(* *)
(**************************************************************************)
-include "turing/simple_machines.ma".
-include "turing/multi_universal/compare.ma".
-include "turing/multi_universal/par_test.ma".
-include "turing/multi_universal/moves_2.ma".
+include "turing/auxiliary_multi_machines.ma".
-lemma eq_vec_change_vec : ∀sig,n.∀v1,v2:Vector sig n.∀i,t,d.
- nth i ? v2 d = t →
- (∀j.i ≠ j → nth j ? v1 d = nth j ? v2 d) →
- v2 = change_vec ?? v1 t i.
-#sig #n #v1 #v2 #i #t #d #H1 #H2 @(eq_vec … d)
-#i0 #Hlt cases (decidable_eq_nat i0 i) #Hii0
-[ >Hii0 >nth_change_vec //
-| >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @H2 @sym_not_eq // ]
-qed.
-
-lemma right_mk_tape :
- ∀sig,ls,c,rs.(c = None ? → ls = [ ] ∨ rs = [ ]) → right ? (mk_tape sig ls c rs) = rs.
-#sig #ls #c #rs cases c // cases ls
-[ cases rs //
-| #l0 #ls0 #H normalize cases (H (refl ??)) #H1 [ destruct (H1) | >H1 % ] ]
-qed.
-
-lemma left_mk_tape : ∀sig,ls,c,rs.left ? (mk_tape sig ls c rs) = ls.
-#sig #ls #c #rs cases c // cases ls // cases rs //
-qed.
-
-lemma current_mk_tape : ∀sig,ls,c,rs.current ? (mk_tape sig ls c rs) = c.
-#sig #ls #c #rs cases c // cases ls // cases rs //
-qed.
-
-lemma length_tail : ∀A,l.0 < |l| → |tail A l| < |l|.
-#A * normalize //
-qed.
-
-(*
-[ ] → [ ], l2, 1
-a::al →
- [ ] → [ ], l1, 2
- b::bl → match rec(al,bl)
- x, y, 1 → b::x, y, 1
- x, y, 2 → a::x, y, 2
-*)
-
-lemma lists_length_split :
- ∀A.∀l1,l2:list A.(∃la,lb.(|la| = |l1| ∧ l2 = la@lb) ∨ (|la| = |l2| ∧ l1 = la@lb)).
-#A #l1 elim l1
-[ #l2 %{[ ]} %{l2} % % %
-| #hd1 #tl1 #IH *
- [ %{[ ]} %{(hd1::tl1)} %2 % %
- | #hd2 #tl2 cases (IH tl2) #x * #y *
- [ * #IH1 #IH2 %{(hd2::x)} %{y} % normalize % //
- | * #IH1 #IH2 %{(hd1::x)} %{y} %2 normalize % // ]
- ]
-]
-qed.
-
-definition option_cons ≝ λsig.λc:option sig.λl.
- match c with [ None ⇒ l | Some c0 ⇒ c0::l ].
-
-lemma opt_cons_tail_expand : ∀A,l.l = option_cons A (option_hd ? l) (tail ? l).
-#A * //
-qed.
-
-definition match_test ≝ λsrc,dst.λsig:DeqSet.λn.λv:Vector ? n.
- match (nth src (option sig) v (None ?)) with
- [ None ⇒ false
- | Some x ⇒ notb (nth dst (DeqOption sig) v (None ?) == None ?) ].
-
+(* rewind *)
definition rewind ≝ λsrc,dst,sig,n.
parmove src dst sig n L · mmove src sig n R · mmove dst sig n R.
∀ls0,y,rs0.nth dst ? int (niltape ?) = midtape sig ls0 y rs0 →
outt = int).
-(*
-theorem accRealize_to_Realize :
- ∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc.
- M ⊨ [ acc: Rtrue, Rfalse ] → M ⊨ Rtrue ∪ Rfalse.
-#sig #n #M #Rtrue #Rfalse #acc #HR #t
-cases (HR t) #k * #outc * * #Hloop
-#Htrue #Hfalse %{k} %{outc} % //
-cases (true_or_false (cstate sig (states sig n M) n outc == acc)) #Hcase
-[ % @Htrue @(\P Hcase) | %2 @Hfalse @(\Pf Hcase) ]
-qed.
-*)
-
lemma sem_rewind_strong : ∀src,dst,sig,n.
src ≠ dst → src < S n → dst < S n →
rewind src dst sig n ⊨ R_rewind_strong src dst sig n.
#ta #tb * * * #H1 #H2 #H3 #H4 % /2 by /
qed.
+(* match step *)
+
+definition match_test ≝ λsrc,dst.λsig:DeqSet.λn.λv:Vector ? n.
+ match (nth src (option sig) v (None ?)) with
+ [ None ⇒ false
+ | Some x ⇒ notb (nth dst (DeqOption sig) v (None ?) == None ?) ].
+
definition match_step ≝ λsrc,dst,sig,n.
compare src dst sig n ·
(ifTM ?? (partest sig n (match_test src dst sig ?))
(single_finalTM ??
- (rewind src dst sig n · (inject_TM ? (move_r ?) n dst)))
+ (rewind src dst sig n · mmove dst ?? R))
(nop …)
partest1).
(acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?))
(sem_seq …
(sem_rewind ???? Hneq Hsrc Hdst)
- (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
- (sem_nop …)))
+ (sem_move_multi … R ?))
+ (sem_nop …))) /2/
[ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?))))
cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%);
[ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %]
lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
>change_vec_commute // >change_vec_change_vec
>change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
- >Hte in Htb; * * #_ >nth_change_vec // #Htb1
- lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2 %
- [ @(eq_vec … (niltape ?)) #i #Hi
- cases (true_or_false ((dst : DeqNat) == i)) #Hdsti
- [ <(\P Hdsti) >Htb1 >nth_change_vec // >Hmidta_dst
- >Hrs0 >reverse_reverse cases xs [|#r1 #rs1] %
- | <Htb2 [|@(\Pf Hdsti)] >nth_change_vec_neq [| @(\Pf Hdsti)]
- >Hrs0 >reverse_reverse >nth_change_vec_neq in ⊢ (???%);
- <Hrs <Hmidta_src [|@(\Pf Hdsti)] >change_vec_same % ]
+ >Hte in Htb; whd in ⊢ (%→?); #Htb >Htb %
+ [ >change_vec_change_vec >nth_change_vec //
+ >reverse_reverse <Hrs <Hmidta_src >change_vec_same <Hrs0 <Hmidta_dst
+ %
| >Hmidta_dst %{s'} % [%] #_
>Hrs0 %{xs} %{ci} %{rs''} %{ls0} %{cj} %{rs0'} % // % //
]
| lapply (\Pf Hss0) -Hss0 #Hss0 #Htc cut (tc = ta)
[@Htc % % @(not_to_not ??? Hss0) #H destruct (H) %]
-Htc #Htc destruct (Htc) #_ * #td * whd in ⊢ (%→?); * #_
- #Htd destruct (Htd) * #te * * #_ #Hte * * #_ #Htb1 #Htb2
+ #Htd destruct (Htd) * #te * * #_ #Hte whd in ⊢ (%→?); #Htb
#s1 #rs1 >Hmidta_src #H destruct (H)
lapply (Hte … Hmidta_src … Hmidta_dst) -Hte #Hte destruct (Hte) %
- [ @(eq_vec … (niltape ?)) #i #Hi
- cases (true_or_false ((dst : DeqNat) == i)) #Hdsti
- [ <(\P Hdsti) >(Htb1 … Hmidta_dst) >nth_change_vec // >Hmidta_dst
- cases rs0 [|#r2 #rs2] %
- | <Htb2 [|@(\Pf Hdsti)] >nth_change_vec_neq [| @(\Pf Hdsti)] % ]
+ [ >Htb %
| >Hs0 %{s0} % // #H destruct (H) @False_ind cases (Hss0) /2/ ]
]
]
(acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?))
(sem_seq …
(sem_rewind_strong ???? Hneq Hsrc Hdst)
- (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
- (sem_nop …)))
+ (sem_move_multi … R ?))
+ (sem_nop …))) [/2/]
[ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?))))
cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%);
[ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %]
lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
>change_vec_commute [|//] >change_vec_change_vec
>change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
- >Hte * * #_ >nth_change_vec [|//] >reverse_reverse
- #H lapply (H … (refl ??)) -H #Htb1 #Htb2
+ >Hte whd in ⊢ (%→?); >change_vec_change_vec >nth_change_vec [|//]
+ >reverse_reverse #Htb
cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta (midtape sig [ ] s0 (xs@ci::rs'')) src) (mk_tape sig (s0::lsb) (option_hd sig (xs@cj::rs0')) (tail sig (xs@cj::rs0'))) dst)
- [ @(eq_vec_change_vec … (niltape ?))
- [@Htb1| #j #Hj <Htb2 // >(nth_change_vec_neq ??????? Hj) % ] ]
- -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec [|//]
+ [ >Htb @eq_f3 // cases (xs@cj::rs0') // ]
+ -Htb #Htb >Htb whd >nth_change_vec [|//]
>nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec [|//]
>right_mk_tape [|cases xs [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H)]
normalize in match (left ??);
>H1 in Hlen'; >H2 whd in ⊢ (??%%→?); #Hlen'
>length_reverse >length_reverse destruct (Hlen') //
| /2 by refl, trans_eq/ ] -Hte
- #Hte #_ * * #_ >Hte >nth_change_vec [|//] #Htb1 #Htb2
+ #Hte #_ whd in ⊢ (%→?); #Htb
cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
(mk_tape sig (hda::lsb) (option_hd ? (reverse sig (reverse sig xs@s0::reverse sig tla)@cj::rs0')) (tail ? (reverse sig (reverse sig xs@s0::reverse sig tla)@cj::rs0'))) dst)
(midtape ? [ ] hdb (reverse sig (reverse sig xs@s0::reverse sig tlb)@ci::rs'')) src)
- [ >change_vec_commute [|@sym_not_eq //] @(eq_vec_change_vec … (niltape ?))
- [ @Htb1 %
- | #j #Hj <Htb2 // >change_vec_commute // >change_vec_change_vec
- >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
- >nth_change_vec_neq in ⊢ (???%); // ] ]
- -Htb1 -Htb2 #Htb >Htb whd
+ [ >Htb >Hte >nth_change_vec // >change_vec_change_vec >change_vec_commute [|//]
+ >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
+ >change_vec_change_vec >change_vec_commute [|//]
+ @eq_f3 // cases (reverse sig (reverse sig xs@s0::reverse sig tla)@cj::rs0') // ]
+ -Htb #Htb >Htb whd
>nth_change_vec [|//] >nth_change_vec_neq [|//] >nth_change_vec [|//]
>right_mk_tape
[| cases (reverse sig (reverse sig xs@s0::reverse sig tla))
lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
>change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
>change_vec_change_vec #Hte #_
- >Hte * * #_ >nth_change_vec [|//] >reverse_reverse
- #H lapply (H … (refl ??)) -H #Htb1 #Htb2
+ >Hte whd in ⊢ (%→?); >nth_change_vec [|//] >reverse_reverse #Htb
cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta (mk_tape ? [s0] (option_hd ? (xs@cj::rs0')) (tail ? (xs@cj::rs0'))) dst)
(midtape ? lsb s0 (xs@ci::rs'')) src)
- [ >change_vec_commute [|@sym_not_eq //] @(eq_vec_change_vec … (niltape ?))
- [ @Htb1
- | #j #Hj <Htb2 // >nth_change_vec_neq in ⊢ (???%); // ] ]
- -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec [|//]
+ [ >Htb >change_vec_change_vec >change_vec_commute [|//]
+ @eq_f3 // <Hrs0 cases rs0 // ]
+ -Htb #Htb >Htb whd >nth_change_vec [|//]
>nth_change_vec_neq [|//] >nth_change_vec [|//]
>right_mk_tape
[| cases xs [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
>H1 in Hlen'; >H2 whd in ⊢ (??%%→?); #Hlen'
>length_reverse >length_reverse destruct (Hlen') //
| /2 by refl, trans_eq/ ] -Hte
- #Hte #_ * * #_ >Hte >nth_change_vec_neq [|//] >nth_change_vec [|//] #Htb1 #Htb2
+ #Hte #_ whd in ⊢ (%→?); >Hte >nth_change_vec_neq [|//] >nth_change_vec [|//] #Htb
cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
(mk_tape sig [hdb] (option_hd ? (reverse sig (reverse sig xs@s0::reverse sig tlb)@cj::rs0')) (tail ? (reverse sig (reverse sig xs@s0::reverse sig tlb)@cj::rs0'))) dst)
(midtape ? lsb hda (reverse sig (reverse sig xs@s0::reverse sig tla)@ci::rs'')) src)
- [ >change_vec_commute [|@sym_not_eq //] @(eq_vec_change_vec … (niltape ?))
- [ @Htb1 %
- | #j #Hj <Htb2 [|//] >change_vec_change_vec
- >change_vec_commute [|@sym_not_eq //]
- >change_vec_change_vec >nth_change_vec_neq in ⊢ (???%); // ] ]
- -Htb1 -Htb2 #Htb >Htb whd
+ [ >Htb >change_vec_change_vec >change_vec_commute [|//]
+ >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
+ >change_vec_change_vec >change_vec_commute [|//]
+ @eq_f3 // cases (reverse sig (reverse sig xs@s0::reverse sig tlb)@cj::rs0') // ]
+ -Htb #Htb >Htb whd
>nth_change_vec [|//] >nth_change_vec_neq [|//] >nth_change_vec [|//]
>right_mk_tape
[| cases (reverse sig (reverse sig xs@s0::reverse sig tlb))
cut (|reverse ? lsa| = |reverse ? ls|) [ // ] #Hlen'
@(list_cases2 … Hlen')
[ #H1 #H2 >H1 >H2 -H1 -H2 #_ #_ normalize in match (reverse ? [ ]); #Hte #_
- lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte) * * #_
- >Hmidta_dst #Htb1 lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2
+ lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte)
+ whd in ⊢ (%→?); >Hmidta_dst #Htb
cut (tb = change_vec ?? ta (mk_tape ? (s0::lsa@lsb) (option_hd ? rs0) (tail ? rs0)) dst)
- [@(eq_vec_change_vec … (niltape ?)) [@Htb1|@Htb2] ]
- -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec [|//]
+ [ >Htb cases rs0 // ]
+ -Htb #Htb >Htb whd >nth_change_vec [|//]
>nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst
>right_mk_tape
[| cases rs0 [ #_ %2 % | #x0 #xs0 normalize in ⊢ (??%?→?); #H destruct (H)] ]
lapply (Hte ???? (refl ??) ? s0 ? (reverse ? tla) ?? (refl ??))
[ >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) //
normalize #H destruct (H) // ] #Hte #_ #_ #_
- * * #_ >Hte >nth_change_vec [|//] #Htb1 #Htb2
+ whd in ⊢ (%→?); >Hte >change_vec_change_vec >nth_change_vec // #Htb
cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
(mk_tape sig (hda::lsb) (option_hd ? (reverse sig (reverse sig tla)@s0::rs0)) (tail ? (reverse sig (reverse sig tla)@s0::rs0))) dst)
(midtape ? [ ] hdb (reverse sig (reverse sig tlb)@s::rs)) src)
- [ >change_vec_commute [|@sym_not_eq //] @(eq_vec_change_vec … (niltape ?))
- [ @Htb1 //
- | #j #Hj <Htb2 // >nth_change_vec_neq in ⊢ (???%); // ]]
- -Htb1 -Htb2 #Htb >Htb whd
+ [ >Htb >change_vec_commute [|//] @eq_f3 // cases (reverse sig (reverse sig tla)@s0::rs0) // ]
+ -Htb #Htb >Htb whd
>nth_change_vec [|//] >nth_change_vec_neq [|//] >nth_change_vec [|//]
>right_mk_tape
[| cases (reverse sig (reverse sig tla))
@(list_cases2 … Hlen')
[ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #_ #_ #Hte
lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte)
- * * #_ >Hmidta_dst #Htb1 lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2
+ whd in ⊢ (%→?); #Htb whd >Hmidta_dst
cut (tb = change_vec (tape sig) (S n) ta (mk_tape ? (s0::ls0) (option_hd ? rs0) (tail ? rs0)) dst)
- [ @(eq_vec_change_vec … (niltape ?)) // @Htb2 ]
- -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec [|//]
+ [ >Htb >Hmidta_dst cases rs0 // ]
+ -Htb #Htb >Htb whd >nth_change_vec [|//]
>nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst
>current_mk_tape >right_mk_tape
[| cases rs0 [ #_ %2 % | #x0 #xs0 normalize in ⊢ (??%?→?); #H destruct (H) ]]
| >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) //
normalize #H destruct (H) //
| /2 by refl, trans_eq/ ] -Hte
- #Hte * * #_ >Hte >nth_change_vec_neq [|//] >nth_change_vec [|//] #Htb1 #Htb2
+ #Hte whd in ⊢ (%→?); >Hte >nth_change_vec_neq [|//] >nth_change_vec [|//] #Htb
cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
(mk_tape sig [hdb] (option_hd ? (reverse sig (reverse sig tlb)@s0::rs0)) (tail ? (reverse sig (reverse sig tlb)@s0::rs0))) dst)
(midtape ? lsb hda (reverse sig (reverse sig tla)@s::rs)) src)
- [ >change_vec_commute [|@sym_not_eq //] @(eq_vec_change_vec … (niltape ?))
- [ @Htb1 %
- | #j #Hj <Htb2 // >change_vec_commute [|@sym_not_eq //]
- >nth_change_vec_neq in ⊢ (???%); // ]]
- -Htb1 -Htb2 #Htb >Htb whd
+ [ >Htb >change_vec_commute [|//] >change_vec_change_vec
+ @eq_f3 // cases (reverse sig (reverse sig tlb)@s0::rs0) // ]
+ -Htb #Htb >Htb whd
>nth_change_vec [|//] >nth_change_vec_neq [|//] >nth_change_vec [|//]
>right_mk_tape
[| cases (reverse ? (reverse ? tlb)) [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
projects / helm.git / blobdiff