--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A||
+ \ / This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ V_____________________________________________________________*)
+
+include "basics/vectors.ma".
+(* include "basics/relations.ma". *)
+
+record tape (sig:FinSet): Type[0] ≝
+{ left : list sig;
+ right: list sig
+}.
+
+inductive move : Type[0] ≝
+| L : move
+| R : move
+.
+
+(* 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,states:FinSet): Type[0] ≝
+{ cstate : states;
+ 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 (states 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. 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_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 (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).
+
+
+definition accRealize ≝ λsig.λM:TM sig.λacc:states sig M.λRtrue,Rfalse:relation (tape sig).
+∀t.∃i.∃outc.
+ loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) (initc sig M t) = Some ? outc ∧
+ (cstate ?? outc = acc → Rtrue t (ctape ?? outc)) ∧
+ (cstate ?? outc ≠ acc → Rfalse 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,S1,S2,c.match c with
+ [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inl … s) t ].
+
+definition lift_confR ≝
+ λsig,S1,S2,c.match c with
+ [ mk_config s t ⇒ mk_config sig (FinSum S1 S2) (inr … s) t ].
+
+definition halt_liftL ≝
+ λS1,S2,halt.λs:FinSum S1 S2.
+ match s with
+ [ inl s1 ⇒ halt s1
+ | inr _ ⇒ true ]. (* should be vacuous in all cases we use halt_liftL *)
+
+definition halt_liftR ≝
+ λS1,S2,halt.λs:FinSum S1 S2.
+ match s with
+ [ inl _ ⇒ false
+ | inr s2 ⇒ halt s2 ].
+
+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.
+ 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 trans_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〉.
+#sig #M1 * #Q2 #T2 #init2 #halt2 #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_confR : ∀sig,M1,M2,c0.
+ 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)
+[% | //]
+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)
+[% | //]
+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) %
+ | normalize <Hhlift // @IH ]
+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 ?? (halt sig M1) (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 ⊢ (??%? → ??%?);
+ cases (true_or_false (halt ?? (cstate sig (states ? M1) c0))) #Hc0 >Hc0
+ [ >(?: halt_liftL ?? (halt sig M1) (cstate sig ? (lift_confL … c0)) = true)
+ [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2) %
+ | <Hc0 cases c0 // ]
+ | >(?: halt_liftL ?? (halt sig M1) (cstate ?? (lift_confL … c0)) = false)
+ [whd in ⊢ (??%? → ??%?); #Hc2 <(IH ? Hc2) @eq_f
+ @step_lift_confL //
+ | <Hc0 cases c0 // ]
+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 ⊢ (??%? → ??%?);
+ cases (true_or_false (halt ?? (cstate sig ? c0))) #Hc0 >Hc0
+ [ >(?: halt ? (seq sig M1 M2) (cstate sig ? (lift_confR … c0)) = true)
+ [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2) %
+ | <Hc0 cases c0 // ]
+ | >(?: halt ? (seq sig M1 M2) (cstate sig ? (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 %
+qed.
+
+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,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).
+#sig #M1 #M2 #R1 #R2 #HR1 #HR2 #t #i #outc #Hloop
+cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
+cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
+@(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
+%
+[@(loop_split ???????????
+ (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) ]
+ || #c0 #Hhalt <step_lift_confL //
+ | #x <p_halt_liftL %
+ |6:cases outc1 #s1 #t1 %
+ |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2)
+ [ * #s2 #t2 %
+ | #c0 #Hhalt <step_lift_confR // ]
+ |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 ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
+ % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
+]
+qed.
+
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