(* *)
(**************************************************************************)
-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".
-definition Rtc_multi_true ≝
- λalpha,test,n,i.λt1,t2:Vector ? (S n).
- (∃c. current alpha (nth i ? t1 (niltape ?)) = Some ? c ∧ test c = true) ∧ t2 = t1.
-
-definition Rtc_multi_false ≝
- λalpha,test,n,i.λt1,t2:Vector ? (S n).
- (∀c. current alpha (nth i ? t1 (niltape ?)) = Some ? c → test c = false) ∧ t2 = t1.
-
-lemma sem_test_char_multi :
- ∀alpha,test,n,i.i ≤ n →
- inject_TM ? (test_char ? test) n i ⊨
- [ tc_true : Rtc_multi_true alpha test n i, Rtc_multi_false alpha test n i ].
-#alpha #test #n #i #Hin #int
-cases (acc_sem_inject … Hin (sem_test_char alpha test) int)
-#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
-[ @Hloop
-| #Hqtrue lapply (Htrue Hqtrue) * * * #c *
- #Hcur #Htestc #Hnth_i #Hnth_j %
- [ %{c} % //
- | @(eq_vec … (niltape ?)) #i0 #Hi0
- cases (decidable_eq_nat i0 i) #Hi0i
- [ >Hi0i @Hnth_i
- | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
-| #Hqfalse lapply (Hfalse Hqfalse) * * #Htestc #Hnth_i #Hnth_j %
- [ @Htestc
- | @(eq_vec … (niltape ?)) #i0 #Hi0
- cases (decidable_eq_nat i0 i) #Hi0i
- [ >Hi0i @Hnth_i
- | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
-qed.
-
-definition Rm_test_null_true ≝
- λalpha,n,i.λt1,t2:Vector ? (S n).
- current alpha (nth i ? t1 (niltape ?)) ≠ None ? ∧ t2 = t1.
-
-definition Rm_test_null_false ≝
- λalpha,n,i.λt1,t2:Vector ? (S n).
- current alpha (nth i ? t1 (niltape ?)) = None ? ∧ t2 = t1.
-
-lemma sem_test_null_multi : ∀alpha,n,i.i ≤ n →
- inject_TM ? (test_null ?) n i ⊨
- [ tc_true : Rm_test_null_true alpha n i, Rm_test_null_false alpha n i ].
-#alpha #n #i #Hin #int
-cases (acc_sem_inject … Hin (sem_test_null alpha) int)
-#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
-[ @Hloop
-| #Hqtrue lapply (Htrue Hqtrue) * * #Hcur #Hnth_i #Hnth_j % //
- @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) #Hi0i
- [ >Hi0i @sym_eq @Hnth_i | @sym_eq @Hnth_j @sym_not_eq // ] ]
-| #Hqfalse lapply (Hfalse Hqfalse) * * #Hcur #Hnth_i #Hnth_j %
- [ @Hcur
- | @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) //
- #Hi0i @sym_eq @Hnth_j @sym_not_eq // ] ]
-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 ?) ].
-
-definition mmove_states ≝ initN 2.
-
-definition mmove0 : mmove_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
-definition mmove1 : mmove_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
-
-definition trans_mmove ≝
- λi,sig,n,D.
- λp:mmove_states × (Vector (option sig) (S n)).
- let 〈q,a〉 ≝ p in match (pi1 … q) with
- [ O ⇒ 〈mmove1,change_vec ? (S n) (null_action ? n) (〈None ?,D〉) i〉
- | S _ ⇒ 〈mmove1,null_action sig n〉 ].
-
-definition mmove ≝
- λi,sig,n,D.
- mk_mTM sig n mmove_states (trans_mmove i sig n D)
- mmove0 (λq.q == mmove1).
-
-definition Rm_multi ≝
- λalpha,n,i,D.λt1,t2:Vector ? (S n).
- t2 = change_vec ? (S n) t1 (tape_move alpha (nth i ? t1 (niltape ?)) D) i.
-
-lemma sem_move_multi :
- ∀alpha,n,i,D.i ≤ n →
- mmove i alpha n D ⊨ Rm_multi alpha n i D.
-#alpha #n #i #D #Hin #int %{2}
-%{(mk_mconfig ? mmove_states n mmove1 ?)}
-[| %
- [ whd in ⊢ (??%?); @eq_f whd in ⊢ (??%?); @eq_f %
- | whd >tape_move_multi_def
- <(change_vec_same … (ctapes …) i (niltape ?))
- >pmap_change <tape_move_multi_def >tape_move_null_action % ] ]
- qed.
-
+(* rewind *)
definition rewind ≝ λsrc,dst,sig,n.
parmove src dst sig n L · mmove src sig n R · mmove dst sig n R.
(midtape sig ls0 y0 (reverse ? target@y::rs0)) src) ∧
(∀x,rs.nth src ? int (niltape ?) = midtape sig [] x rs →
∀ls0,y,rs0.nth dst ? int (niltape ?) = midtape sig ls0 y rs0 →
+ outt = int) ∧
+ (∀x,rs.nth dst ? int (niltape ?) = midtape sig [] x rs →
+ ∀ls0,y,rs0.nth src ? int (niltape ?) = midtape sig ls0 y rs0 →
outt = int).
definition R_rewind ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
∀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.
@(sem_seq_app sig n ????? (sem_parmoveL src dst sig n Hneq Hsrc Hdst) ?)
[| @(sem_seq_app sig n ????? (sem_move_multi … R ?) (sem_move_multi … R ?)) //
@le_S_S_to_le // ]
-#ta #tb * #tc * * * #Htc1 #Htc2 #_ * #td * whd in ⊢ (%→%→?); #Htd #Htb % [ %
+#ta #tb * #tc * * * #Htc1 #Htc2 #_ * #td * whd in ⊢ (%→%→?); #Htd #Htb % [ % [ %
[ #x #x0 #xs #rs #Hmidta_src #ls0 #y #y0 #target #rs0 #Hlen #Hmidta_dst
>(Htc1 ??? Hmidta_src ls0 y (target@[y0]) rs0 ??) in Htd;
[|>Hmidta_dst //
>nth_change_vec // >change_vec_change_vec
whd in match (tape_move ???);whd in match (tape_move ???); <Hmidta_src
<Hls0 <Hmidta_dst >change_vec_same >change_vec_same //
-]]
+ ] ]
+| #x #rs #Hmidta_dst #ls0 #y #rs0 #Hmidta_src
+ lapply (Htc2 … Hmidta_dst … (refl ??) Hmidta_src) -Htc2 #Htc >Htc in Htd;
+ >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
+ >nth_change_vec // lapply (refl ? ls0) cases ls0 in ⊢ (???%→%);
+ [ #Hls0 destruct (Hls0) #Htd >Htd in Htb;
+ >nth_change_vec // >change_vec_change_vec
+ whd in match (tape_move ???);whd in match (tape_move ???);
+ <Hmidta_src <Hmidta_dst >change_vec_same >change_vec_same //
+ | #l1 #ls1 #Hls0 destruct (Hls0) #Htd >Htd in Htb;
+ >nth_change_vec // >change_vec_change_vec
+ whd in match (tape_move ???); whd in match (tape_move ???); <Hmidta_src
+ <Hmidta_dst >change_vec_same >change_vec_same //
+ ]
+]
qed.
lemma sem_rewind : ∀src,dst,sig,n.
src ≠ dst → src < S n → dst < S n →
rewind src dst sig n ⊨ R_rewind src dst sig n.
#src #dst #sig #n #Hneq #Hsrc #Hdst @(Realize_to_Realize … (sem_rewind_strong …)) //
-#ta #tb * * #H1 #H2 #H3 % /2/
+#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/ ]
]
]
]
qed.
-axiom daemon : ∀P:Prop.P.
-
-(* XXX: move to turing (or mono) *)
-definition option_cons ≝ λsig.λc:option sig.λl.
- match c with [ None ⇒ l | Some c0 ⇒ c0::l ].
-
definition R_match_step_true_naive ≝
λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
|left ? (nth src ? outt (niltape ?))| +
|left ? (nth src ? int (niltape ?))| +
|option_cons ? (current ? (nth dst ? int (niltape ?))) (right ? (nth dst ? int (niltape ?)))|.
-axiom right_mk_tape : ∀sig,ls,c,rs.right ? (mk_tape sig ls c rs) = rs.
-axiom left_mk_tape : ∀sig,ls,c,rs.left ? (mk_tape sig ls c rs) = ls.
-axiom current_mk_tape : ∀sig,ls,c,rs.current ? (mk_tape sig ls c rs) = c.
-axiom length_tail : ∀A,l.0 < |l| → |tail A l| < |l|.
-axiom lists_length_split :
- ∀A.∀l1,l2:list A.(∃la,lb.(|la| = |l1| ∧ l2 = la@lb) ∨ (|la| = |l2| ∧ l1 = la@lb)).
-axiom opt_cons_tail_expand : ∀A,l.l = option_cons A (option_hd ? l) (tail ? l).
-
lemma sem_match_step_termination :
∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
match_step src dst sig n ⊨
(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 %]
>nth_current_chars >nth_current_chars >Hcurtc_dst
cases (current ? (nth src …))
[normalize in ⊢ (%→?); #H destruct (H)
- | #x >nth_change_vec // cases (reverse ? rs0)
+ | #x >nth_change_vec [|@Hdst] cases (reverse ? rs0)
[ normalize in ⊢ (%→?); #H destruct (H)
| #r1 #rs1 normalize in ⊢ (%→?); #H destruct (H) ] ]
| * #rs0' * #_ #Hcurtc_src * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
]]
normalize in ⊢ (%→?); #H destruct (H) ]
| * #xs * #ci * #cj * #rs'' * #rs0' * * * #Hcicj #Hrs #Hrs0
- #Htc * #td * * #Hmatch #Htd destruct (Htd) * #te * *
- >Htc >change_vec_commute // >nth_change_vec //
- >change_vec_commute [|@sym_not_eq //] >nth_change_vec //
+ #Htc * #td * * #Hmatch #Htd destruct (Htd) * #te * * *
+ >Htc >change_vec_commute [|//] >nth_change_vec [|//]
+ >change_vec_commute [|@sym_not_eq //] >nth_change_vec [|//]
cases (lists_length_split ? ls ls0) #lsa * #lsb * * #Hlen #Hlsalsb
destruct (Hlsalsb) *
[ #Hte #_ #_ <(reverse_reverse … ls) in Hte; <(reverse_reverse … lsa)
cut (|reverse ? lsa| = |reverse ? ls|) [ // ] #Hlen'
@(list_cases2 … Hlen')
- [ #H1 #H2 >H1 >H2 -H1 -H2 normalize in match (reverse ? [ ]); #Hte
+ [ #H1 #H2 >H1 >H2 -H1 -H2 normalize in match (reverse ? [ ]); #Hte #_
lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
- >change_vec_commute // >change_vec_change_vec
+ >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
- 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)
- [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
- >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
- >right_mk_tape normalize in match (left ??);
+ >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)
+ [ >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 ??);
>Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand
whd in match (option_cons ???); >Hrs0
normalize in ⊢ (?(?%)%); //
>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)
- [@daemon] -Htb1 -Htb2 #Htb >Htb whd
- >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
- >right_mk_tape >Hmidta_src >Hmidta_dst
+ [ >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))
+ [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
+ >Hmidta_src >Hmidta_dst
whd in match (left ??); whd in match (left ??); whd in match (right ??);
>current_mk_tape <opt_cons_tail_expand whd in match (option_cons ???);
>Hrs0 >length_append whd in ⊢ (??(??%)); >length_append >length_reverse
[ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #Hte
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
+ >change_vec_change_vec #Hte #_
+ >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)
- [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
- >nth_change_vec_neq // >nth_change_vec //
- >right_mk_tape normalize in match (left ??);
+ [ >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) ]
+ normalize in match (left ??);
>Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand >Hrs0
>length_append normalize >length_append >length_append
<(reverse_reverse ? lsa) >H1 normalize //
>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)
- [@daemon] -Htb1 -Htb2 #Htb >Htb whd
- >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
- >right_mk_tape >Hmidta_src >Hmidta_dst
+ [ >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))
+ [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
+ >Hmidta_src >Hmidta_dst
whd in match (left ??); whd in match (left ??); whd in match (right ??);
>current_mk_tape <opt_cons_tail_expand
whd in match (option_cons ???);
| 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 * * * >Hmidta_src >Hmidta_dst
+ #Htd destruct (Htd) * #te * * * * >Hmidta_src >Hmidta_dst
cases (lists_length_split ? ls ls0) #lsa * #lsb * * #Hlen #Hlsalsb
destruct (Hlsalsb)
[ <(reverse_reverse … ls) <(reverse_reverse … lsa)
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
+ [ #H1 #H2 >H1 >H2 -H1 -H2 #_ #_ normalize in match (reverse ? [ ]); #Hte #_
+ 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)
- [@daemon] -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 normalize in match (left ??); normalize in match (right ??);
+ >right_mk_tape
+ [| cases rs0 [ #_ %2 % | #x0 #xs0 normalize in ⊢ (??%?→?); #H destruct (H)] ]
+ normalize in match (left ??); normalize in match (right ??);
>Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand
normalize //
| #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
>reverse_cons >reverse_cons >associative_append #Hte
lapply (Hte ???? (refl ??) ? s0 ? (reverse ? tla) ?? (refl ??))
- [ lapply
- lsb cj hda (reverse ? xs@s0::reverse ? tla) rs0' ??)
- [ /2 by cons_injective_l, nil/
- | >length_append >length_append @eq_f @(eq_f ?? S)
- >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
+ [ >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) //
+ normalize #H destruct (H) // ] #Hte #_ #_ #_
+ 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 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)
- [@daemon] -Htb1 -Htb2 #Htb >Htb whd
- >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
- >right_mk_tape >Hmidta_src >Hmidta_dst
+ (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)
+ [ >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))
+ [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ]
+ >Hmidta_src >Hmidta_dst
whd in match (left ??); whd in match (left ??); whd in match (right ??);
- >length_tail >Hrs0 >length_append >length_append >length_reverse
- >length_append >commutative_plus in match (|reverse ??| + ?);
- whd in match (|?::?|); >length_reverse >length_reverse
- <(length_reverse ? ls) <Hlen' >H1 normalize // ]
- | #_ #Hte #_ <(reverse_reverse … ls0) in Hte; <(reverse_reverse … lsa)
- cut (|reverse ? lsa| = |reverse ? ls0|) [ // ] #Hlen'
- @(list_cases2 … Hlen')
- [ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #Hte
- 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
- (* 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) *)
- 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)
- [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
- >nth_change_vec_neq // >nth_change_vec //
- >right_mk_tape normalize in match (left ??);
- >Hmidta_src >Hmidta_dst >length_tail >Hrs0 >length_append
- >commutative_plus in match (pred ?); normalize
- >length_append >(?:|lsa| = O)
- [ normalize <plus_n_Sm <plus_n_Sm // | <(length_reverse ? lsa) >H1 % ]
- | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
- >reverse_cons >reverse_cons #Hte
- lapply (Hte cj hdb (reverse ? xs@s0::reverse ? tlb) rs0' ?
- lsb ci hda (reverse ? xs@s0::reverse ? tla) rs'' ??)
- [ /2 by cons_injective_l, nil/
- | >length_append >length_append @eq_f @(eq_f ?? S)
- >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
- 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)
- [@daemon] -Htb1 -Htb2 #Htb >Htb whd
- >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
- >right_mk_tape >Hmidta_src >Hmidta_dst
- whd in match (left ??); whd in match (left ??); whd in match (right ??);
- >length_tail >Hrs0 >length_append >length_append >length_reverse
- >length_append >commutative_plus in match (|reverse ??| + ?);
- whd in match (|?::?|); >length_reverse >length_reverse >Hlen
- <(length_reverse ? ls0) >H2 whd in match (|?::?|);
- >length_append normalize //
- ]
- ]
-
- #Hte1 #H2 #H3 #H4 >Hmidta_src >Hmidta_dst #Hte
- lapply (Hte … (refl ??) … (refl ??)) * * #_ #Htb1 #Htb2
- #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)] % ]
- | >Hs0 %{s0} % // #H destruct (H) @False_ind cases (Hss0) /2/ ] *)
- ]
- ]
- ]
+ >current_mk_tape <opt_cons_tail_expand >length_append
+ >length_reverse >length_reverse <(length_reverse ? ls) <Hlen'
+ >H1 normalize // ]
+ | #_ <(reverse_reverse … ls0) <(reverse_reverse … lsa)
+ cut (|reverse ? lsa| = |reverse ? ls0|) [ // ] #Hlen'
+ @(list_cases2 … Hlen')
+ [ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #_ #_ #Hte
+ lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte)
+ 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)
+ [ >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) ]]
+ normalize in ⊢ (??%); <opt_cons_tail_expand
+ normalize //
+ | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
+ >reverse_cons >reverse_cons #Hte #_ #_
+ lapply (Hte s0 hdb (reverse ? tlb) rs0 ?
+ lsb s hda (reverse ? tla) rs ??)
+ [ /2 by cons_injective_l, nil/
+ | >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) //
+ normalize #H destruct (H) //
+ | /2 by refl, trans_eq/ ] -Hte
+ #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)
+ [ >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) ]
+ >Hmidta_src >Hmidta_dst
+ whd in match (left ??); whd in match (left ??); whd in match (right ??);
+ >current_mk_tape <opt_cons_tail_expand >length_append
+ normalize in ⊢ (??%); >length_append >reverse_reverse
+ <(length_reverse ? lsa) >Hlen' >H2 normalize //
+ ]
+ ]
+ ]
+ ]
+ ]
| #ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd destruct (Htd)
whd in ⊢ (%→?); #Htb destruct (Htb) #ls #x #xs #Hta_src
lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
normalize #H destruct (H) ] ] ]
qed.
+(* lemma WF_to_WF_f : ∀A,B,R,f,b. WF A R (f b) → WF B (λx,y.R (f x) (f y)) b. *)
+let rec WF_to_WF_f A B R f b (Hwf: WF A R (f b)) on Hwf: WF B (λx,y.R (f x) (f y)) b ≝
+ match Hwf return (λa0,r.f b = a0 → WF B (λx,y:B. R (f x) (f y)) b) with
+ [ wf a Hwfa ⇒ λHeq.? ] (refl ??).
+% #b1 #HRb @WF_to_WF_f @Hwfa <Heq @HRb
+qed.
+
+lemma lt_WF : ∀n.WF ? lt n.
+#n @(nat_elim1 n) -n #n #IH % @IH
+qed.
-definition Pre_match_m ≝
- λsrc,sig,n.λt: Vector (tape sig) (S n).
- ∃x,xs.
- nth src (tape sig) t (niltape sig) = midtape ? [] x xs.
-
lemma terminate_match_m :
∀src,dst,sig,n,t.
src ≠ dst → src < S n → dst < S n →
- Pre_match_m src sig n t →
match_m src dst sig n ↓ t.
-#src #dst #sig #n #t #Hneq #Hsrc #Hdst * #start * #xs
-#Hmid_src
-@(terminate_while … (sem_match_step src dst sig n Hneq Hsrc Hdst)) //
-<(change_vec_same … t dst (niltape ?))
-lapply (refl ? (nth dst (tape sig) t (niltape ?)))
-cases (nth dst (tape sig) t (niltape ?)) in ⊢ (???%→?);
-[ #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
- >Hmid_src #HR cases (HR ?? (refl ??)) -HR
- >nth_change_vec // >Htape_dst #_ * #s0 * normalize in ⊢ (%→?); #H destruct (H)
-| #x0 #xs0 #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
- >Hmid_src #HR cases (HR ?? (refl ??)) -HR
- >nth_change_vec // >Htape_dst #_ normalize in ⊢ (%→?);
- * #s0 * #H destruct (H)
-| #x0 #xs0 #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
- >Hmid_src #HR cases (HR ?? (refl ??)) -HR
- >nth_change_vec // >Htape_dst #_ normalize in ⊢ (%→?);
- * #s0 * #H destruct (H)
-| #ls #s #rs lapply s -s lapply ls -ls lapply Hmid_src lapply t -t elim rs
- [#t #Hmid_src #ls #s #Hmid_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
- >Hmid_src >nth_change_vec // >Hmid_dst #HR cases (HR ?? (refl ??)) -HR
- >change_vec_change_vec #Ht1 #_ % #t2 whd in ⊢ (%→?);
- >Ht1 >nth_change_vec_neq [|@sym_not_eq //] >Hmid_src #HR
- cases (HR ?? (refl ??)) -HR #_
- >nth_change_vec // * #s1 * normalize in ⊢ (%→?); #H destruct (H)
- |#r0 #rs0 #IH #t #Hmid_src #ls #s #Hmid_dst % #t1 whd in ⊢ (%→?);
- >nth_change_vec_neq [|@sym_not_eq //] >Hmid_src
- #Htrue cases (Htrue ?? (refl ??)) -Htrue >change_vec_change_vec
- >nth_change_vec // >Hmid_dst whd in match (tape_move_mono ???); #Ht1
- * #s0 * whd in ⊢ (??%?→?); #H destruct (H) #_ >Ht1
- lapply (IH t1 ? (s0::ls) r0 ?)
- [ >Ht1 >nth_change_vec //
- | >Ht1 >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src
- | >Ht1 >nth_change_vec // ]
- ]
-]
+#src #dst #sig #n #ta #Hneq #Hsrc #Hdst
+@(terminate_while … (sem_match_step_termination src dst sig n Hneq Hsrc Hdst)) //
+letin f ≝ (λt0:Vector (tape sig) (S n).|left ? (nth src (tape ?) t0 (niltape ?))|
+ +|option_cons ? (current ? (nth dst (tape ?) t0 (niltape ?)))
+ (right ? (nth dst (tape ?) t0 (niltape ?)))|)
+change with (λx,y.f x < f y) in ⊢ (??%?); @WF_to_WF_f @lt_WF
+qed.
+
+lemma sem_match_m : ∀src,dst,sig,n.
+src ≠ dst → src < S n → dst < S n →
+ match_m src dst sig n \vDash R_match_m src dst sig n.
+#src #dst #sig #n #Hneq #Hsrc #Hdst @WRealize_to_Realize [/2/| @wsem_match_m // ]
qed.
\ No newline at end of file
projects / helm.git / blobdiff