]
qed.
+lemma dec_test: ∀alpha,rs,test.
+ decidable (∀c.memb alpha c rs = true → test c = false).
+#alpha #rs #test elim rs
+ [%1 #n normalize #H destruct
+ |#a #tl cases (true_or_false (test a)) #Ha
+ [#_ %2 % #Hall @(absurd ?? not_eq_true_false) <Ha
+ @Hall @memb_hd
+ |* [#Hall %1 #c #memc cases (orb_true_l … memc)
+ [#eqca >(\P eqca) @Ha |@Hall]
+ |#Hnall %2 @(not_to_not … Hnall) #Hall #c #memc @Hall @memb_cons //
+ ]
+ qed.
+
definition R_adv_to_mark_r ≝ λalpha,test,t1,t2.
+ (current ? t1 = None ? → t1 = t2) ∧
∀ls,c,rs.
(t1 = midtape alpha ls c rs →
((test c = true ∧ t2 = t1) ∨
(test c = false ∧
- ∀rs1,b,rs2. rs = rs1@b::rs2 →
+ (∀rs1,b,rs2. rs = rs1@b::rs2 →
test b = true → (∀x.memb ? x rs1 = true → test x = false) →
- t2 = midtape ? (reverse ? rs1@c::ls) b rs2))).
+ t2 = midtape ? (reverse ? rs1@c::ls) b rs2) ∧
+ ((∀x.memb ? x rs = true → test x = false) →
+ ∀a,l.reverse ? (c::rs) = a::l →
+ t2 = rightof alpha a (l@ls))))).
definition adv_to_mark_r ≝
λalpha,test.whileTM alpha (atmr_step alpha test) atm2.
#alpha #test #t #i #outc #Hloop
lapply (sem_while … (sem_atmr_step alpha test) t i outc Hloop) [%]
-Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
-[ #tapea * #Htapea *
- [ #H1 #ls #c #rs #H2 >H2 in H1; whd in ⊢ (??%? → ?);
- #Hfalse destruct (Hfalse)
- | * #a * #Ha #Htest #ls #c #rs #H2 %
- >H2 in Ha; whd in ⊢ (??%? → ?); #Heq destruct (Heq) % //
- <Htapea //
- ]
-| #tapea #tapeb #tapec #Hleft #Hright #IH #HRfalse
- lapply (IH HRfalse) -IH #IH
- #ls #c #rs #Htapea %2
- cases Hleft #ls0 * #a0 * #rs0 * * #Htapea' #Htest #Htapeb
-
- >Htapea' in Htapea; #Htapea destruct (Htapea) % // *
- [ #b #rs2 #Hrs >Hrs in Htapeb; #Htapeb #Htestb #_
- cases (IH … Htapeb)
- [ * #_ #Houtc >Houtc >Htapeb %
- | * #Hfalse >Hfalse in Htestb; #Htestb destruct (Htestb) ]
- | #r1 #rs1 #b #rs2 #Hrs >Hrs in Htapeb; #Htapeb #Htestb #Hmemb
- cases (IH … Htapeb)
- [ * #Hfalse >(Hmemb …) in Hfalse;
- [ #Hft destruct (Hft)
- | @memb_hd ]
- | * #Htestr1 #H1 >reverse_cons >associative_append
- @H1 // #x #Hx @Hmemb @memb_cons //
+[ * #Htapea *
+ [ #H1 %
+ [#_ @Htapea
+ |#ls #c #rs #H2 >H2 in H1; whd in ⊢ (??%? → ?);
+ #Hfalse destruct (Hfalse)
+ ]
+ | * #a * #Ha #Htest %
+ [ >Ha #H destruct (H);
+ | #ls #c #rs #H2 %
+ >H2 in Ha; whd in ⊢ (??%? → ?); #Heq destruct (Heq) % //
+ <Htapea //
]
]
+| #tapeb #tapec #Hleft #Hright #IH #HRfalse
+ lapply (IH HRfalse) -IH #IH %
+ [cases Hleft #ls * #a * #rs * * #Htapea #_ #_ >Htapea
+ whd in ⊢((??%?)→?); #H destruct (H);
+ |#ls #c #rs #Htapea %2
+ cases Hleft #ls0 * #a0 * #rs0 * * #Htapea' #Htest #Htapeb
+ >Htapea' in Htapea; #Htapea destruct (Htapea) % [ % // ]
+ [*
+ [ #b #rs2 #Hrs >Hrs in Htapeb; #Htapeb #Htestb #_
+ cases (proj2 ?? IH … Htapeb)
+ [ * #_ #Houtc >Houtc >Htapeb %
+ | * * >Htestb #Hfalse destruct (Hfalse) ]
+ | #r1 #rs1 #b #rs2 #Hrs >Hrs in Htapeb; #Htapeb #Htestb #Hmemb
+ cases (proj2 ?? IH … Htapeb)
+ [ * #Hfalse >(Hmemb …) in Hfalse;
+ [ #Hft destruct (Hft)
+ | @memb_hd ]
+ | * * #Htestr1 #H1 #_ >reverse_cons >associative_append
+ @H1 // #x #Hx @Hmemb @memb_cons //
+ ]
+ ]
+ |cases rs in Htapeb; normalize in ⊢ (%→?);
+ [#Htapeb #_ #a0 #l whd in ⊢ ((??%?)→?); #Hrev destruct (Hrev)
+ >Htapeb in IH; #IH cases (proj1 ?? IH … (refl …)) //
+ |#r1 #rs1 #Htapeb #Hmemb
+ cases (proj2 ?? IH … Htapeb) [ * >Hmemb [ #Hfalse destruct(Hfalse) ] @memb_hd ]
+ * #_ #H1 #a #l <(reverse_reverse … l) cases (reverse … l)
+ [#H cut (c::r1::rs1 = [a])
+ [<(reverse_reverse … (c::r1::rs1)) >H //]
+ #Hrev destruct (Hrev)
+ |#a1 #l2 >reverse_cons >reverse_cons >reverse_cons
+ #Hrev cut ([c] = [a1])
+ [@(append_l2_injective_r ?? (a::reverse … l2) … Hrev) //]
+ #Ha <Ha >associative_append @H1
+ [#x #membx @Hmemb @memb_cons @membx
+ |<(append_l1_injective_r ?? (a::reverse … l2) … Hrev) //
+ ]
qed.
lemma terminate_adv_to_mark_r :
ms0 (λq.q == ms1).
definition R_mark ≝ λalpha,t1,t2.
- ∀ls,c,b,rs.
- t1 = midtape (FinProd … alpha FinBool) ls 〈c,b〉 rs →
- t2 = midtape ? ls 〈c,true〉 rs.
+ (∀ls,c,b,rs.
+ t1 = midtape (FinProd … alpha FinBool) ls 〈c,b〉 rs →
+ t2 = midtape ? ls 〈c,true〉 rs) ∧
+ (current ? t1 = None ? → t2 = t1).
lemma sem_mark :
∀alpha.Realize ? (mark alpha) (R_mark alpha).
#alpha #intape @(ex_intro ?? 2) cases intape
[ @ex_intro
- [| % [ % | #ls #c #b #rs #Hfalse destruct ] ]
+ [| % [ % | % [#ls #c #b #rs #Hfalse destruct | // ]]]
|#a #al @ex_intro
- [| % [ % | #ls #c #b #rs #Hfalse destruct ] ]
+ [| % [ % | % [#ls #c #b #rs #Hfalse destruct | // ]]]
|#a #al @ex_intro
- [| % [ % | #ls #c #b #rs #Hfalse destruct ] ]
+ [| % [ % | % [#ls #c #b #rs #Hfalse destruct ] // ]]
| #ls * #c #b #rs
- @ex_intro [| % [ % | #ls0 #c0 #b0 #rs0 #H1 destruct (H1) % ] ] ]
+ @ex_intro [| % [ % | %
+ [#ls0 #c0 #b0 #rs0 #H1 destruct (H1) %
+ | whd in ⊢ ((??%?)→?); #H1 destruct (H1)]]]
qed.
clear0 (λq.q == clear1).
definition R_clear_mark ≝ λalpha,t1,t2.
+ (current ? t1 = None ? → t1 = t2) ∧
∀ls,c,b,rs.
t1 = midtape (FinProd … alpha FinBool) ls 〈c,b〉 rs →
t2 = midtape ? ls 〈c,false〉 rs.
∀alpha.Realize ? (clear_mark alpha) (R_clear_mark alpha).
#alpha #intape @(ex_intro ?? 2) cases intape
[ @ex_intro
- [| % [ % | #ls #c #b #rs #Hfalse destruct ] ]
+ [| % [ % | % [#_ %|#ls #c #b #rs #Hfalse destruct ]]]
|#a #al @ex_intro
- [| % [ % | #ls #c #b #rs #Hfalse destruct ] ]
+ [| % [ % | % [#_ %|#ls #c #b #rs #Hfalse destruct ]]]
|#a #al @ex_intro
- [| % [ % | #ls #c #b #rs #Hfalse destruct ] ]
+ [| % [ % | % [#_ %|#ls #c #b #rs #Hfalse destruct ]]]
| #ls * #c #b #rs
- @ex_intro [| % [ % | #ls0 #c0 #b0 #rs0 #H1 destruct (H1) % ] ] ]
+ @ex_intro [| % [ % | %
+ [whd in ⊢ (??%?→?); #H destruct| #ls0 #c0 #b0 #rs0 #H1 destruct (H1) % ]]]]
qed.
(* ADVANCE MARK RIGHT machine
clear_mark alpha · move_r ? · mark alpha.
definition R_adv_mark_r ≝ λalpha,t1,t2.
- ∀ls,c,d,b,rs.
- t1 = midtape (FinProd … alpha FinBool) ls 〈c,true〉 (〈d,b〉::rs) →
- t2 = midtape ? (〈c,false〉::ls) 〈d,true〉 rs.
+ (∀ls,c.
+ (∀d,b,rs.
+ t1 = midtape (FinProd … alpha FinBool) ls 〈c,true〉 (〈d,b〉::rs) →
+ t2 = midtape ? (〈c,false〉::ls) 〈d,true〉 rs) ∧
+ (t1 = midtape (FinProd … alpha FinBool) ls 〈c,true〉 [ ] →
+ t2 = rightof ? 〈c,false〉 ls)) ∧
+ (current ? t1 = None ? → t1 = t2).
lemma sem_adv_mark_r :
∀alpha.Realize ? (adv_mark_r alpha) (R_adv_mark_r alpha).
-#alpha #intape
-cases (sem_seq ????? (sem_clear_mark …)
- (sem_seq ????? (sem_move_r ?) (sem_mark alpha)) intape)
-#k * #outc * #Hloop whd in ⊢ (%→?);
-* #ta * whd in ⊢ (%→?); #Hs1 * #tb * whd in ⊢ (%→%→?); #Hs2 #Hs3
-@(ex_intro ?? k) @(ex_intro ?? outc) %
-[ @Hloop
-| -Hloop #ls #c #d #b #rs #Hintape @(Hs3 … b)
- @(Hs2 ls 〈c,false〉 (〈d,b〉::rs))
- @(Hs1 … Hintape)
-]
+#alpha
+@(sem_seq_app … (sem_clear_mark …)
+ (sem_seq ????? (sem_move_r ?) (sem_mark alpha)) …)
+#intape #outtape whd in ⊢ (%→?); * #ta *
+whd in ⊢ (%→?); #Hs1 whd in ⊢ (%→?); * #tb * #Hs2 whd in ⊢ (%→?); #Hs3 %
+ [#ls #c %
+ [#d #b #rs #Hintape @(proj1 … Hs3 ?? b ?)
+ @(proj2 … Hs2 ls 〈c,false〉 (〈d,b〉::rs))
+ @(proj2 ?? Hs1 … Hintape)
+ |#Hintape lapply (proj2 ?? Hs1 … Hintape) #Hta lapply (proj2 … Hs2 … Hta)
+ whd in ⊢ ((???%)→?); #Htb <Htb @(proj2 … Hs3) >Htb //
+ ]
+ |#Hcur lapply(proj1 ?? Hs1 … Hcur) #Hta >Hta >Hta in Hcur; #Hcur
+ lapply (proj1 ?? Hs2 … Hcur) #Htb >Htb >Htb in Hcur; #Hcur
+ @sym_eq @(proj2 ?? Hs3) @Hcur
+ ]
qed.
(* ADVANCE TO MARK (left)
axiomatized
*)
+definition atml_step ≝
+ λalpha:FinSet.λtest:alpha→bool.
+ mk_TM alpha atm_states
+ (λp.let 〈q,a〉 ≝ p in
+ match a with
+ [ None ⇒ 〈atm1, None ?〉
+ | Some a' ⇒
+ match test a' with
+ [ true ⇒ 〈atm1,None ?〉
+ | false ⇒ 〈atm2,Some ? 〈a',L〉〉 ]])
+ atm0 (λx.notb (x == atm0)).
+
+definition Ratml_step_true ≝
+ λalpha,test,t1,t2.
+ ∃ls,a,rs.
+ t1 = midtape alpha ls a rs ∧ test a = false ∧
+ t2 = mk_tape alpha (tail ? ls) (option_hd ? ls) (a :: rs).
+
+definition Ratml_step_false ≝
+ λalpha,test,t1,t2.
+ t1 = t2 ∧
+ (current alpha t1 = None ? ∨
+ (∃a.current ? t1 = Some ? a ∧ test a = true)).
+
+lemma atml_q0_q1 :
+ ∀alpha,test,ls,a0,rs. test a0 = true →
+ step alpha (atml_step alpha test)
+ (mk_config ?? atm0 (midtape … ls a0 rs)) =
+ mk_config alpha (states ? (atml_step alpha test)) atm1
+ (midtape … ls a0 rs).
+#alpha #test #ls #a0 #ts #Htest whd in ⊢ (??%?);
+whd in match (trans … 〈?,?〉); >Htest %
+qed.
+
+lemma atml_q0_q2 :
+ ∀alpha,test,ls,a0,rs. test a0 = false →
+ step alpha (atml_step alpha test)
+ (mk_config ?? atm0 (midtape … ls a0 rs)) =
+ mk_config alpha (states ? (atml_step alpha test)) atm2
+ (mk_tape … (tail ? ls) (option_hd ? ls) (a0 :: rs)).
+#alpha #test #ls #a0 #rs #Htest whd in ⊢ (??%?);
+whd in match (trans … 〈?,?〉); >Htest cases ls //
+qed.
+
+lemma sem_atml_step :
+ ∀alpha,test.
+ accRealize alpha (atml_step alpha test)
+ atm2 (Ratml_step_true alpha test) (Ratml_step_false alpha test).
+#alpha #test *
+[ @(ex_intro ?? 2)
+ @(ex_intro ?? (mk_config ?? atm1 (niltape ?))) %
+ [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
+| #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? atm1 (leftof ? a al)))
+ % [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
+| #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? atm1 (rightof ? a al)))
+ % [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
+| #ls #c #rs @(ex_intro ?? 2)
+ cases (true_or_false (test c)) #Htest
+ [ @(ex_intro ?? (mk_config ?? atm1 ?))
+ [| %
+ [ %
+ [ whd in ⊢ (??%?); >atml_q0_q1 //
+ | whd in ⊢ ((??%%)→?); #Hfalse destruct ]
+ | #_ % // %2 @(ex_intro ?? c) % // ]
+ ]
+ | @(ex_intro ?? (mk_config ?? atm2 (mk_tape ? (tail ? ls) (option_hd ? ls) (c::rs))))
+ %
+ [ %
+ [ whd in ⊢ (??%?); >atml_q0_q2 //
+ | #_ @(ex_intro ?? ls) @(ex_intro ?? c) @(ex_intro ?? rs)
+ % // % //
+ ]
+ | #Hfalse @False_ind @(absurd ?? Hfalse) %
+ ]
+ ]
+]
+qed.
definition R_adv_to_mark_l ≝ λalpha,test,t1,t2.
+ (current ? t1 = None ? → t1 = t2) ∧
∀ls,c,rs.
(t1 = midtape alpha ls c rs →
- ((test c = true â\88§ t2 = t1) â\88¨
- (test c = false â\88§
- ∀ls1,b,ls2. ls = ls1@b::ls2 →
+ ((test c = true â\86\92 t2 = t1) â\88§
+ (test c = false â\86\92
+ (∀ls1,b,ls2. ls = ls1@b::ls2 →
test b = true → (∀x.memb ? x ls1 = true → test x = false) →
- t2 = midtape ? ls2 b (reverse ? ls1@c::rs)))).
+ t2 = midtape ? ls2 b (reverse ? ls1@c::rs)) ∧
+ ((∀x.memb ? x ls = true → test x = false) →
+ ∀a,l. reverse ? (c::ls) = a::l → t2 = leftof ? a (l@rs))
+ ))).
-axiom adv_to_mark_l : ∀alpha:FinSet.(alpha → bool) → TM alpha.
-(* definition adv_to_mark_l ≝
- λalpha,test.whileTM alpha (atml_step alpha test) 2. *)
+definition adv_to_mark_l ≝
+ λalpha,test.whileTM alpha (atml_step alpha test) atm2.
-axiom wsem_adv_to_mark_l :
+lemma wsem_adv_to_mark_l :
∀alpha,test.
WRealize alpha (adv_to_mark_l alpha test) (R_adv_to_mark_l alpha test).
-(*
#alpha #test #t #i #outc #Hloop
-lapply (sem_while … (sem_atmr_step alpha test) t i outc Hloop) [%]
+lapply (sem_while … (sem_atml_step alpha test) t i outc Hloop) [%]
-Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
-[ #tapea * #Htapea *
- [ #H1 #ls #c #rs #H2 >H2 in H1; whd in ⊢ (??%? → ?);
- #Hfalse destruct (Hfalse)
- | * #a * #Ha #Htest #ls #c #rs #H2 %
- >H2 in Ha; whd in ⊢ (??%? → ?); #Heq destruct (Heq) % //
- <Htapea //
+[ * #Htapea *
+ [ #H1 %
+ [#_ @Htapea
+ |#ls #c #rs #H2 >H2 in H1; whd in ⊢ (??%? → ?);
+ #Hfalse destruct (Hfalse)
+ ]
+ | * #a * #Ha #Htest %
+ [>Ha #H destruct (H);
+ |#ls #c #rs #H2 %
+ [#Hc <Htapea //
+ |#Hc @False_ind >H2 in Ha; whd in ⊢ ((??%?)→?);
+ #H destruct (H) /2/
+ ]
+ ]
]
-| #tapea #tapeb #tapec #Hleft #Hright #IH #HRfalse
- lapply (IH HRfalse) -IH #IH
- #ls #c #rs #Htapea %2
- cases Hleft #ls0 * #a0 * #rs0 * * #Htapea' #Htest #Htapeb
-
- >Htapea' in Htapea; #Htapea destruct (Htapea) % // *
- [ #b #rs2 #Hrs >Hrs in Htapeb; #Htapeb #Htestb #_
- cases (IH … Htapeb)
- [ * #_ #Houtc >Houtc >Htapeb %
- | * #Hfalse >Hfalse in Htestb; #Htestb destruct (Htestb) ]
- | #r1 #rs1 #b #rs2 #Hrs >Hrs in Htapeb; #Htapeb #Htestb #Hmemb
- cases (IH … Htapeb)
- [ * #Hfalse >(Hmemb …) in Hfalse;
- [ #Hft destruct (Hft)
- | @memb_hd ]
- | * #Htestr1 #H1 >reverse_cons >associative_append
- @H1 // #x #Hx @Hmemb @memb_cons //
+| #tapeb #tapec #Hleft #Hright #IH #HRfalse
+ lapply (IH HRfalse) -IH #IH %
+ [cases Hleft #ls0 * #a0 * #rs0 * * #Htapea #_ #_ >Htapea
+ whd in ⊢ ((??%?)→?); #H destruct (H)
+ |#ls #c #rs #Htapea %
+ [#Hc cases Hleft #ls0 * #a0 * #rs0 * * #Htapea' #Htest @False_ind
+ >Htapea' in Htapea; #H destruct /2/
+ |cases Hleft #ls0 * #a * #rs0 *
+ * #Htapea1 >Htapea in Htapea1; #H destruct (H) #_ #Htapeb
+ #Hc %
+ [*
+ [#b #ls2 #Hls >Hls in Htapeb; #Htapeb #Htestb #_
+ cases (proj2 ?? IH … Htapeb) #H1 #_ >H1 // >Htapeb %
+ |#l1 #ls1 #b #ls2 #Hls >Hls in Htapeb; #Htapeb #Htestb #Hmemb
+ cases (proj2 ?? IH … Htapeb) #_ #H1 >reverse_cons >associative_append
+ @(proj1 ?? (H1 ?) … (refl …) Htestb …)
+ [@Hmemb @memb_hd
+ |#x #memx @Hmemb @memb_cons @memx
+ ]
+ ]
+ |cases ls0 in Htapeb; normalize in ⊢ (%→?);
+ [#Htapeb #Htest #a0 #l whd in ⊢ ((??%?)→?); #Hrev destruct (Hrev)
+ >Htapeb in IH; #IH cases (proj1 ?? IH … (refl …)) //
+ |#l1 #ls1 #Htapeb
+ cases (proj2 ?? IH … Htapeb) #_ #H1 #Htest #a0 #l
+ <(reverse_reverse … l) cases (reverse … l)
+ [#H cut (a::l1::ls1 = [a0])
+ [<(reverse_reverse … (a::l1::ls1)) >H //]
+ #Hrev destruct (Hrev)
+ |#a1 #l2 >reverse_cons >reverse_cons >reverse_cons
+ #Hrev cut ([a] = [a1])
+ [@(append_l2_injective_r ?? (a0::reverse … l2) … Hrev) //]
+ #Ha <Ha >associative_append @(proj2 ?? (H1 ?))
+ [@Htest @memb_hd
+ |#x #membx @Htest @memb_cons @membx
+ |<(append_l1_injective_r ?? (a0::reverse … l2) … Hrev) //
+ ]
+ ]
+ ]
+ ]
]
]
qed.
-*)
-axiom terminate_adv_to_mark_l :
+lemma terminate_adv_to_mark_l :
∀alpha,test.
∀t.Terminate alpha (adv_to_mark_l alpha test) t.
-(*
#alpha #test #t
-@(terminate_while … (sem_atmr_step alpha test))
+@(terminate_while … (sem_atml_step alpha test))
[ %
| cases t
[ % #t1 * #ls0 * #c0 * #rs0 * * #Hfalse destruct (Hfalse)
|2,3: #a0 #al0 % #t1 * #ls0 * #c0 * #rs0 * * #Hfalse destruct (Hfalse)
- | #ls #c #rs generalize in match c; -c generalize in match ls; -ls
- elim rs
- [#ls #c % #t1 * #ls0 * #c0 * #rs0 * *
+ | #ls elim ls
+ [#c #rs % #t1 * #ls0 * #c0 * #rs0 * *
#H1 destruct (H1) #Hc0 #Ht1 normalize in Ht1;
% #t2 * #ls1 * #c1 * #rs1 * * >Ht1
normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
- | #r0 #rs0 #IH #ls #c % #t1 * #ls0 * #c0 * #rs0 * *
+ | #rs0 #r0 #IH #ls #c % #t1 * #ls0 * #c0 * #rs0 * *
#H1 destruct (H1) #Hc0 #Ht1 normalize in Ht1;
>Ht1 @IH
]
]
]
qed.
-*)
lemma sem_adv_to_mark_l :
∀alpha,test.
adv_to_mark_l (FinProd alpha FinBool) (is_marked alpha) ·
adv_mark_r alpha.
+definition R_adv_both_marks ≝ λalpha,t1,t2.
+ ∀ls,x0,rs.
+ t1 = midtape (FinProd … alpha FinBool) ls 〈x0,true〉 rs →
+ (rs = [ ] → (* first case: rs empty *)
+ t2 = rightof (FinProd … alpha FinBool) 〈x0,false〉 ls) ∧
+ (∀l0,x,a,a0,b,l1,l1',l2.
+ ls = (l1@〈x,true〉::l0) →
+ (∀c.memb ? c l1 = true → is_marked ? c = false) →
+ rs = (〈a0,b〉::l2) →
+ reverse ? (〈x0,false〉::l1) = 〈a,false〉::l1' →
+ t2 = midtape ? (〈x,false〉::l0) 〈a,true〉 (l1'@〈a0,true〉::l2)).
+
+lemma sem_adv_both_marks :
+ ∀alpha.Realize ? (adv_both_marks alpha) (R_adv_both_marks alpha).
+#alpha
+@(sem_seq_app … (sem_adv_mark_r …)
+ (sem_seq ????? (sem_move_l …)
+ (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
+ (sem_adv_mark_r alpha))) …)
+#intape #outtape * #tapea * #Hta * #tb * #Htb * #tc * #Htc #Hout
+#ls #c #rs #Hintape %
+ [#Hrs >Hrs in Hintape; #Hintape lapply (proj2 ?? (proj1 ?? Hta … ) … Hintape) -Hta #Hta
+ lapply (proj1 … Htb) >Hta -Htb #Htb lapply (Htb (refl …)) -Htb #Htb
+ lapply (proj1 ?? Htc) <Htb -Htc #Htc lapply (Htc (refl …)) -Htc #Htc
+ @sym_eq >Htc @(proj2 ?? Hout …) <Htc %
+ |#l0 #x #a #a0 #b #l1 #l1' #l2 #Hls #Hmarks #Hrs #Hrev
+ >Hrs in Hintape; >Hls #Hintape
+ @(proj1 ?? (proj1 ?? Hout … ) ? false) -Hout
+ lapply (proj1 … (proj1 … Hta …) … Hintape) #Htapea
+ lapply (proj2 … Htb … Htapea) -Htb
+ whd in match (mk_tape ????) ; #Htapeb
+ lapply (proj1 ?? (proj2 ?? (proj2 ?? Htc … Htapeb) (refl …))) -Htc #Htc
+ change with ((?::?)@?) in match (cons ???); <Hrev >reverse_cons
+ >associative_append @Htc [%|%|@Hmarks]
+ ]
+qed.
+
+(*
definition R_adv_both_marks ≝
λalpha,t1,t2.
∀l0,x,a,l1,x0,a0,l2. (∀c.memb ? c l1 = true → is_marked ? c = false) →
- t1 = midtape (FinProd … alpha FinBool)
+ (t1 = midtape (FinProd … alpha FinBool)
(l1@〈a,false〉::〈x,true〉::l0) 〈x0,true〉 (〈a0,false〉::l2) →
- t2 = midtape ? (〈x,false〉::l0) 〈a,true〉 (reverse ? l1@〈x0,false〉::〈a0,true〉::l2).
+ t2 = midtape ? (〈x,false〉::l0) 〈a,true〉 (reverse ? l1@〈x0,false〉::〈a0,true〉::l2)) ∧
+ (t1 = midtape (FinProd … alpha FinBool)
+ (l1@〈a,false〉::〈x,true〉::l0) 〈x0,true〉 [] →
+ t2 = rightof ? 〈x0,false〉 (l1@〈a,false〉::〈x,true〉::l0)).
lemma sem_adv_both_marks :
∀alpha.Realize ? (adv_both_marks alpha) (R_adv_both_marks alpha).
| -Hloop #l0 #x #a #l1 #x0 #a0 #l2 #Hl1 #Hintape
@(Hs4 … false) -Hs4
lapply (Hs1 … Hintape) #Hta
- lapply (Hs2 … Hta) #Htb
+ lapply (proj2 … Hs2 … Hta) #Htb
cases (Hs3 … Htb)
[ * #Hfalse normalize in Hfalse; destruct (Hfalse)
| * #_ -Hs3 #Hs3
| >associative_append %
| >reverse_append #Htc @Htc ]
]
-qed.
+qed. *)
(*
MATCH AND ADVANCE(f)
definition R_match_and_adv ≝
λalpha,f,t1,t2.
- ∀l0,x,a,l1,x0,a0,l2. (∀c.memb ? c l1 = true → is_marked ? c = false) →
- t1 = midtape (FinProd … alpha FinBool)
- (l1@〈a,false〉::〈x,true〉::l0) 〈x0,true〉 (〈a0,false〉::l2) →
- (f 〈x0,true〉 = true ∧ t2 = midtape ? (〈x,false〉::l0) 〈a,true〉 (reverse ? l1@〈x0,false〉::〈a0,true〉::l2))
- ∨ (f 〈x0,true〉 = false ∧
- t2 = midtape ? (l1@〈a,false〉::〈x,true〉::l0) 〈x0,false〉 (〈a0,false〉::l2)).
+ ∀ls,x0,rs.
+ t1 = midtape (FinProd … alpha FinBool) ls 〈x0,true〉 rs →
+ ((* first case: (f 〈x0,true〉 = false) *)
+ (f 〈x0,true〉 = false) →
+ t2 = midtape (FinProd … alpha FinBool) ls 〈x0,false〉 rs) ∧
+ ((f 〈x0,true〉 = true) → rs = [ ] → (* second case: rs empty *)
+ t2 = rightof (FinProd … alpha FinBool) 〈x0,false〉 ls) ∧
+ ((f 〈x0,true〉 = true) →
+ ∀l0,x,a,a0,b,l1,l1',l2.
+ (* third case: we expect to have a mark on the left! *)
+ ls = (l1@〈x,true〉::l0) →
+ (∀c.memb ? c l1 = true → is_marked ? c = false) →
+ rs = 〈a0,b〉::l2 →
+ reverse ? (〈x0,false〉::l1) = 〈a,false〉::l1' →
+ t2 = midtape ? (〈x,false〉::l0) 〈a,true〉 (l1'@〈a0,true〉::l2)).
lemma sem_match_and_adv :
∀alpha,f.Realize ? (match_and_adv alpha f) (R_match_and_adv alpha f).
cases (sem_if ? (test_char ? f) … tc_true (sem_test_char ? f) (sem_adv_both_marks alpha) (sem_clear_mark ?) intape)
#k * #outc * #Hloop #Hif @(ex_intro ?? k) @(ex_intro ?? outc)
% [ @Hloop ] -Hloop
-cases Hif
-[ * #ta * whd in ⊢ (%→%→?); #Hta #Houtc
- #l0 #x #a #l1 #x0 #a0 #l2 #Hl1 #Hintape
- >Hintape in Hta; #Hta cases (Hta … (refl ??)) -Hta #Hf #Hta % %
- [ @Hf | @Houtc [ @Hl1 | @Hta ] ]
-| * #ta * whd in ⊢ (%→%→?); #Hta #Houtc
- #l0 #x #a #l1 #x0 #a0 #l2 #Hl1 #Hintape
- >Hintape in Hta; #Hta cases (Hta … (refl ??)) -Hta #Hf #Hta %2 %
- [ @Hf | >(Houtc … Hta) % ]
+(*
+@(sem_if_app … (sem_test_char ? f) (sem_adv_both_marks alpha) (sem_clear_mark ?))
+#intape #outape #Htb * #H *)
+cases Hif -Hif
+[ * #ta * whd in ⊢ (%→%→?); * * #c * #Hcurrent #fc #Hta #Houtc
+ #ls #x #rs #Hintape >Hintape in Hcurrent; whd in ⊢ ((??%?)→?); #H destruct (H) %
+ [%[>fc #H destruct (H)
+ |#_ #Hrs >Hrs in Hintape; #Hintape >Hintape in Hta; #Hta
+ cases (Houtc … Hta) #Houtc #_ @Houtc //
+ ]
+ |#l0 #x0 #a #a0 #b #l1 #l1' #l2 #Hls #Hmarks #Hrs #Hrev >Hintape in Hta; #Hta
+ @(proj2 ?? (Houtc … Hta) … Hls Hmarks Hrs Hrev)
+ ]
+| * #ta * * #H #Hta * #_ #Houtc #ls #c #rs #Hintape
+ >Hintape in H; #H lapply(H … (refl …)) #fc %
+ [%[#_ >Hintape in Hta; #Hta @(Houtc … Hta)
+ |>fc #H destruct (H)
+ ]
+ |>fc #H destruct (H)
+ ]
]
qed.
-(*
- if x = c
- then move_right; ----
- adv_to_mark_r;
- if current (* x0 *) = 0
- then advance_mark ----
- adv_to_mark_l;
- advance_mark
- else STOP
- else M
-*)
+definition R_match_and_adv_of ≝
+ λalpha,t1,t2.current (FinProd … alpha FinBool) t1 = None ? → t2 = t1.
+
+lemma sem_match_and_adv_of :
+ ∀alpha,f.Realize ? (match_and_adv alpha f) (R_match_and_adv_of alpha).
+#alpha #f #intape
+cases (sem_if ? (test_char ? f) … tc_true (sem_test_char ? f) (sem_adv_both_marks alpha) (sem_clear_mark ?) intape)
+#k * #outc * #Hloop #Hif @(ex_intro ?? k) @(ex_intro ?? outc)
+% [ @Hloop ] -Hloop
+cases Hif
+[ * #ta * whd in ⊢ (%→%→?); #Hta #Houtc #Hcur
+ cases Hta * #x >Hcur * #Hfalse destruct (Hfalse)
+| * #ta * whd in ⊢ (%→%→?); * #_ #Hta * #Houtc #_ <Hta #Hcur >(Houtc Hcur) % ]
+qed.
+
+lemma sem_match_and_adv_full :
+ ∀alpha,f.Realize ? (match_and_adv alpha f)
+ (R_match_and_adv alpha f ∩ R_match_and_adv_of alpha).
+#alpha #f #intape cases (sem_match_and_adv ? f intape)
+#i * #outc * #Hloop #HR1 %{i} %{outc} % // % //
+cases (sem_match_and_adv_of ? f intape) #i0 * #outc0 * #Hloop0 #HR2
+>(loop_eq … Hloop Hloop0) //
+qed.
definition comp_step_subcase ≝ λalpha,c,elseM.
ifTM ? (test_char ? (λx.x == c))
definition R_comp_step_subcase ≝
λalpha,c,RelseM,t1,t2.
- ∀l0,x,rs.t1 = midtape (FinProd … alpha FinBool) l0 〈x,true〉 rs →
- (〈x,true〉 = c ∧
- ∀a,l1,x0,a0,l2. (∀c.memb ? c l1 = true → is_marked ? c = false) →
- rs = 〈a,false〉::l1@〈x0,true〉::〈a0,false〉::l2 →
- ((x = x0 ∧
- t2 = midtape ? (〈x,false〉::l0) 〈a,true〉 (l1@〈x0,false〉::〈a0,true〉::l2)) ∨
- (x ≠ x0 ∧
- t2 = midtape (FinProd … alpha FinBool)
- (reverse ? l1@〈a,false〉::〈x,true〉::l0) 〈x0,false〉 (〈a0,false〉::l2)))) ∨
- (〈x,true〉 ≠ c ∧ RelseM t1 t2).
+ ∀ls,x,rs.t1 = midtape (FinProd … alpha FinBool) ls 〈x,true〉 rs →
+ (〈x,true〉 = c →
+ ((* test true but no marks in rs *)
+ (∀c.memb ? c rs = true → is_marked ? c = false) →
+ ∀a,l. (a::l) = reverse ? (〈x,true〉::rs) →
+ t2 = rightof (FinProd alpha FinBool) a (l@ls)) ∧
+ ∀l1,x0,l2.
+ (∀c.memb ? c l1 = true → is_marked ? c = false) →
+ rs = l1@〈x0,true〉::l2 →
+ (x = x0 →
+ l2 = [ ] → (* test true but l2 is empty *)
+ t2 = rightof ? 〈x0,false〉 ((reverse ? l1)@〈x,true〉::ls)) ∧
+ (x = x0 →
+ ∀a,a0,b,l1',l2'. (* test true and l2 is not empty *)
+ 〈a,false〉::l1' = l1@[〈x0,false〉] →
+ l2 = 〈a0,b〉::l2' →
+ t2 = midtape ? (〈x,false〉::ls) 〈a,true〉 (l1'@〈a0,true〉::l2')) ∧
+ (x ≠ x0 →(* test false *)
+ t2 = midtape (FinProd … alpha FinBool) ((reverse ? l1)@〈x,true〉::ls) 〈x0,false〉 l2)) ∧
+ (〈x,true〉 ≠ c → RelseM t1 t2).
+
+lemma dec_marked: ∀alpha,rs.
+ decidable (∀c.memb ? c rs = true → is_marked alpha c = false).
+#alpha #rs elim rs
+ [%1 #n normalize #H destruct
+ |#a #tl cases (true_or_false (is_marked ? a)) #Ha
+ [#_ %2 % #Hall @(absurd ?? not_eq_true_false) <Ha
+ @Hall @memb_hd
+ |* [#Hall %1 #c #memc cases (orb_true_l … memc)
+ [#eqca >(\P eqca) @Ha |@Hall]
+ |#Hnall %2 @(not_to_not … Hnall) #Hall #c #memc @Hall @memb_cons //
+ ]
+ qed.
+
+(* axiom daemon:∀P:Prop.P. *)
lemma sem_comp_step_subcase :
∀alpha,c,elseM,RelseM.
(sem_test_char ? (λx.x == c))
(sem_seq ????? (sem_move_r …)
(sem_seq ????? (sem_adv_to_mark_r ? (is_marked alpha))
- (sem_match_and_adv ? (λx.x == c)))) Helse intape)
+ (sem_match_and_adv_full ? (λx.x == c)))) Helse intape)
#k * #outc * #Hloop #HR @(ex_intro ?? k) @(ex_intro ?? outc)
% [ @Hloop ] -Hloop cases HR -HR
-[ * #ta * whd in ⊢ (%→?); #Hta * #tb * whd in ⊢ (%→?); #Htb
- * #tc * whd in ⊢ (%→?); #Htc whd in ⊢ (%→?); #Houtc
- #l0 #x #rs #Hintape cases (true_or_false (〈x,true〉==c)) #Hc
- [ % % [ @(\P Hc) ]
- #a #l1 #x0 #a0 #l2 #Hl1 #Hrs >Hrs in Hintape; #Hintape
- >Hintape in Hta; #Hta cases (Hta ? (refl ??)) -Hta
- #Hx #Hta lapply (Htb … Hta) -Htb #Htb
- cases (Htc … Htb) [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
- -Htc * #_ #Htc lapply (Htc l1 〈x0,true〉 (〈a0,false〉::l2) (refl ??) (refl ??) Hl1)
- -Htc #Htc cases (Houtc ???????? Htc) -Houtc
- [ * #Hx0 #Houtc %
- % [ <(\P Hx0) in Hx; #Hx lapply (\P Hx) #Hx' destruct (Hx') %
- | >Houtc >reverse_reverse % ]
- | * #Hx0 #Houtc %2
- % [ <(\P Hx) in Hx0; #Hx0 lapply (\Pf Hx0) @not_to_not #Hx' >Hx' %
- | >Houtc % ]
- | #x #membx @Hl1 <(reverse_reverse …l1) @memb_reverse @membx ]
- | %2 % [ @(\Pf Hc) ]
- >Hintape in Hta; #Hta cases (Hta ? (refl ??)) -Hta #Hx #Hta
- >Hx in Hc;#Hc destruct (Hc) ]
-| * #ta * whd in ⊢ (%→?); #Hta #Helse #ls #c0 #rs #Hintape %2
- >Hintape in Hta; #Hta cases (Hta ? (refl ??)) -Hta #Hc #Hta %
- [ @(\Pf Hc) | <Hta @Helse ]
-]
+ [* #ta * whd in ⊢ (%→?); * * #cin * #Hcin #Hcintrue #Hta * #tb * whd in ⊢ (%→?); #Htb
+ * #tc * whd in ⊢ (%→?); #Htc * whd in ⊢ (%→%→?); #Houtc #Houtc1
+ #ls #x #rs #Hintape >Hintape in Hcin; whd in ⊢ ((??%?)→?); #H destruct (H) %
+ [#_ cases (dec_marked ? rs) #Hdec
+ [%
+ [#_ #a #l1
+ >Hintape in Hta; #Hta
+ lapply (proj2 ?? Htb … Hta) -Htb -Hta cases rs in Hdec;
+ (* by cases on rs *)
+ [#_ whd in ⊢ ((???%)→?); #Htb >Htb in Htc; #Htc
+ lapply (proj1 ?? Htc (refl …)) -Htc #Htc <Htc in Houtc1; #Houtc1
+ normalize in ⊢ (???%→?); #Hl1 destruct(Hl1) @(Houtc1 (refl …))
+ |#r0 #rs0 #Hdec whd in ⊢ ((???%)→?); #Htb >Htb in Htc; #Htc
+ >reverse_cons >reverse_cons #Hl1
+ cases (proj2 ?? Htc … (refl …))
+ [* >(Hdec …) [ #Hfalse destruct(Hfalse) ] @memb_hd
+ |* #_ -Htc #Htc cut (∃l2.l1 = l2@[〈x,true〉])
+ [generalize in match Hl1; -Hl1 <(reverse_reverse … l1)
+ cases (reverse ? l1)
+ [#Hl1 cut ([a]=〈x,true〉::r0::rs0)
+ [ <(reverse_reverse … (〈x,true〉::r0::rs0))
+ >reverse_cons >reverse_cons <Hl1 %
+ | #Hfalse destruct(Hfalse)]
+ |#a0 #l10 >reverse_cons #Heq
+ lapply (append_l2_injective_r ? (a::reverse ? l10) ???? Heq) //
+ #Ha0 destruct(Ha0) /2/ ]
+ |* #l2 #Hl2 >Hl2 in Hl1; #Hl1
+ lapply (append_l1_injective_r ? (a::l2) … Hl1) // -Hl1 #Hl1
+ >reverse_cons in Htc; #Htc lapply (Htc … (sym_eq … Hl1))
+ [ #x0 #Hmemb @Hdec @memb_cons @Hmemb ]
+ -Htc #Htc >Htc in Houtc1; #Houtc1 >associative_append @Houtc1 %
+ ]
+ ]
+ ]
+ |#l1 #x0 #l2 #_ #Hrs @False_ind
+ @(absurd ?? not_eq_true_false)
+ change with (is_marked ? 〈x0,true〉) in match true;
+ @Hdec >Hrs @memb_append_l2 @memb_hd
+ ]
+ |% [#H @False_ind @(absurd …H Hdec)]
+ (* by cases on l1 *) *
+ [#x0 #l2 #Hdec normalize in ⊢ (%→?); #Hrs >Hrs in Hintape; #Hintape
+ >Hintape in Hta; (* * * #x1 * whd in ⊢ ((??%?)→?); #H destruct (H) #Hx *)
+ #Hta lapply (proj2 … Htb … Hta) -Htb -Hta
+ whd in match (mk_tape ????); whd in match (tail ??); #Htb cases Htc -Htc
+ #_ #Htc cases (Htc … Htb) -Htc
+ [2: * * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
+ * * #Htc >Htb in Htc; -Htb #Htc cases (Houtc … Htc) -Houtc *
+ #H1 #H2 #H3 cases (true_or_false (x==x0)) #eqxx0
+ [>(\P eqxx0) % [2: #H @False_ind /2/] %
+ [#_ #Hl2 >(H2 … Hl2) <(\P eqxx0) [% | @Hcintrue]
+ |#_ #a #a0 #b #l1' #l2' normalize in ⊢ (%→?); #Hdes destruct (Hdes)
+ #Hl2 @(H3 … Hdec … Hl2) <(\P eqxx0) [@Hcintrue | % | @reverse_single]
+ ]
+ |% [% #eqx @False_ind lapply (\Pf eqxx0) #Habs @(absurd … eqx Habs)]
+ #_ @H1 @(\bf ?) @(not_to_not ??? (\Pf eqxx0)) <(\P Hcintrue)
+ #Hdes destruct (Hdes) %
+ ]
+ |#l1hd #l1tl #x0 #l2 #Hdec normalize in ⊢ (%→?); #Hrs >Hrs in Hintape; #Hintape
+ >Hintape in Hta; (* * * #x1 * whd in ⊢ ((??%?)→?); #H destruct (H) #Hx *)
+ #Hta lapply (proj2 … Htb … Hta) -Htb -Hta
+ whd in match (mk_tape ????); whd in match (tail ??); #Htb cases Htc -Htc
+ #_ #Htc cases (Htc … Htb) -Htc
+ [* #Hfalse @False_ind >(Hdec … (memb_hd …)) in Hfalse; #H destruct]
+ * * #_ #Htc lapply (Htc … (refl …) (refl …) ?) -Htc
+ [#x1 #membx1 @Hdec @memb_cons @membx1] #Htc
+ cases (Houtc … Htc) -Houtc *
+ #H1 #H2 #H3 #_ cases (true_or_false (x==x0)) #eqxx0
+ [>(\P eqxx0) % [2: #H @False_ind /2/] %
+ [#_ #Hl2 >(H2 … Hl2) <(\P eqxx0)
+ [>reverse_cons >associative_append % | @Hcintrue]
+ |#_ #a #a0 #b #l1' #l2' normalize in ⊢ (%→?); #Hdes (* destruct (Hdes) *)
+ #Hl2 @(H3 ?????? (reverse … (l1hd::l1tl)) … Hl2) <(\P eqxx0)
+ [@Hcintrue
+ |>reverse_cons >associative_append %
+ |#c0 #memc @Hdec <(reverse_reverse ? (l1hd::l1tl)) @memb_reverse @memc
+ |>Hdes >reverse_cons >reverse_reverse >(\P eqxx0) %
+ ]
+ ]
+ |% [% #eqx @False_ind lapply (\Pf eqxx0) #Habs @(absurd … eqx Habs)]
+ #_ >reverse_cons >associative_append @H1 @(\bf ?)
+ @(not_to_not ??? (\Pf eqxx0)) <(\P Hcintrue) #Hdes
+ destruct (Hdes) %
+ ]
+ ]
+ ]
+ |>(\P Hcintrue) * #Hfalse @False_ind @Hfalse %
+ ]
+ | * #ta * * #Hcur #Hta #Houtc
+ #l0 #x #rs #Hintape >Hintape in Hcur; #Hcur lapply (Hcur ? (refl …)) -Hcur #Hc %
+ [ #Hfalse >Hfalse in Hc; #Hc cases (\Pf Hc) #Hc @False_ind @Hc %
+ | -Hc #Hc <Hintape <Hta @Houtc ] ]
qed.
(*
(clear_mark …)))))
(nop ?)
tc_true.
+
+(* da spostare *)
+
+lemma mem_append : ∀A,x,l1,l2. mem A x (l1@l2) →
+ mem A x l1 ∨ mem A x l2.
+#A #x #l1 elim l1 normalize [/2/]
+#a #tl #Hind #l2 * [#eqxa %1 /2/ |#memx cases (Hind … memx) /3/]
+qed.
+
+let rec split_on A (l:list A) f acc on l ≝
+ match l with
+ [ nil ⇒ 〈acc,nil ?〉
+ | cons a tl ⇒
+ if f a then 〈acc,a::tl〉 else split_on A tl f (a::acc)
+ ].
-definition R_comp_step_true ≝
- λt1,t2.
- ∀l0,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) l0 c rs →
- ∃c'. c = 〈c',true〉 ∧
- ((bit_or_null c' = true ∧
- ∀a,l1,c0,a0,l2.
- rs = 〈a,false〉::l1@〈c0,true〉::〈a0,false〉::l2 →
+lemma split_on_spec: ∀A:DeqSet.∀l,f,acc,res1,res2.
+ split_on A l f acc = 〈res1,res2〉 →
+ (∃l1. res1 = l1@acc ∧
+ reverse ? l1@res2 = l ∧
+ ∀x. memb ? x l1 =true → f x = false) ∧
+ ∀a,tl. res2 = a::tl → f a = true.
+#A #l #f elim l
+ [#acc #res1 #res2 normalize in ⊢ (%→?); #H destruct %
+ [@(ex_intro … []) % normalize [% % | #x #H destruct]
+ |#a #tl #H destruct
+ ]
+ |#a #tl #Hind #acc #res1 #res2 normalize in ⊢ (%→?);
+ cases (true_or_false (f a)) #Hfa >Hfa normalize in ⊢ (%→?);
+ #H destruct
+ [% [@(ex_intro … []) % normalize [% % | #x #H destruct]
+ |#a1 #tl1 #H destruct (H) //]
+ |cases (Hind (a::acc) res1 res2 H) * #l1 * *
+ #Hres1 #Htl #Hfalse #Htrue % [2:@Htrue] @(ex_intro … (l1@[a])) %
+ [% [>associative_append @Hres1 | >reverse_append <Htl % ]
+ |#x #Hmemx cases (memb_append ???? Hmemx)
+ [@Hfalse | #H >(memb_single … H) //]
+ ]
+ ]
+ ]
+qed.
+
+axiom mem_reverse: ∀A,l,x. mem A x (reverse ? l) → mem A x l.
+
+lemma split_on_spec_ex: ∀A:DeqSet.∀l,f.∃l1,l2.
+ l1@l2 = l ∧ (∀x:A. memb ? x l1 = true → f x = false) ∧
+ ∀a,tl. l2 = a::tl → f a = true.
+#A #l #f @(ex_intro … (reverse … (\fst (split_on A l f []))))
+@(ex_intro … (\snd (split_on A l f [])))
+cases (split_on_spec A l f [ ] ?? (eq_pair_fst_snd …)) * #l1 * *
+>append_nil #Hl1 >Hl1 #Hl #Hfalse #Htrue %
+ [% [@Hl|#x #memx @Hfalse <(reverse_reverse … l1) @memb_reverse //] | @Htrue]
+qed.
+
+(* versione esistenziale *)
+
+definition R_comp_step_true ≝ λt1,t2.
+ ∃ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls 〈c,true〉 rs ∧
+ ((* bit_or_null c = false *)
+ (bit_or_null c = false → t2 = midtape ? ls 〈c,false〉 rs) ∧
+ (* no marks in rs *)
+ (bit_or_null c = true →
+ (∀c.memb ? c rs = true → is_marked ? c = false) →
+ ∀a,l. (a::l) = reverse ? (〈c,true〉::rs) →
+ t2 = rightof (FinProd FSUnialpha FinBool) a (l@ls)) ∧
+ (∀l1,c0,l2.
+ bit_or_null c = true →
(∀c.memb ? c l1 = true → is_marked ? c = false) →
- (c0 = c' ∧
- t2 = midtape ? (〈c',false〉::l0) 〈a,true〉 (l1@〈c0,false〉::〈a0,true〉::l2)) ∨
- (c0 ≠ c' ∧
+ rs = l1@〈c0,true〉::l2 →
+ (c = c0 →
+ l2 = [ ] → (* test true but l2 is empty *)
+ t2 = rightof ? 〈c0,false〉 ((reverse ? l1)@〈c,true〉::ls)) ∧
+ (c = c0 →
+ ∀a,a0,b,l1',l2'. (* test true and l2 is not empty *)
+ 〈a,false〉::l1' = l1@[〈c0,false〉] →
+ l2 = 〈a0,b〉::l2' →
+ t2 = midtape ? (〈c,false〉::ls) 〈a,true〉 (l1'@〈a0,true〉::l2')) ∧
+ (c ≠ c0 →(* test false *)
t2 = midtape (FinProd … FSUnialpha FinBool)
- (reverse ? l1@〈a,false〉::〈c',true〉::l0) 〈c0,false〉 (〈a0,false〉::l2))) ∨
- (bit_or_null c' = false ∧ t2 = midtape ? l0 〈c',false〉 rs)).
+ ((reverse ? l1)@〈c,true〉::ls) 〈c0,false〉 l2))).
definition R_comp_step_false ≝
λt1,t2.
∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
is_marked ? c = false ∧ t2 = t1.
+
+lemma is_marked_to_exists: ∀alpha,c. is_marked alpha c = true →
+ ∃c'. c = 〈c',true〉.
+#alpha * #c * [#_ @(ex_intro … c) //| normalize #H destruct]
+qed.
+
+lemma exists_current: ∀alpha,c,t.
+ current alpha t = Some alpha c → ∃ls,rs. t= midtape ? ls c rs.
+#alpha #c *
+ [whd in ⊢ (??%?→?); #H destruct
+ |#a #l whd in ⊢ (??%?→?); #H destruct
+ |#a #l whd in ⊢ (??%?→?); #H destruct
+ |#ls #c1 #rs whd in ⊢ (??%?→?); #H destruct
+ @(ex_intro … ls) @(ex_intro … rs) //
+ ]
+qed.
lemma sem_comp_step :
accRealize ? comp_step (inr … (inl … (inr … start_nop)))
R_comp_step_true R_comp_step_false.
-#intape
-cases (acc_sem_if … (sem_test_char ? (is_marked ?))
+@(acc_sem_if_app … (sem_test_char ? (is_marked ?))
(sem_comp_step_subcase FSUnialpha 〈bit false,true〉 ??
(sem_comp_step_subcase FSUnialpha 〈bit true,true〉 ??
(sem_comp_step_subcase FSUnialpha 〈null,true〉 ??
(sem_clear_mark …))))
- (sem_nop …) intape)
-#k * #outc * * #Hloop #H1 #H2
-@(ex_intro ?? k) @(ex_intro ?? outc) %
-[ % [@Hloop ] ] -Hloop
-[ #Hstate lapply (H1 Hstate) -H1 -Hstate -H2 *
- #ta * whd in ⊢ (%→?); #Hleft #Hright #ls #c #rs #Hintape
- >Hintape in Hleft; #Hleft cases (Hleft ? (refl ??)) -Hleft
- cases c in Hintape; #c' #b #Hintape whd in ⊢ (??%?→?);
- #Hb >Hb #Hta @(ex_intro ?? c') % //
- cases (Hright … Hta)
- [ * #Hc' #H1 % % [destruct (Hc') % ]
- #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
- cases (H1 … Hl1 Hrs)
- [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
- | * #Hneq #Houtc %2 %
- [ @sym_not_eq //
- | @Houtc ]
- ]
- | * #Hc #Helse1 cases (Helse1 … Hta)
- [ * #Hc' #H1 % % [destruct (Hc') % ]
- #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
- cases (H1 … Hl1 Hrs)
- [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
- | * #Hneq #Houtc %2 %
- [ @sym_not_eq //
- | @Houtc ]
+ (sem_nop …) …)
+[#intape #outape #ta #Hta #Htb cases Hta * #cm * #Hcur
+ cases (exists_current … Hcur) #ls * #rs #Hintape #cmark
+ cases (is_marked_to_exists … cmark) #c #Hcm
+ >Hintape >Hcm -Hintape -Hcm #Hta
+ @(ex_intro … ls) @(ex_intro … c) @(ex_intro …rs) % [//] lapply Hta -Hta
+ (* #ls #c #rs #Hintape whd in Hta;
+ >Hintape in Hta; * #_ -Hintape forse non serve *)
+ cases (true_or_false (c==bit false)) #Hc
+ [>(\P Hc) #Hta %
+ [%[whd in ⊢ ((??%?)→?); #Hdes destruct
+ |#Hc @(proj1 ?? (proj1 ?? (Htb … Hta) (refl …)))
+ ]
+ |#l1 #c0 #l2 #Hc @(proj2 ?? (proj1 ?? (Htb … Hta) (refl …)))
+ ]
+ |cases (true_or_false (c==bit true)) #Hc1
+ [>(\P Hc1) #Hta
+ cut (〈bit true, true〉 ≠ 〈bit false, true〉) [% #Hdes destruct] #Hneq %
+ [%[whd in ⊢ ((??%?)→?); #Hdes destruct
+ |#Hc @(proj1 … (proj1 ?? (proj2 ?? (Htb … Hta) Hneq … Hta) (refl …)))
+ ]
+ |#l1 #c0 #l2 #Hc @(proj2 ?? (proj1 ?? (proj2 ?? (Htb … Hta) Hneq … Hta)(refl …)))
]
- | * #Hc' #Helse2 cases (Helse2 … Hta)
- [ * #Hc'' #H1 % % [destruct (Hc'') % ]
- #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
- cases (H1 … Hl1 Hrs)
- [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
- | * #Hneq #Houtc %2 %
- [ @sym_not_eq //
- | @Houtc ]
+ |cases (true_or_false (c==null)) #Hc2
+ [>(\P Hc2) #Hta
+ cut (〈null, true〉 ≠ 〈bit false, true〉) [% #Hdes destruct] #Hneq
+ cut (〈null, true〉 ≠ 〈bit true, true〉) [% #Hdes destruct] #Hneq1 %
+ [%[whd in ⊢ ((??%?)→?); #Hdes destruct
+ |#Hc @(proj1 … (proj1 ?? (proj2 ?? (proj2 ?? (Htb … Hta) Hneq … Hta) Hneq1 … Hta) (refl …)))
+ ]
+ |#l1 #c0 #l2 #Hc @(proj2 ?? (proj1 ?? (proj2 ?? (proj2 ?? (Htb … Hta) Hneq … Hta) Hneq1 … Hta) (refl …)))
]
- | * #Hc'' whd in ⊢ (%→?); #Helse3 %2 %
- [ generalize in match Hc''; generalize in match Hc'; generalize in match Hc;
- cases c'
- [ * [ #_ #Hfalse @False_ind @(absurd ?? Hfalse) %
- | #Hfalse @False_ind @(absurd ?? Hfalse) % ]
- | #_ #_ #Hfalse @False_ind @(absurd ?? Hfalse) %
- |*: #_ #_ #_ % ]
- | @(Helse3 … Hta)
+ |#Hta cut (bit_or_null c = false)
+ [lapply Hc; lapply Hc1; lapply Hc2 -Hc -Hc1 -Hc2
+ cases c normalize [* normalize /2/] /2/] #Hcut %
+ [%[cases (Htb … Hta) #_ -Htb #Htb
+ cases (Htb … Hta) [2: % #H destruct (H) normalize in Hc; destruct] #_ -Htb #Htb
+ cases (Htb … Hta) [2: % #H destruct (H) normalize in Hc1; destruct] #_ -Htb #Htb
+ lapply (Htb ?) [% #H destruct (H) normalize in Hc2; destruct]
+ * #_ #Houttape #_ @(Houttape … Hta)
+ |>Hcut #H destruct
+ ]
+ |#l1 #c0 #l2 >Hcut #H destruct
]
]
]
]
-| #Hstate lapply (H2 Hstate) -H1 -Hstate -H2 *
- #ta * whd in ⊢ (%→%→?); #Hleft #Hright #ls #c #rs #Hintape
- >Hintape in Hleft; #Hleft
- cases (Hleft ? (refl ??)) -Hleft
- #Hc #Hta % // >Hright //
+|#intape #outape #ta #Hta #Htb #ls #c #rs #Hintape
+ >Hintape in Hta; whd in ⊢ (%→?); * #Hmark #Hta % [@Hmark //]
+ whd in Htb; >Htb //
]
qed.
+(* compare *)
+
definition compare ≝
whileTM ? comp_step (inr … (inl … (inr … start_nop))).
t2 = midtape ? l0 〈grid,false〉 (l1@〈comma,true〉::l2)) ∨
(b = bit x ∧ b = c ∧ bs = b0s
*)
+
+definition list_cases2: ∀A.∀P:list A →list A →Prop.∀l1,l2. |l1| = |l2| →
+P [ ] [ ] → (∀a1,a2:A.∀tl1,tl2. |tl1| = |tl2| → P (a1::tl1) (a2::tl2)) → P l1 l2.
+#A #P #l1 #l2 #Hlen lapply Hlen @(list_ind2 … Hlen) //
+#tl1 #tl2 #hd1 #hd2 #Hind normalize #HlenS #H1 #H2 @H2 //
+qed.
+
definition R_compare :=
λt1,t2.
∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
(∀c'.bit_or_null c' = false → c = 〈c',true〉 → t2 = midtape ? ls 〈c',false〉 rs) ∧
(∀c'. c = 〈c',false〉 → t2 = t1) ∧
+(* forse manca il caso no marks in rs *)
∀b,b0,bs,b0s,l1,l2.
|bs| = |b0s| →
(∀c.memb (FinProd … FSUnialpha FinBool) c bs = true → bit_or_null (\fst c) = true) →
#t #i #outc #Hloop
lapply (sem_while ?????? sem_comp_step t i outc Hloop) [%]
-Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
-[ #tapea whd in ⊢ (%→?); #Rfalse #ls #c #rs #Htapea %
+[ whd in ⊢ (%→?); #Rfalse #ls #c #rs #Htapea %
[ %
[ #c' #Hc' #Hc lapply (Rfalse … Htapea) -Rfalse * >Hc
whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse)
| #b #b0 #bs #b0s #l1 #l2 #Hlen #Hbs1 #Hb0s1 #Hbs2 #Hb0s2 #Hl1 #Hc
cases (Rfalse … Htapea) -Rfalse >Hc whd in ⊢ (??%?→?);#Hfalse destruct (Hfalse)
]
-| #tapea #tapeb #tapec #Hleft #Hright #IH #Htapec lapply (IH Htapec) -Htapec -IH #IH
- whd in Hleft; #ls #c #rs #Htapea cases (Hleft … Htapea) -Hleft
- #c' * #Hc >Hc cases (true_or_false (bit_or_null c')) #Hc'
- [2: *
- [ * >Hc' #H @False_ind destruct (H)
- | * #_ #Htapeb cases (IH … Htapeb) * #_ #H #_ %
- [%
- [#c1 #Hc1 #Heqc destruct (Heqc) <Htapeb @(H c1) %
- |#c1 #Hfalse destruct (Hfalse)
- ]
- |#b #b0 #bs #b0s #l1 #l2 #_ #_ #_ #_ #_ #_
- #Heq destruct (Heq) >Hc' #Hfalse @False_ind destruct (Hfalse)
+| #tapeb #tapec #Hleft #Hright #IH #Htapec lapply (IH Htapec) -Htapec -IH #IH
+ whd in Hleft; #ls #c #rs #Htapea cases Hleft -Hleft
+ #ls0 * #c' * #rs0 * >Htapea #Hdes destruct (Hdes) * *
+ cases (true_or_false (bit_or_null c')) #Hc'
+ [2: #Htapeb lapply (Htapeb Hc') -Htapeb #Htapeb #_ #_ %
+ [%[#c1 #Hc1 #Heqc destruct (Heqc)
+ cases (IH … Htapeb) * #_ #H #_ <Htapeb @(H … (refl…))
+ |#c1 #Heqc destruct (Heqc)
]
+ |#b #b0 #bs #b0s #l1 #l2 #_ #_ #_ #_ #_ #_
+ #Heq destruct (Heq) >Hc' #Hfalse @False_ind destruct (Hfalse)
]
- |#Hleft %
+ |#_ (* no marks in rs ??? *) #_ #Hleft %
[ %
[ #c'' #Hc'' #Heq destruct (Heq) >Hc'' in Hc'; #H destruct (H)
| #c0 #Hfalse destruct (Hfalse)
]
|#b #b0 #bs #b0s #l1 #l2 #Hlen #Hbs1 #Hb0s1 #Hbs2 #Hb0s2 #Hl1
- #Heq destruct (Heq) #_ #Hrs cases Hleft -Hleft
- [2: * >Hc' #Hfalse @False_ind destruct ] * #_
- @(list_cases2 … Hlen)
- [ #Hbs #Hb0s generalize in match Hrs; >Hbs in ⊢ (%→?); >Hb0s in ⊢ (%→?);
- -Hrs #Hrs normalize in Hrs; #Hleft cases (Hleft ????? Hrs ?) -Hleft
- [ * #Heqb #Htapeb cases (IH … Htapeb) -IH * #IH #_ #_
- % %
- [ >Heqb >Hbs >Hb0s %
- | >Hbs >Hb0s @IH %
- ]
- |* #Hneqb #Htapeb %2
- @(ex_intro … [ ]) @(ex_intro … b)
- @(ex_intro … b0) @(ex_intro … [ ])
- @(ex_intro … [ ]) %
- [ % [ % [@sym_not_eq //| >Hbs %] | >Hb0s %]
- | cases (IH … Htapeb) -IH * #_ #IH #_ >(IH ? (refl ??))
- @Htapeb
+ #Heq destruct (Heq) #_ >append_cons; <associative_append #Hrs
+ cases (Hleft … Hc' … Hrs) -Hleft
+ [2: #c1 #memc1 cases (memb_append … memc1) -memc1 #memc1
+ [cases (memb_append … memc1) -memc1 #memc1
+ [@Hbs2 @memc1 |>(memb_single … memc1) %]
+ |@Hl1 @memc1
+ ]
+ |* (* manca il caso in cui dopo una parte uguale il
+ secondo nastro finisca ... ???? *)
+ #_ cases (true_or_false (b==b0)) #eqbb0
+ [2: #_ #Htapeb %2 lapply (Htapeb … (\Pf eqbb0)) -Htapeb #Htapeb
+ cases (IH … Htapeb) * #_ #Hout #_
+ @(ex_intro … []) @(ex_intro … b) @(ex_intro … b0)
+ @(ex_intro … bs) @(ex_intro … b0s) %
+ [%[%[@(\Pf eqbb0) | %] | %]
+ |>(Hout … (refl …)) -Hout >Htapeb @eq_f3 [2,3:%]
+ >reverse_append >reverse_append >associative_append
+ >associative_append %
+ ]
+ |lapply Hbs1 lapply Hb0s1 lapply Hbs2 lapply Hb0s2 lapply Hrs
+ -Hbs1 -Hb0s1 -Hbs2 -Hb0s2 -Hrs
+ @(list_cases2 … Hlen)
+ [#Hrs #_ #_ #_ #_ >associative_append >associative_append #Htapeb #_
+ lapply (Htapeb … (\P eqbb0) … (refl …) (refl …)) -Htapeb #Htapeb
+ cases (IH … Htapeb) -IH * #Hout #_ #_ %1 %
+ [>(\P eqbb0) %
+ |>(Hout grid (refl …) (refl …)) @eq_f
+ normalize >associative_append %
]
- | @Hl1 ]
- | * #b' #bitb' * #b0' #bitb0' #bs' #b0s' #Hbs #Hb0s
- generalize in match Hrs; >Hbs in ⊢ (%→?); >Hb0s in ⊢ (%→?);
- cut (bit_or_null b' = true ∧ bit_or_null b0' = true ∧
- bitb' = false ∧ bitb0' = false)
- [ % [ % [ % [ >Hbs in Hbs1; #Hbs1 @(Hbs1 〈b',bitb'〉) @memb_hd
- | >Hb0s in Hb0s1; #Hb0s1 @(Hb0s1 〈b0',bitb0'〉) @memb_hd ]
- | >Hbs in Hbs2; #Hbs2 @(Hbs2 〈b',bitb'〉) @memb_hd ]
- | >Hb0s in Hb0s2; #Hb0s2 @(Hb0s2 〈b0',bitb0'〉) @memb_hd ]
- | * * * #Ha #Hb #Hc #Hd >Hc >Hd
- #Hrs #Hleft
- cases (Hleft b' (bs'@〈grid,false〉::l1) b0 b0'
- (b0s'@〈comma,false〉::l2) ??) -Hleft
- [ 3: >Hrs normalize @eq_f >associative_append %
- | * #Hb0 #Htapeb cases (IH …Htapeb) -IH * #_ #_ #IH
- cases (IH b' b0' bs' b0s' (l1@[〈b0,false〉]) l2 ??????? Ha ?) -IH
- [ * #Heq #Houtc % %
- [ >Hb0 @eq_f >Hbs in Heq; >Hb0s in ⊢ (%→?); #Heq
- destruct (Heq) >Hb0s >Hc >Hd %
- | >Houtc >Hbs >Hb0s >Hc >Hd >reverse_cons >associative_append
- >associative_append %
+ |* #a1 #ba1 * #a2 #ba2 #tl1 #tl2 #HlenS #Hrs #Hb0s2 #Hbs2 #Hb0s1 #Hbs1
+ cut (ba1 = false) [@(Hbs2 〈a1,ba1〉) @memb_hd] #Hba1 >Hba1
+ >associative_append >associative_append #Htapeb #_
+ lapply (Htapeb … (\P eqbb0) … (refl …) (refl …)) -Htapeb #Htapeb
+ cases (IH … Htapeb) -IH * #_ #_
+ cut (ba2=false) [@(Hb0s2 〈a2,ba2〉) @memb_hd] #Hba2 >Hba2
+ #IH cases(IH a1 a2 ?? (l1@[〈b0,false〉]) l2 HlenS ????? (refl …) ??)
+ [3:#x #memx @Hbs1 @memb_cons @memx
+ |4:#x #memx @Hb0s1 @memb_cons @memx
+ |5:#x #memx @Hbs2 @memb_cons @memx
+ |6:#x #memx @Hb0s2 @memb_cons @memx
+ |7:#x #memx cases (memb_append …memx) -memx #memx
+ [@Hl1 @memx | >(memb_single … memx) %]
+ |8:@(Hbs1 〈a1,ba1〉) @memb_hd
+ |9: >associative_append >associative_append %
+ |-IH -Hbs1 -Hb0s1 -Hbs2 -Hrs *
+ #Ha1a2 #Houtc %1 %
+ [>(\P eqbb0) @eq_f destruct (Ha1a2) %
+ |>Houtc @eq_f3
+ [>reverse_cons >associative_append %
+ |%
+ |>associative_append %
+ ]
]
- | * #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #H4 %2
- @(ex_intro … (〈b,false〉::la)) @(ex_intro … c') @(ex_intro … d')
- @(ex_intro … lb) @(ex_intro … lc)
- % [ % [ % // >Hbs >Hc >H2 % | >Hb0s >Hd >H3 >Hb0 % ]
- | >H4 >Hbs >Hb0s >Hc >Hd >Hb0 >reverse_append
- >reverse_cons >reverse_cons
- >associative_append >associative_append
- >associative_append >associative_append %
+ |-IH -Hbs1 -Hb0s1 -Hbs2 -Hrs *
+ #la * #c' * #d' * #lb * #lc * * *
+ #Hcd #H1 #H2 #Houtc %2
+ @(ex_intro … (〈b,false〉::la)) @(ex_intro … c') @(ex_intro … d')
+ @(ex_intro … lb) @(ex_intro … lc) %
+ [%[%[@Hcd | >H1 %] |>(\P eqbb0) >Hba2 >H2 %]
+ |>Houtc @eq_f3
+ [>(\P eqbb0) >reverse_append >reverse_cons
+ >reverse_cons >associative_append >associative_append
+ >associative_append >associative_append %
+ |%
+ |%
]
- | generalize in match Hlen; >Hbs >Hb0s
- normalize #Hlen destruct (Hlen) @e0
- | #c0 #Hc0 @Hbs1 >Hbs @memb_cons //
- | #c0 #Hc0 @Hb0s1 >Hb0s @memb_cons //
- | #c0 #Hc0 @Hbs2 >Hbs @memb_cons //
- | #c0 #Hc0 @Hb0s2 >Hb0s @memb_cons //
- | #c0 #Hc0 cases (memb_append … Hc0)
- [ @Hl1 | #Hc0' >(memb_single … Hc0') % ]
- | %
- | >associative_append >associative_append % ]
- | * #Hneq #Htapeb %2
- @(ex_intro … []) @(ex_intro … b) @(ex_intro … b0)
- @(ex_intro … bs) @(ex_intro … b0s) %
- [ % // % // @sym_not_eq //
- | >Hbs >Hb0s >Hc >Hd >reverse_cons >associative_append
- >reverse_append in Htapeb; >reverse_cons
- >associative_append >associative_append
- #Htapeb <Htapeb
- cases (IH … Htapeb) -Htapeb -IH * #_ #IH #_ @(IH ? (refl ??))
- ]
- | #c1 #Hc1 cases (memb_append … Hc1) #Hyp
- [ @Hbs2 >Hbs @memb_cons @Hyp
- | cases (orb_true_l … Hyp)
- [ #Hyp2 >(\P Hyp2) %
- | @Hl1
- ]
- ]
- ]
-]]]]]
-qed.
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+]
+qed.
+
+lemma WF_cst_niltape:
+ WF ? (inv ? R_comp_step_true) (niltape (FinProd FSUnialpha FinBool)).
+@wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * #H destruct
+qed.
+
+lemma WF_cst_rightof:
+ ∀a,ls. WF ? (inv ? R_comp_step_true) (rightof (FinProd FSUnialpha FinBool) a ls).
+#a #ls @wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * #H destruct
+qed.
+
+lemma WF_cst_leftof:
+ ∀a,ls. WF ? (inv ? R_comp_step_true) (leftof (FinProd FSUnialpha FinBool) a ls).
+#a #ls @wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * #H destruct
+qed.
+
+lemma WF_cst_midtape_false:
+ ∀ls,c,rs. WF ? (inv ? R_comp_step_true)
+ (midtape (FinProd … FSUnialpha FinBool) ls 〈c,false〉 rs).
+#ls #c #rs @wf #t1 whd in ⊢ (%→?); * #ls' * #c' * #rs' * #H destruct
+qed.
+
+(* da spostare *)
+lemma not_nil_to_exists:∀A.∀l: list A. l ≠ [ ] →
+ ∃a,tl. a::tl = l.
+ #A * [* #H @False_ind @H // | #a #tl #_ @(ex_intro … a) @(ex_intro … tl) //]
+ qed.
+
+lemma terminate_compare:
+ ∀t. Terminate ? compare t.
+#t @(terminate_while … sem_comp_step) [%]
+cases t // #ls * #c * //
+#rs
+(* we cannot proceed by structural induction on the right tape,
+ since compare moves the marks! *)
+cut (∃len. |rs| = len) [/2/]
+* #len lapply rs lapply c lapply ls -ls -c -rs elim len
+ [#ls #c #rs #Hlen >(lenght_to_nil … Hlen) @wf #t1 whd in ⊢ (%→?); * #ls0 * #c0 * #rs0 * #Hmid destruct (Hmid)
+ * * #H1 #H2 #_ cases (true_or_false (bit_or_null c0)) #Hc0
+ [>(H2 Hc0 … (refl …)) // #x whd in ⊢ ((??%?)→?); #Hdes destruct
+ |>(H1 Hc0) //
+ ]
+ |-len #len #Hind #ls #c #rs #Hlen @wf #t1 whd in ⊢ (%→?); * #ls0 * #c0 * #rs0 * #Hmid destruct (Hmid)
+ * * #H1 #H2 #H3 cases (true_or_false (bit_or_null c0)) #Hc0
+ [-H1 cases (split_on_spec_ex ? rs0 (is_marked ?)) #rs1 * #rs2
+ cases rs2
+ [(* no marks in right tape *)
+ * * >append_nil #H >H -H #Hmarks #_
+ cases (not_nil_to_exists ? (reverse (FSUnialpha×bool) (〈c0,true〉::rs0)) ?)
+ [2: % >reverse_cons #H cases (nil_to_nil … H) #_ #H1 destruct]
+ #a0 * #tl #H4 >(H2 Hc0 Hmarks a0 tl H4) //
+ |(* the first marked element is a0 *)
+ * #a0 #a0b #rs3 * * #H4 #H5 #H6 lapply (H3 ? a0 rs3 … Hc0 H5 ?)
+ [<H4 @eq_f @eq_f2 [@eq_f @(H6 〈a0,a0b〉 … (refl …)) | %]
+ |cases (true_or_false (c0==a0)) #eqc0a0 (* comparing a0 with c0 *)
+ [* * (* we check if we have elements at the right of a0 *)
+ lapply H4 -H4 cases rs3
+ [#_ #Ht1 #_ #_ >(Ht1 (\P eqc0a0) (refl …)) //
+ |(* a1 will be marked *)
+ cases (not_nil_to_exists ? (rs1@[〈a0,false〉]) ?)
+ [2: % #H cases (nil_to_nil … H) #_ #H1 destruct]
+ * #a2 #a2b * #tl2 #H7 * #a1 #a1b #rs4 #H4 #_ #Ht1 #_
+ cut (a2b =false)
+ [lapply (memb_hd ? 〈a2,a2b〉 tl2) >H7 #mema2
+ cases (memb_append … mema2)
+ [@H5 |#H lapply(memb_single … H) #H2 destruct %]
+ ]
+ #Ha2b >Ha2b in H7; #H7
+ >(Ht1 (\P eqc0a0) … H7 (refl …)) @Hind -Hind -Ht1 -Ha2b -H2 -H3 -H5 -H6
+ <H4 in Hlen; >length_append normalize <plus_n_Sm #Hlen1
+ >length_append normalize <(injective_S … Hlen1) @eq_f2 //
+ cut (|〈a2,false〉::tl2|=|rs1@[〈a0,false〉]|) [>H7 %]
+ >length_append normalize <plus_n_Sm <plus_n_O //
+ ]
+ |(* c0 =/= a0 *) * * #_ #_ #Ht1 >(Ht1 (\Pf eqc0a0)) //
+ ]
+ ]
+ ]
+ |>(H1 Hc0) //
+ ]
+qed.
+
+lemma sem_compare : Realize ? compare R_compare.
+/2/ qed.
-axiom sem_compare : Realize ? compare R_compare.
\ No newline at end of file
+(* new *)
+definition R_compare_new :=
+ λt1,t2.
+ ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
+ (∀c'.bit_or_null c' = false → c = 〈c',true〉 → t2 = midtape ? ls 〈c',false〉 rs) ∧
+ (∀c'. c = 〈c',false〉 → t2 = t1) ∧
+(* forse manca il caso no marks in rs *)
+ ∀b,b0,bs,b0s,comma,l1,l2.
+ |bs| = |b0s| →
+ (∀c.memb (FinProd … FSUnialpha FinBool) c bs = true → bit_or_null (\fst c) = true) →
+ (∀c.memb ? c bs = true → is_marked ? c = false) →
+ (∀c.memb ? c b0s = true → is_marked ? c = false) →
+ (∀c.memb ? c l1 = true → is_marked ? c = false) →
+ c = 〈b,true〉 → bit_or_null b = true →
+ rs = bs@〈grid,false〉::l1@〈b0,true〉::b0s@〈comma,false〉::l2 →
+ (〈b,true〉::bs = 〈b0,true〉::b0s ∧
+ t2 = midtape ? (reverse ? bs@〈b,false〉::ls)
+ 〈grid,false〉 (l1@〈b0,false〉::b0s@〈comma,true〉::l2)) ∨
+ (∃la,c',d',lb,lc.c' ≠ d' ∧
+ 〈b,false〉::bs = la@〈c',false〉::lb ∧
+ 〈b0,false〉::b0s = la@〈d',false〉::lc ∧
+ t2 = midtape (FinProd … FSUnialpha FinBool) (reverse ? la@
+ reverse ? l1@
+ 〈grid,false〉::
+ reverse ? lb@
+ 〈c',true〉::
+ reverse ? la@ls)
+ 〈d',false〉 (lc@〈comma,false〉::l2)).
+
+lemma wsem_compare_new : WRealize ? compare R_compare_new.
+#t #i #outc #Hloop
+lapply (sem_while ?????? sem_comp_step t i outc Hloop) [%]
+-Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
+[ #tapea whd in ⊢ (%→?); #Rfalse #ls #c #rs #Htapea %
+ [ %
+ [ #c' #Hc' #Hc lapply (Rfalse … Htapea) -Rfalse * >Hc
+ whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse)
+ | #c' #Hc lapply (Rfalse … Htapea) -Rfalse * #_
+ #Htrue @Htrue ]
+ | #b #b0 #bs #b0s #l1 #l2 #Hlen #Hbs1 #Hb0s1 #Hbs2 #Hb0s2 #Hl1 #Hc
+ cases (Rfalse … Htapea) -Rfalse >Hc whd in ⊢ (??%?→?);#Hfalse destruct (Hfalse)
+ ]
+| #tapea #tapeb #tapec #Hleft #Hright #IH #Htapec lapply (IH Htapec) -Htapec -IH #IH
+ whd in Hleft; #ls #c #rs #Htapea cases Hleft -Hleft
+ #ls0 * #c' * #rs0 * >Htapea #Hdes destruct (Hdes) * *
+ cases (true_or_false (bit_or_null c')) #Hc'
+ [2: #Htapeb lapply (Htapeb Hc') -Htapeb #Htapeb #_ #_ %
+ [%[#c1 #Hc1 #Heqc destruct (Heqc)
+ cases (IH … Htapeb) * #_ #H #_ <Htapeb @(H … (refl…))
+ |#c1 #Heqc destruct (Heqc)
+ ]
+ |#b #b0 #bs #b0s #comma #l1 #l2 #_ #_ #_ #_ #_
+ #Heq destruct (Heq) >Hc' #Hfalse @False_ind destruct (Hfalse)
+ ]
+ |#_ (* no marks in rs ??? *) #_ #Hleft %
+ [ %
+ [ #c'' #Hc'' #Heq destruct (Heq) >Hc'' in Hc'; #H destruct (H)
+ | #c0 #Hfalse destruct (Hfalse)
+ ]
+ |#b #b0 #bs #b0s #comma #l1 #l2 #Hlen #Hbs1 #Hbs2 #Hb0s2 #Hl1
+ #Heq destruct (Heq) #_ >append_cons; <associative_append #Hrs
+ cases (Hleft … Hc' … Hrs) -Hleft
+ [2: #c1 #memc1 cases (memb_append … memc1) -memc1 #memc1
+ [cases (memb_append … memc1) -memc1 #memc1
+ [@Hbs2 @memc1 |>(memb_single … memc1) %]
+ |@Hl1 @memc1
+ ]
+ |* (* manca il caso in cui dopo una parte uguale il
+ secondo nastro finisca ... ???? *)
+ #_ cases (true_or_false (b==b0)) #eqbb0
+ [2: #_ #Htapeb %2 lapply (Htapeb … (\Pf eqbb0)) -Htapeb #Htapeb
+ cases (IH … Htapeb) * #_ #Hout #_
+ @(ex_intro … []) @(ex_intro … b) @(ex_intro … b0)
+ @(ex_intro … bs) @(ex_intro … b0s) %
+ [%[%[@(\Pf eqbb0) | %] | %]
+ |>(Hout … (refl …)) -Hout >Htapeb @eq_f3 [2,3:%]
+ >reverse_append >reverse_append >associative_append
+ >associative_append %
+ ]
+ |lapply Hbs1 lapply Hbs2 lapply Hb0s2 lapply Hrs
+ -Hbs1 -Hbs2 -Hb0s2 -Hrs
+ @(list_cases2 … Hlen)
+ [#Hrs #_ #_ #_ >associative_append >associative_append #Htapeb #_
+ lapply (Htapeb … (\P eqbb0) … (refl …) (refl …)) -Htapeb #Htapeb
+ cases (IH … Htapeb) -IH * #Hout #_ #_ %1 %
+ [>(\P eqbb0) %
+ |>(Hout grid (refl …) (refl …)) @eq_f
+ normalize >associative_append %
+ ]
+ |* #a1 #ba1 * #a2 #ba2 #tl1 #tl2 #HlenS #Hrs #Hb0s2 #Hbs2 #Hbs1
+ cut (ba1 = false) [@(Hbs2 〈a1,ba1〉) @memb_hd] #Hba1 >Hba1
+ >associative_append >associative_append #Htapeb #_
+ lapply (Htapeb … (\P eqbb0) … (refl …) (refl …)) -Htapeb #Htapeb
+ cases (IH … Htapeb) -IH * #_ #_
+ cut (ba2=false) [@(Hb0s2 〈a2,ba2〉) @memb_hd] #Hba2 >Hba2
+ #IH cases(IH a1 a2 ??? (l1@[〈b0,false〉]) l2 HlenS ???? (refl …) ??)
+ [4:#x #memx @Hbs1 @memb_cons @memx
+ |5:#x #memx @Hbs2 @memb_cons @memx
+ |6:#x #memx @Hb0s2 @memb_cons @memx
+ |7:#x #memx cases (memb_append …memx) -memx #memx
+ [@Hl1 @memx | >(memb_single … memx) %]
+ |8:@(Hbs1 〈a1,ba1〉) @memb_hd
+ |9: >associative_append >associative_append %
+ |-IH -Hbs1 -Hbs2 -Hrs *
+ #Ha1a2 #Houtc %1 %
+ [>(\P eqbb0) @eq_f destruct (Ha1a2) %
+ |>Houtc @eq_f3
+ [>reverse_cons >associative_append %
+ |%
+ |>associative_append %
+ ]
+ ]
+ |-IH -Hbs1 -Hbs2 -Hrs *
+ #la * #c' * #d' * #lb * #lc * * *
+ #Hcd #H1 #H2 #Houtc %2
+ @(ex_intro … (〈b,false〉::la)) @(ex_intro … c') @(ex_intro … d')
+ @(ex_intro … lb) @(ex_intro … lc) %
+ [%[%[@Hcd | >H1 %] |>(\P eqbb0) >Hba2 >H2 %]
+ |>Houtc @eq_f3
+ [>(\P eqbb0) >reverse_append >reverse_cons
+ >reverse_cons >associative_append >associative_append
+ >associative_append >associative_append %
+ |%
+ |%
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+]
+qed.
\ No newline at end of file
projects / helm.git / blobdiff