include "basics/vectors.ma".
(* include "basics/relations.ma". *)
-record tape (sig:FinSet): Type[0] ≝
-{ left : list sig;
- right: list sig
-}.
+(******************************** tape ****************************************)
+
+(* A tape is essentially a triple 〈left,current,right〉 where however the current
+symbol could be missing. This may happen for three different reasons: both tapes
+are empty; we are on the left extremity of a non-empty tape (left overflow), or
+we are on the right extremity of a non-empty tape (right overflow). *)
+
+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 ? ].
+
+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 ] ].
inductive move : Type[0] ≝
-| L : move
-| R : move
-.
+ | L : move | R : move | N : move.
-(* We do not distinuish an input tape *)
+(********************************** machine ***********************************)
record TM (sig:FinSet): Type[1] ≝
{ states : FinSet;
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_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
+ 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))
- ]
- ].
+ | 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)
+ ] ].
+
+record config (sig,states:FinSet): Type[0] ≝
+{ cstate : states;
+ ctape: tape sig
+}.
+
+lemma config_expand: ∀sig,Q,c.
+ c = mk_config sig Q (cstate ?? c) (ctape ?? c).
+#sig #Q * //
+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.
-definition step ≝ λsig.λM:TM sig.λc:config sig M.
- let current_char ≝ option_hd ? (right ? (ctape ?? c)) in
+definition step ≝ λsig.λM:TM sig.λc:config sig (states sig M).
+ 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).
-
+
+(******************************** loop ****************************************)
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_S_true :
+ ∀A,n,f,p,a. p a = true →
+ loop A (S n) f p a = Some ? a.
+#A #n #f #p #a #pa normalize >pa //
+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.
+
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.
]
qed.
-lemma loop_split : ∀A,f,p,q.(∀b. p b = false → q b = false) →
+lemma loop_merge : ∀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 →
[#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) //
+ |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) //
- ]
- ]
+lemma loop_split : ∀A,f,p,q.(∀b. q b = true → p b = true) →
+ ∀k,a1,a2.
+ loop A k f q a1 = Some ? a2 →
+ ∃k1,a3.
+ loop A k1 f p a1 = Some ? a3 ∧
+ loop A (S(k-k1)) f q a3 = Some ? a2.
+#A #f #p #q #Hpq #k elim k
+ [#a1 #a2 normalize #Heq destruct
+ |#i #Hind #a1 #a2 normalize
+ cases (true_or_false (q a1)) #Hqa1 >Hqa1 normalize
+ [ #Ha1a2 destruct
+ @(ex_intro … 1) @(ex_intro … a2) %
+ [normalize >(Hpq …Hqa1) // |>Hqa1 //]
+ |#Hloop cases (true_or_false (p a1)) #Hpa1
+ [@(ex_intro … 1) @(ex_intro … a1) %
+ [normalize >Hpa1 // |>Hqa1 <Hloop normalize //]
+ |cases (Hind …Hloop) #k2 * #a3 * #Hloop1 #Hloop2
+ @(ex_intro … (S k2)) @(ex_intro … a3) %
+ [normalize >Hpa1 normalize // | @Hloop2 ]
+ ]
+ ]
+ ]
+qed.
+
+lemma loop_eq : ∀sig,f,q,i,j,a,x,y.
+ loop sig i f q a = Some ? x → loop sig j f q a = Some ? y → x = y.
+#sig #f #q #i #j @(nat_elim2 … i j)
+[ #n #a #x #y normalize #Hfalse destruct (Hfalse)
+| #n #a #x #y #H1 normalize #Hfalse destruct (Hfalse)
+| #n1 #n2 #IH #a #x #y normalize cases (q a) normalize
+ [ #H1 #H2 destruct %
+ | /2/ ]
+]
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 elim k
+ [#a #b normalize #Hfalse destruct
+ |#k0 #IH #a #b whd in ⊢ (??%? → ?); cases (true_or_false (p a)) #Hpa
+ [ >Hpa normalize #H1 destruct // | >Hpa normalize @IH ]
+ ]
+qed.
+
+lemma loop_lift : ∀A,B,k,lift,f,g,h,hlift,c1,c2.
+ (∀x.hlift (lift x) = h x) →
+ (∀x.h x = false → lift (f x) = g (lift x)) →
+ loop A k f h c1 = Some ? c2 →
+ loop B k g hlift (lift c1) = Some ? (lift … c2).
+#A #B #k #lift #f #g #h #hlift #c1 #c2 #Hfg #Hhlift
+generalize in match c1; elim k
+[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
+|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
+ cases (true_or_false (h c0)) #Hc0 >Hfg >Hc0 normalize
+ [ #Heq destruct (Heq) % | <Hhlift // @IH ]
+qed.
+
+(************************** Realizability *************************************)
+definition loopM ≝ λsig,M,i,cin.
+ loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) cin.
definition initc ≝ λsig.λM:TM sig.λt.
- mk_config sig M (start sig M) t.
+ mk_config sig (states 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).
+ loopM sig M i (initc sig M t) = Some ? outc ∧ R t (ctape ?? outc).
+
+definition WRealize ≝ λsig.λM:TM sig.λR:relation (tape sig).
+∀t,i,outc.
+ loopM sig M i (initc sig M t) = Some ? outc → R t (ctape ?? outc).
+
+definition Terminate ≝ λsig.λM:TM sig.λt. ∃i,outc.
+ loopM sig M i (initc sig M t) = Some ? outc.
+
+notation "M \vDash R" non associative with precedence 45 for @{ 'models $M $R}.
+interpretation "realizability" 'models M R = (Realize ? M R).
-(* Compositions *)
+notation "M \VDash R" non associative with precedence 45 for @{ 'wmodels $M $R}.
+interpretation "weak realizability" 'wmodels M R = (WRealize ? M R).
+
+interpretation "termination" 'fintersects M t = (Terminate ? M t).
+
+lemma WRealize_to_Realize : ∀sig.∀M: TM sig.∀R.
+ (∀t.M ↓ t) → M ⊫ R → M ⊨ R.
+#sig #M #R #HT #HW #t cases (HT … t) #i * #outc #Hloop
+@(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
+qed.
+
+theorem Realize_to_WRealize : ∀sig.∀M:TM sig.∀R.
+ M ⊨ R → M ⊫ R.
+#sig #M #R #H1 #inc #i #outc #Hloop
+cases (H1 inc) #k * #outc1 * #Hloop1 #HR >(loop_eq … Hloop Hloop1) //
+qed.
+
+definition accRealize ≝ λsig.λM:TM sig.λacc:states sig M.λRtrue,Rfalse.
+∀t.∃i.∃outc.
+ loopM sig M i (initc sig M t) = Some ? outc ∧
+ (cstate ?? outc = acc → Rtrue t (ctape ?? outc)) ∧
+ (cstate ?? outc ≠ acc → Rfalse t (ctape ?? outc)).
+
+(******************************** NOP Machine *********************************)
+
+(* NO OPERATION
+ t1 = t2 *)
+
+definition nop_states ≝ initN 1.
+definition start_nop : initN 1 ≝ mk_Sig ?? 0 (le_n … 1).
+
+definition nop ≝
+ λalpha:FinSet.mk_TM alpha nop_states
+ (λp.let 〈q,a〉 ≝ p in 〈q,None ?〉)
+ start_nop (λ_.true).
+
+definition R_nop ≝ λalpha.λt1,t2:tape alpha.t2 = t1.
+
+lemma sem_nop :
+ ∀alpha.nop alpha ⊨ R_nop alpha.
+#alpha #intape @(ex_intro ?? 1)
+@(ex_intro … (mk_config ?? start_nop intape)) % %
+qed.
+
+(************************** Sequential Composition ****************************)
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〉
+ 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.
(FinSum (states sig M1) (states sig M2))
(seq_trans sig M1 M2)
(inl … (start sig M1))
- (λs.match s with
+ (λs.match s with
[ inl _ ⇒ false | inr s2 ⇒ halt sig M2 s2]).
+notation "a · b" non associative with precedence 65 for @{ 'middot $a $b}.
+interpretation "sequential composition" 'middot a b = (seq ? a b).
+
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 ].
+ λsig,S1,S2,c.match c with
+ [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (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 ].
+ λsig,S1,S2,c.match c with
+ [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inr … s) t ].
definition halt_liftL ≝
- λsig.λM1,M2:TM sig.λs:FinSum (states ? M1) (states ? M2).
+ λS1,S2,halt.λs:FinSum S1 S2.
match s with
- [ inl s1 ⇒ halt sig M1 s1
+ [ inl s1 ⇒ halt 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).
+ λS1,S2,halt.λs:FinSum S1 S2.
match s with
[ inl _ ⇒ false
- | inr s2 ⇒ halt sig M2 s2 ].
+ | inr s2 ⇒ halt 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 %
+lemma p_halt_liftL : ∀sig,S1,S2,halt,c.
+ halt (cstate sig S1 c) =
+ halt_liftL S1 S2 halt (cstate … (lift_confL … c)).
+#sig #S1 #S2 #halt #c cases c #s #t %
qed.
-lemma trans_liftL : ∀sig,M1,M2,s,a,news,move.
+lemma trans_seq_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〉.
#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
qed.
-lemma trans_liftR : ∀sig,M1,M2,s,a,news,move.
+lemma trans_seq_liftR : ∀sig,M1,M2,s,a,news,move.
halt ? M2 s = false →
trans sig M2 〈s,a〉 = 〈news,move〉 →
trans sig (seq sig M1 M2) 〈inr … s,a〉 = 〈inr … 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_confR : ∀sig,M1,M2,c0.
+lemma step_seq_liftR : ∀sig,M1,M2,c0.
halt ? M2 (cstate ?? c0) = false →
- step sig (seq sig M1 M2) (lift_confR sig M1 M2 c0) =
- lift_confR sig M1 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)
-[% | //]
+ 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 #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_seq_liftR … Heq) //
qed.
-lemma step_lift_confL : ∀sig,M1,M2,c0.
+lemma step_seq_liftL : ∀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)
-[% | //]
+ 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 #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_seq_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.
-
-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 generalize in match c1;
-elim k
-[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
-|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
- lapply (refl ? (halt ?? (cstate sig M2 c0)))
- cases (halt ?? (cstate sig M2 c0)) in ⊢ (???% → ?); #Hc0 >Hc0
- [ >(?: halt ?? (cstate sig (seq ? M1 M2) (lift_confR … c0)) = true)
- [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2) %
- | <Hc0 cases c0 // ]
- | >(?: halt ?? (cstate sig (seq ? M1 M2) (lift_confR … c0)) = false)
- [whd in ⊢ (??%? → ??%?); #Hc2 <(IH ? Hc2) @eq_f
- @step_lift_confR //
- | <Hc0 cases c0 // ]
- ]
-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 elim k
-[#a #b normalize #Hfalse destruct
-|#k0 #IH #a #b whd in ⊢ (??%? → ?); cases (true_or_false (p a)) #Hpa
- [ >Hpa normalize #H1 destruct //
- | >Hpa normalize @IH
- ]
-]
-qed.
-
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 %
+#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 %
+lemma eq_ctape_lift_conf_L : ∀sig,S1,S2,outc.
+ ctape sig (FinSum S1 S2) (lift_confL … outc) = ctape … outc.
+#sig #S1 #S2 #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 %
+lemma eq_ctape_lift_conf_R : ∀sig,S1,S2,outc.
+ ctape sig (FinSum S1 S2) (lift_confR … outc) = ctape … outc.
+#sig #S1 #S2 #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).
+theorem sem_seq: ∀sig.∀M1,M2:TM sig.∀R1,R2.
+ M1 ⊨ R1 → M2 ⊨ R2 → 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
+cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
@(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
%
-[@(loop_split ??????????? (loop_liftL … Hloop1))
- [* *
+[@(loop_merge ???????????
+ (loop_lift ??? (lift_confL sig (states sig M1) (states sig M2))
+ (step sig M1) (step sig (seq sig M1 M2))
+ (λc.halt sig M1 (cstate … c))
+ (λc.halt_liftL ?? (halt sig M1) (cstate … c)) … 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) ]
+ || #c0 #Hhalt <step_seq_liftL //
+ | #x <p_halt_liftL %
+ |6:cases outc1 #s1 #t1 %
+ |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2)
+ [ * #s2 #t2 %
+ | #c0 #Hhalt <step_seq_liftR // ]
+ |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 whd in ⊢ (???%); //
+ | @(loop_Some ?????? Hloop10) ]
]
-| @(ex_intro … (ctape ? (seq sig M1 M2) (lift_confL … outc1)))
- % //
+| @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
+ % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
]
qed.
+theorem sem_seq_app: ∀sig.∀M1,M2:TM sig.∀R1,R2,R3.
+ M1 ⊨ R1 → M2 ⊨ R2 → R1 ∘ R2 ⊆ R3 → M1 · M2 ⊨ R3.
+#sig #M1 #M2 #R1 #R2 #R3 #HR1 #HR2 #Hsub
+#t cases (sem_seq … HR1 HR2 t)
+#k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
+% [@Hloop |@Hsub @Houtc]
+qed.
projects / helm.git / blobdiff