rfalse #t #Hacctrue #HM #HWF elim HWF
+#t1 #H #Hind cases (HM … t1) #i * #outc * * #Hloop
+#Htrue #Hfalse cases (true_or_false (cstate … outc == acc)) #Hcase
+ [cases (Hind ? (Htrue … (\P Hcase))) #iwhile * #outcfinal
+ #Hloopwhile @(ex_intro … (i+iwhile))
+ @(ex_intro … outcfinal) @(loop_merge … outc … Hloopwhile)
+ [@(λc.halt sig M (cstate … c))
+ |* #s0 #t0 normalize cases (s0 == acc) normalize
+ [ cases (halt sig M s0) //
+ | cases (halt sig M s0) normalize //
]
- |#_ #H1 #H2 % // normalize >(\P Hc) % ]
- | @(ex_intro ?? 4)
- cases lt
- [ @ex_intro
- [|%
- [ %
- [ >loop_S_false //
- >cmove_q0_q1 //
- | normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
- ]
- | normalize in ⊢ (%→?); #_ #H1 @False_ind @(absurd ?? H1) %
- ]
- ]
- | #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/
- ]
+ |@(loop_lift ?? i (λc.c) ?
+ (step ? (whileTM ? M acc)) ?
+ (λc.halt sig M (cstate ?? c)) ??
+ ?? Hloop)
+ [ #x %
+ | * #s #t #Hx whd in ⊢ (??%%); >while_trans_false
+ [%
+ |% #Hfalse <Hfalse in Hacctrue; >Hx #H0 destruct ]
]
+ |@step_while_acc @(\P Hcase)
+ |>(\P Hcase) @halt_while_acc
]
- ]
-]
-qed.
-
-definition R_while_cmove ≝
- λalpha,sep,t1,t2.
- ∀a,b,ls,rs,rs'. b ≠ sep → memb ? sep rs = false →
- t1 = midtape alpha (a::ls) b (rs@sep::rs') →
- t2 = midtape alpha (a::reverse ? rs@b::ls) sep rs'.
-
-lemma star_cases :
- ∀A,R,x,y.star A R x y → x = y ∨ ∃z.R x z ∧ star A R z y.
-#A #R #x #y #Hstar elim Hstar
-[ #b #c #Hleft #Hright *
- [ #H1 %2 @(ex_intro ?? c) % //
- | * #x0 * #H1 #H2 %2 @(ex_intro ?? x0) % /2/ ]
-| /2/ ]
-qed.
-
-axiom star_ind_r :
- ∀A:Type[0].∀R:relation A.∀Q:A → A → Prop.
- (∀a.Q a a) →
- (∀a,b,c.R a b → star A R b c → Q b c → Q a c) →
- ∀x,y.star A R x y → Q x y.
-(* #A #R #Q #H1 #H2 #x #y #H0 elim H0
-[ #b #c #Hleft #Hright #IH
- cases (star_cases ???? Hleft)
- [ #Hx @H2 //
- | * #z * #H3 #H4 @(H2 … H3) /2/
-[
-|
-generalize in match (λb.H2 x b y); elim H0
-[#b #c #Hleft #Hright #H2' #H3 @H3
- @(H3 b)
- elim H0
-[ #b #c #Hleft #Hright #IH //
-| *)
-
-lemma sem_move_char :
- ∀alpha,sep.
- WRealize alpha (whileTM alpha (move_char alpha sep) 〈3,sep〉)
- (R_while_cmove alpha sep).
-#alpha #sep #inc #i #outc #Hloop
-lapply (sem_while … (sem_move_char alpha sep) inc i outc Hloop) [%]
-* #t1 * #Hstar @(star_ind_r ??????? Hstar)
-[ #a whd in ⊢ (% → ?); #H1 #a #b #ls #rs #rs' #Hb #Hrs #Hinc
- >Hinc in H1; normalize in ⊢ (% → ?); #H1
- cases (H1 ??)
- [ #Hfalse @False_ind @(absurd ?? Hb) destruct %
- |% #H2 destruct (H2)
- |% #H2 destruct ]
-| #a #b #c #Hstar1 #HRtrue #IH #HRfalse
- lapply (IH HRfalse) -IH whd in ⊢ (%→%); #IH
- #a0 #b0 #ls #rs #rs' #Hb0 #Hrs #Ha
- whd in Hstar1;
- lapply (Hstar1 … Hb0 Ha) #Hb
- @(IH … Hb0 Hrs) >Hb whd in HRfalse;
- [
- inc Rtrue* b Rtrue c Rfalse outc
-
-|
-]
-qed.
-
- #H1
- #a #b #ls #rs #rs #Hb #Hrs #Hinc
- >Hinc in H2;whd in ⊢ ((??%? → ?) → ?);
-
-lapply (H inc i outc Hloop) *
-
-
-(* (*
-
-(* 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) //
+ |@(ex_intro … i) @(ex_intro … outc)
+ @(loop_lift_acc ?? i (λc.c) ?????? (λc.cstate ?? c == acc) ???? Hloop)
+ [#x #Hx >(\P Hx) //
+ |#x @halt_while_not_acc
+ |#x #H whd in ⊢ (??%%); >while_trans_false [%]
+ % #eqx >eqx in H; >Hacctrue #H destruct
+ |@Hcase
]
]
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) //
+theorem terminate_while_guarded: ∀sig,M,acc,Pre,Rtrue,Rfalse.
+ halt sig M acc = true →
+ accGRealize sig M acc Pre Rtrue Rfalse →
+ (∀t1,t2. Pre t1 → Rtrue t1 t2 → Pre t2) → ∀t.
+ WF ? (inv … Rtrue) t → Pre t → whileTM sig M acc ↓ t.
+#sig #M #acc #Pre #Rtrue #Rfalse #Hacctrue #HM #Hinv #t #HWF elim HWF
+#t1 #H #Hind #HPre cases (HM … t1 HPre) #i * #outc * * #Hloop
+#Htrue #Hfalse cases (true_or_false (cstate … outc == acc)) #Hcase
+ [cases (Hind ? (Htrue … (\P Hcase)) ?)
+ [2: @(Hinv … HPre) @Htrue @(\P Hcase)]
+ #iwhile * #outcfinal
+ #Hloopwhile @(ex_intro … (i+iwhile))
+ @(ex_intro … outcfinal) @(loop_merge … outc … Hloopwhile)
+ [@(λc.halt sig M (cstate … c))
+ |* #s0 #t0 normalize cases (s0 == acc) normalize
+ [ cases (halt sig M s0) //
+ | cases (halt sig M s0) normalize //
+ ]
+ |@(loop_lift ?? i (λc.c) ?
+ (step ? (whileTM ? M acc)) ?
+ (λc.halt sig M (cstate ?? c)) ??
+ ?? Hloop)
+ [ #x %
+ | * #s #t #Hx whd in ⊢ (??%%); >while_trans_false
+ [%
+ |% #Hfalse <Hfalse in Hacctrue; >Hx #H0 destruct ]
+ ]
+ |@step_while_acc @(\P Hcase)
+ |>(\P Hcase) @halt_while_acc
+ ]
+ |@(ex_intro … i) @(ex_intro … outc)
+ @(loop_lift_acc ?? i (λc.c) ?????? (λc.cstate ?? c == acc) ???? Hloop)
+ [#x #Hx >(\P Hx) //
+ |#x @halt_while_not_acc
+ |#x #H whd in ⊢ (??%%); >while_trans_false [%]
+ % #eqx >eqx in H; >Hacctrue #H destruct
+ |@Hcase
]
]
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).
-*)
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projects / helm.git / blobdiff