(* *)
(**************************************************************************)
+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".
+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 rewind ≝ λsrc,dst,sig,n.
+ parmove src dst sig n L · mmove src sig n R · mmove dst sig n R.
-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 R_rewind_strong ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ (∀x,x0,xs,rs.
+ nth src ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
+ ∀ls0,y,y0,target,rs0.|xs| = |target| →
+ nth dst ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) src)
+ (midtape sig ls0 y0 (reverse ? target@y::rs0)) dst) ∧
+ (∀x,x0,xs,rs.
+ nth dst ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
+ ∀ls0,y,y0,target,rs0.|xs| = |target| →
+ nth src ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) dst)
+ (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 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.
+definition R_rewind ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ (∀x,x0,xs,rs.
+ nth src ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
+ ∀ls0,y,y0,target,rs0.|xs| = |target| →
+ nth dst ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) src)
+ (midtape sig ls0 y0 (reverse ? target@y::rs0)) dst) ∧
+ (∀x,rs.nth src ? int (niltape ?) = midtape sig [] x rs →
+ ∀ls0,y,rs0.nth dst ? int (niltape ?) = midtape sig ls0 y rs0 →
+ outt = int).
-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 // ] ]
+(*
+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 comp_list: ∀S:DeqSet. ∀l1,l2:list S.∀is_endc. ∃l,tl1,tl2.
- l1 = l@tl1 ∧ l2 = l@tl2 ∧ (∀c.c ∈ l = true → is_endc c = false) ∧
- ∀a,tla. tl1 = a::tla →
- is_endc a = true ∨ (is_endc a = false ∧∀b,tlb.tl2 = b::tlb → a≠b).
-#S #l1 #l2 #is_endc elim l1 in l2;
-[ #l2 %{[ ]} %{[ ]} %{l2} normalize %
- [ % [ % // | #c #H destruct (H) ] | #a #tla #H destruct (H) ]
-| #x #l3 #IH cases (true_or_false (is_endc x)) #Hendcx
- [ #l %{[ ]} %{(x::l3)} %{l} normalize
- % [ % [ % // | #c #H destruct (H) ] | #a #tla #H destruct (H) >Hendcx % % ]
- | *
- [ %{[ ]} %{(x::l3)} %{[ ]} normalize %
- [ % [ % // | #c #H destruct (H) ]
- | #a #tla #H destruct (H) cases (is_endc a)
- [ % % | %2 % // #b #tlb #H destruct (H) ]
- ]
- | #y #l4 cases (true_or_false (x==y)) #Hxy
- [ lapply (\P Hxy) -Hxy #Hxy destruct (Hxy)
- cases (IH l4) -IH #l * #tl1 * #tl2 * * * #Hl3 #Hl4 #Hl #IH
- %{(y::l)} %{tl1} %{tl2} normalize
- % [ % [ % //
- | #c cases (true_or_false (c==y)) #Hcy >Hcy normalize
- [ >(\P Hcy) //
- | @Hl ]
- ]
- | #a #tla #Htl1 @(IH … Htl1) ]
- | %{[ ]} %{(x::l3)} %{(y::l4)} normalize %
- [ % [ % // | #c #H destruct (H) ]
- | #a #tla #H destruct (H) cases (is_endc a)
- [ % % | %2 % // #b #tlb #H destruct (H) @(\Pf Hxy) ]
- ]
- ]
- ]
+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.
+#src #dst #sig #n #Hneq #Hsrc #Hdst
+@(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 % [ % [ %
+[ #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 //
+ |>length_append >length_append >Hlen % ]
+ >change_vec_commute [|@sym_not_eq //]
+ >change_vec_change_vec
+ >nth_change_vec_neq [|@sym_not_eq //]
+ >nth_change_vec // >reverse_append >reverse_single
+ >reverse_append >reverse_single normalize in match (tape_move ???);
+ >rev_append_def >append_nil #Htd >Htd in Htb;
+ >change_vec_change_vec >nth_change_vec //
+ cases ls0 [|#l1 #ls1] normalize in match (tape_move ???); //
+| #x #x0 #xs #rs #Hmidta_dst #ls0 #y #y0 #target #rs0 #Hlen #Hmidta_src
+ >(Htc2 ??? Hmidta_dst ls0 y (target@[y0]) rs0 ??) in Htd;
+ [|>Hmidta_src //
+ |>length_append >length_append >Hlen % ]
+ >change_vec_change_vec
+ >change_vec_commute [|@sym_not_eq //]
+ >nth_change_vec //
+ >reverse_append >reverse_single
+ >reverse_append >reverse_single
+ cases ls0 [|#l1 #ls1] normalize in match (tape_move ???);
+ #Htd >Htd in Htb; >change_vec_change_vec >nth_change_vec //
+ >rev_append_def >change_vec_commute // normalize in match (tape_move ???); // ]
+| #x #rs #Hmidta_src #ls0 #y #rs0 #Hmidta_dst
+ lapply (Htc1 … Hmidta_src … (refl ??) Hmidta_dst) -Htc1 #Htc >Htc in Htd;
+ >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
+ >nth_change_vec_neq [|@sym_not_eq //]
+ >nth_change_vec // lapply (refl ? ls0) cases ls0 in ⊢ (???%→%);
+ [ #Hls0 #Htd >Htd in Htb;
+ >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 //
+ | #l1 #ls1 #Hls0 #Htd >Htd in Htb;
+ >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.
-
-axiom daemon : ∀X:Prop.X.
+qed.
-definition match_test ≝ λsrc,dst.λsig:DeqSet.λn,is_endc.λv:Vector ? n.
- match (nth src (option sig) v (None ?)) with
- [ None ⇒ false
- | Some x ⇒ notb ((is_endc x) ∨ (nth dst (DeqOption sig) v (None ?) == None ?))].
+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 #H4 % /2 by /
+qed.
-definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc.
- compare src dst sig n is_endc ·
- (ifTM ?? (partest sig n (match_test src dst sig ? is_endc))
+definition match_step ≝ λsrc,dst,sig,n.
+ compare src dst sig n ·
+ (ifTM ?? (partest sig n (match_test src dst sig ?))
(single_finalTM ??
- (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst)))
+ (rewind src dst sig n · (inject_TM ? (move_r ?) n dst)))
(nop …)
partest1).
-
+
+(* we assume the src is a midtape
+ we stop
+ if the dst is out of bounds (outt = int)
+ or dst.right is shorter than src.right (outt.current → None)
+ or src.right is a prefix of dst.right (out = just right of the common prefix) *)
definition R_match_step_false ≝
- λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
- ∀ls,x,xs,end,rs.
- nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
- (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
- ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨
+ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ ∀ls,x,xs.
+ nth src ? int (niltape ?) = midtape sig ls x xs →
+ ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨
(∃ls0,rs0,xs0. nth dst ? int (niltape ?) = midtape sig ls0 x rs0 ∧
xs = rs0@xs0 ∧
- current sig (nth dst (tape sig) outt (niltape sig)) = None ?) ∨
+ outt = change_vec ??
+ (change_vec ?? int (mk_tape sig (reverse ? rs0@x::ls) (option_hd ? xs0) (tail ? xs0)) src)
+ (mk_tape ? (reverse ? rs0@x::ls0) (None ?) [ ]) dst) ∨
(∃ls0,rs0.
nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
- ∀rsj,c.
- rs0 = c::rsj →
+ (* ∀rsj,c.
+ rs0 = c::rsj → *)
outt = change_vec ??
- (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
- (midtape sig (reverse ? xs@x::ls0) c rsj) dst).
+ (change_vec ?? int (mk_tape sig (reverse ? xs@x::ls) (None ?) [ ]) src)
+ (mk_tape sig (reverse ? xs@x::ls0) (option_hd ? rs0) (tail ? rs0)) dst).
+(*
+ we assume the src is a midtape [ ] s rs
+ if we iterate
+ then dst.current = Some ? s1
+ and if s ≠ s1 then outt = int.dst.move_right()
+ and if s = s1
+ then int.src.right and int.dst.right have a common prefix
+ and the heads of their suffixes are different
+ and outt = int.dst.move_right().
+
+ *)
definition R_match_step_true ≝
- λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
- ∀s.current sig (nth src (tape sig) int (niltape sig)) = Some ? s →
- current sig (nth dst (tape sig) int (niltape sig)) ≠ None ? ∧
- (is_startc s = true →
- (∀c.c ∈ right ? (nth src (tape sig) int (niltape sig)) = true → is_startc c = false) →
- (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 → s ≠ s1 →
- outt = change_vec ?? int
- (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈s1,R〉)) dst ∧ is_endc s = false) ∧
- (∀ls,x,xs,ci,cj,rs,ls0,rs0.
- nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
- nth dst ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) →
- (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
- ci ≠ cj →
- (outt = change_vec ?? int
- (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false))).
-(* ∧
- (rs0 = [ ] →
- outt = change_vec ??
- (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) src)
- (mk_tape sig (reverse ? xs@x::ls0) (None ?) [ ]) dst)). *)
-
+ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ ∀s,rs.nth src ? int (niltape ?) = midtape ? [ ] s rs →
+ outt = change_vec ?? int
+ (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst ∧
+ (∃s0.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s0 ∧
+ (s0 = s →
+ ∃xs,ci,rs',ls0,cj,rs0.
+ rs = xs@ci::rs' ∧
+ nth dst ? int (niltape ?) = midtape sig ls0 s (xs@cj::rs0) ∧
+ ci ≠ cj)).
+
lemma sem_match_step :
- ∀src,dst,sig,n,is_startc,is_endc.src ≠ dst → src < S n → dst < S n →
- match_step src dst sig n is_startc is_endc ⊨
+ ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
+ match_step src dst sig n ⊨
[ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
- R_match_step_true src dst sig n is_startc is_endc,
- R_match_step_false src dst sig n is_endc ].
-#src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst
-@(acc_sem_seq_app sig n … (sem_compare src dst sig n is_endc Hneq Hsrc Hdst)
- (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ? is_endc))
+ R_match_step_true src dst sig n,
+ R_match_step_false src dst sig n ].
+#src #dst #sig #n #Hneq #Hsrc #Hdst
+@(acc_sem_seq_app sig n … (sem_compare src dst sig n Hneq Hsrc Hdst)
+ (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?))
(sem_seq …
- (sem_parmoveL ???? is_startc Hneq Hsrc Hdst)
+ (sem_rewind ???? Hneq Hsrc Hdst)
(sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
(sem_nop …)))
-[#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd >Htd -Htd
- * #te * #Hte #Htb whd
- #s #Hcurta_src %
- [ lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
- cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
- [| #c #_ % #Hfalse destruct (Hfalse) ]
- #Hcurta_dst >Hcomp1 in Htest; [| %2 %2 //]
- whd in ⊢ (??%?→?); change with (current ? (niltape ?)) in match (None ?);
- <nth_vec_map >Hcurta_src whd in ⊢ (??%?→?); <nth_vec_map
- >Hcurta_dst cases (is_endc s) normalize in ⊢ (%→?); #H destruct (H)
- | #Hstart #Hnotstart %
- [ #s1 #Hcurta_dst #Hneqss1 -Hcomp2
- cut (tc = ta)
- [@Hcomp1 %2 %1 %1 >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) //]
- #H destruct (H) -Hcomp1 cases Hte #_ -Hte #Hte
- cut (te = ta) [@Hte %1 %1 %{s} % //] -Hte #H destruct (H) %
- [cases Htb * #_ #Hmove #Hmove1 @(eq_vec … (niltape … ))
- #i #Hi cases (decidable_eq_nat i dst) #Hidst
- [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
- #ls * #rs #Hta_mid >(Hmove … Hta_mid) >Hta_mid cases rs //
- | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Hmove1 @sym_not_eq // ]
- | whd in Htest:(??%?); >(nth_vec_map ?? (current sig)) in Hcurta_src; #Hcurta_src
- >Hcurta_src in Htest; whd in ⊢ (??%?→?);
- cases (is_endc s) // whd in ⊢ (??%?→?); #H @sym_eq //
- ]
- |#ls #x #xs #ci #cj #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc #Hcicj
- cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) [ * #H destruct (H) ]
- * #cj' * #rs0' * #Hcjrs0 destruct (Hcjrs0) -Hcomp2 #Hcomp2
- lapply (Hcomp2 (or_intror ?? Hcicj)) -Hcomp2 #Htc %
- [ cases Hte -Hte #Hte #_ whd in Hte;
- >Htasrc_mid in Hcurta_src; whd in ⊢ (??%?→?); #H destruct (H)
- lapply (Hte ls ci (reverse ? xs) rs s ??? ls0 cj' (reverse ? xs) s rs0' (refl ??) ?) //
- [ >Htc >nth_change_vec //
- | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid cases (orb_true_l … Hc0) -Hc0 #Hc0
- [@memb_append_l2 >(\P Hc0) @memb_hd
- |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse //
+[ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?))))
+ cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%);
+ [ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %]
+ whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
+ >nth_current_chars >Hcurta_src normalize in ⊢ (%→?); #H destruct (H)
+ | #s #Hs lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
+ cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
+ [ #Hcurta_dst #Hcomp #_ * #td * >Hcomp [| %2 %]
+ whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
+ >nth_current_chars >nth_current_chars >Hs >Hcurta_dst
+ normalize in ⊢ (%→?); #H destruct (H)
+ | #s0 #Hs0
+ cases (current_to_midtape … Hs) #ls * #rs #Hmidta_src >Hmidta_src
+ cases (current_to_midtape … Hs0) #ls0 * #rs0 #Hmidta_dst >Hmidta_dst
+ cases (true_or_false (s == s0)) #Hss0
+ [ lapply (\P Hss0) -Hss0 #Hss0 destruct (Hss0)
+ #_ #Hcomp cases (Hcomp ????? (refl ??) (refl ??)) -Hcomp [ *
+ [ * #rs' * #_ #Hcurtc_dst * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
+ >nth_current_chars >nth_current_chars >Hcurtc_dst
+ cases (current ? (nth src …))
+ [normalize in ⊢ (%→?); #H destruct (H)
+ | #x >nth_change_vec // cases (reverse ? rs0)
+ [ normalize in ⊢ (%→?); #H destruct (H)
+ | #r1 #rs1 normalize in ⊢ (%→?); #H destruct (H) ] ]
+ | * #rs0' * #_ #Hcurtc_src * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
+ >(?:nth src ? (current_chars ?? tc) (None ?) = None ?)
+ [|>nth_current_chars >Hcurtc_src >nth_change_vec_neq
+ [>nth_change_vec [cases (append ???) // | @Hsrc]
+ |@(not_to_not … Hneq) //
+ ]]
+ 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 // #Hte #_ #Htb
+ #s' #rs' >Hmidta_src #H destruct (H)
+ 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 % ]
+ | >Hmidta_dst %{s'} % [%] #_
+ >Hrs0 %{xs} %{ci} %{rs''} %{ls0} %{cj} %{rs0'} % // % //
+ ]
]
- | >Htc >change_vec_commute // >nth_change_vec // ] -Hte
- >Htc >change_vec_commute // >change_vec_change_vec
- >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
- >Hte in Htb; * * #_ >reverse_reverse #Htbdst1 #Htbdst2 -Hte @(eq_vec … (niltape ?))
- #i #Hi cases (decidable_eq_nat i dst) #Hidst
- [ >Hidst >nth_change_vec // >(Htbdst1 ls0 s (xs@cj'::rs0'))
- [| >nth_change_vec // ]
- >Htadst_mid cases xs //
- | >nth_change_vec_neq [|@sym_not_eq // ]
- <Htbdst2 [| @sym_not_eq // ] >nth_change_vec_neq [| @sym_not_eq // ]
- <Htasrc_mid >change_vec_same % ]
- | >Hcurta_src in Htest; whd in ⊢(??%?→?);
- >Htc >change_vec_commute //
- change with (current ? (niltape ?)) in match (None ?);
- <nth_vec_map >nth_change_vec // whd in ⊢ (??%?→?);
- cases (is_endc ci) whd in ⊢ (??%?→?); #H destruct (H) %
+ | 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
+ #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/ ]
+ ]
]
- ]
]
-|#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb
- whd in ⊢ (%→?); #Hout >Hout >Htb whd
- #ls #c_src #xs #end #rs #Hmid_src #Hnotend #Hend
- lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
- cases (current … (nth dst ? intape (niltape ?))) in Hcomp1;
- [#Hcomp1 #_ %1 % % [% | @Hcomp1 %2 %2 % ]
- |#c_dst cases (true_or_false (c_src == c_dst)) #Hceq
- [#_ #Hmid_dst cases (Hmid_dst c_dst (refl …)) -Hmid_dst
- #ls_dst * #rs_dst #Hmid_dst
- cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * *
- #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq >Hrs_dst in Hmid_dst; #Hmid_dst
- cut (∃r1,rs1.rsi = r1::rs1) [@daemon] * #r1 * #rs1 #Hrs1 >Hrs1 in Hrs_src;
- #Hrs_src >Hrs_src in Hmid_src; #Hmid_src <(\P Hceq) in Hmid_dst; #Hmid_dst
- lapply (Hcomp2 ??????? Hmid_src Hmid_dst ?)
- [ #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
- [ >(\P Hc0) @Hnotend @memb_hd | @Hnotendxs1 //] ]
- *
- [ * #Hrsj >Hrsj #Hta % %2 >Hta >nth_change_vec //
- %{ls_dst} %{xs1} cut (∃xs0.xs = xs1@xs0)
- [lapply Hnotendxs1 -Hnotendxs1 lapply Hrs_src lapply xs elim xs1
- [ #l #_ #_ %{l} %
- | #x2 #xs2 #IH *
- [ whd in ⊢ (??%%→?); #H destruct (H) #Hnotendxs2
- >Hnotendxs2 in Hend; [ #H destruct (H) |@memb_hd ]
- | #x2' #xs2' whd in ⊢ (??%%→?); #H destruct (H)
- #Hnotendxs2 cases (IH xs2' e0 ?)
- [ #xs0 #Hxs2 %{xs0} @eq_f //
- |#c #Hc @Hnotendxs2 @memb_cons // ]
- ]
- ]
- ] * #xs0 #Hxs0 %{xs0} % [ %
- [ >Hmid_dst >Hrsj >append_nil %
- | @Hxs0 ]
- | cases (reverse ? xs1) // ]
- | * #cj * #rs2 * #Hrsj #Hta lapply (Hta ?)
- [ cases (Hneq ?? Hrs1) /2/ * #_ #Hr1 %2 @(Hr1 ?? Hrsj) ] -Hta #Hta
- %2 >Hta in Hc; whd in ⊢ (??%?→?);
- change with (current ? (niltape ?)) in match (None ?);
- <nth_vec_map >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
- whd in ⊢ (??%?→?); #Hc cut (is_endc r1 = true)
- [ cases (is_endc r1) in Hc; whd in ⊢ (??%?→?); //
- change with (current ? (niltape ?)) in match (None ?);
- <nth_vec_map >nth_change_vec // normalize #H destruct (H) ]
- #Hendr1 cut (xs = xs1)
- [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1
- -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs
- [ * normalize in ⊢ (%→?); //
- #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1
- lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H)
- | #x2 #xs2 #IH *
- [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc
- >Hnotendc in Hendr1; [| @memb_cons @memb_hd ]
- normalize in ⊢ (%→?); #H destruct (H)
- | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq)
- #Hnotendc #Hnotendcxs1 @eq_f @IH
- [ @(cons_injective_r … Heq)
- | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0
- [ >(\P Hc0) @memb_hd
- | @memb_cons @memb_cons // ]
- | #c #Hc @Hnotendcxs1 @memb_cons // ]
- ]
- ]
- | #Hxsxs1 destruct (Hxsxs1) >Hmid_dst %{ls_dst} %{rsj} % //
- #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0)
- lapply (append_l2_injective … Hrs_src) // #Hrs' destruct (Hrs') %
- ]
- ]
- |#Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls0 * #rs0 #Hdst
- @False_ind lapply (Hcomp1 ?) [%2 %1 %1 >Hmid_src normalize
- @(not_to_not ??? (\Pf Hceq)) #H destruct //] #Hintape >Hintape in Hc;
- whd in ⊢(??%?→?); >Hmid_src
- change with (current ? (niltape ?)) in match (None ?);
- <nth_vec_map >Hmid_src whd in ⊢ (??%?→?);
- >(Hnotend c_src) [|@memb_hd]
- change with (current ? (niltape ?)) in match (None ?);
- <nth_vec_map >Hmid_src whd in ⊢ (??%?→?); >Hdst normalize #H destruct (H)
- ]
- ]
-]
+| #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 ?))))
+ cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
+ [ #Hcurta_dst % % % // @Hcomp1 %2 //
+ | #x0 #Hcurta_dst cases (current_to_midtape … Hcurta_dst) -Hcurta_dst
+ #ls0 * #rs0 #Hta_dst cases (true_or_false (x == x0)) #Hxx0
+ [ lapply (\P Hxx0) -Hxx0 #Hxx0 destruct (Hxx0)
+ | >(?:tc=ta) in Htest;
+ [|@Hcomp1 % % >Hta_src >Hta_dst @(not_to_not ??? (\Pf Hxx0)) normalize
+ #Hxx0' destruct (Hxx0') % ]
+ whd in ⊢ (??%?→?);
+ >nth_current_chars >Hta_src >nth_current_chars >Hta_dst
+ whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ] -Hcomp1
+ cases (Hcomp2 … Hta_src Hta_dst) [ *
+ [ * #rs' * #Hxs #Hcurtc % %2 %{ls0} %{rs0} %{rs'} %
+ [ % // | >Hcurtc % ]
+ | * #rs0' * #Hxs #Htc %2 >Htc %{ls0} %{rs0'} % // ]
+ | * #xs0 * #ci * #cj * #rs' * #rs0' * * *
+ #Hci #Hxs #Hrs0 #Htc @False_ind
+ whd in Htest:(??%?);
+ >(?:nth src ? (current_chars ?? tc) (None ?) = Some ? ci) in Htest;
+ [|>nth_current_chars >Htc >nth_change_vec_neq [|@(not_to_not … Hneq) //]
+ >nth_change_vec //]
+ >(?:nth dst ? (current_chars ?? tc) (None ?) = Some ? cj)
+ [|>nth_current_chars >Htc >nth_change_vec //]
+ normalize #H destruct (H) ] ] ]
qed.
-definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc.
- whileTM … (match_step src dst sig n is_startc is_endc)
+definition match_m ≝ λsrc,dst,sig,n.
+ whileTM … (match_step src dst sig n)
(inr ?? (inr ?? (inl … (inr ?? start_nop)))).
definition R_match_m ≝
- λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
-(* (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧ *)
- ∀ls,x,xs,end,rs.
- nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
- (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
- (∀c0. memb ? c0 (xs@end::rs) = true → is_startc c0 = false) →
- (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧
- (is_startc x = true →
- (∀ls0,x0,rs0.
- nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
- (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
- ∀cj,l2.l1=cj::l2 →
- outt = change_vec ??
- (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
- (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) dst) ∨
- ∀l,l1.x0::rs0 ≠ l@x::xs@l1)).
-
-(*
-definition R_match_m ≝
- λi,j,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
- (((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
- current ? (nth i ? int (niltape ?)) = None ? ∨
- current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
- (∀ls,x,xs,ci,rs,ls0,x0,rs0.
- (∀x. is_startc x ≠ is_endc x) →
- is_startc x = true → is_endc ci = true →
- (∀z. memb ? z (x::xs) = true → is_endc x = false) →
- nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
- nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
- (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
- ∀cj,l2.l1=cj::l2 →
- outt = change_vec ??
- (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
- (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) j) ∨
- ∀l,l1.x0::rs0 ≠ l@x::xs@l1).
-*)
-
-(*
-axiom sub_list_dec: ∀A.∀l,ls:list A.
- ∃l1,l2. l = l1@ls@l2 ∨ ∀l1,l2. l ≠ l1@ls@l2.
-*)
+ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ ∀x,rs.
+ nth src ? int (niltape ?) = midtape sig [ ] x rs →
+ (current sig (nth dst (tape sig) int (niltape sig)) = None ? →
+ right ? (nth dst (tape sig) int (niltape sig)) = [ ] → outt = int) ∧
+ (∀ls0,x0,rs0.
+ nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
+ (∃l,l1.x0::rs0 = l@x::rs@l1 ∧
+ outt = change_vec ??
+ (change_vec ?? int
+ (mk_tape sig (reverse ? rs@[x]) (None ?) [ ]) src)
+ (mk_tape sig ((reverse ? (l@x::rs))@ls0) (option_hd ? l1) (tail ? l1)) dst) ∨
+ ∀l,l1.x0::rs0 ≠ l@x::rs@l1).
lemma not_sub_list_merge :
∀T.∀a,b:list T. (∀l1.a ≠ b@l1) → (∀t,l,l1.a ≠ t::l@b@l1) → ∀l,l1.a ≠ l@b@l1.
qed.
-lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
+lemma wsem_match_m : ∀src,dst,sig,n.
src ≠ dst → src < S n → dst < S n →
- match_m src dst sig n is_startc is_endc ⊫ R_match_m src dst sig n is_startc is_endc.
-#src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
-lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst) … Hloop) //
+ match_m src dst sig n ⊫ R_match_m src dst sig n.
+#src #dst #sig #n #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
+lapply (sem_while … (sem_match_step src dst sig n Hneq Hsrc Hdst) … Hloop) //
-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ #tc #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend #Hnotstart
- cases (Hfalse … Hmid_src Hnotend Hend) -Hfalse
+[ #Hfalse #x #xs #Hmid_src
+ cases (Hfalse … Hmid_src) -Hfalse
[(* current dest = None *) *
[ * #Hcur_dst #Houtc %
[#_ >Houtc //
- |#Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
- normalize in ⊢ (%→?); #H destruct (H)
+ | #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
+ normalize in ⊢ (%→?); #H destruct (H)
]
| * #ls0 * #rs0 * #xs0 * * #Htc_dst #Hrs0 #HNone %
[ >Htc_dst normalize in ⊢ (%→?); #H destruct (H)
- | #Hstart #ls1 #x1 #rs1 >Htc_dst #H destruct (H)
- >Hrs0 cases xs0
+ | #ls1 #x1 #rs1 >Htc_dst #H destruct (H)
+ >Hrs0 >HNone cases xs0
[ % %{[ ]} %{[ ]} % [ >append_nil >append_nil %]
- #cj #ls2 #H destruct (H)
+ @eq_f3 //
+ [ >reverse_append %
+ | >reverse_append >reverse_cons >reverse_append
+ >associative_append >associative_append % ]
| #x2 #xs2 %2 #l #l1 % #Habs lapply (eq_f ?? (length ?) ?? Habs)
>length_append whd in ⊢ (??%(??%)→?); >length_append
>length_append normalize >commutative_plus whd in ⊢ (???%→?);
]
]
]
- |* #ls0 * #rs0 * #Hmid_dst #HFalse %
+ |* #ls0 * #rs0 * #Hmid_dst #Houtc %
[ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
- | #Hstart #ls1 #x1 #rs1 >Hmid_dst #H destruct (H)
- %1 %{[ ]} %{rs0} % [%] #cj #l2 #Hnotnil
- >reverse_cons >associative_append @(HFalse ?? Hnotnil)
+ |#ls1 #x1 #rs1 >Hmid_dst #H destruct (H)
+ %1 %{[ ]} %{rs0} % [%]
+ >reverse_cons >associative_append >Houtc %
]
]
-|#ta #tb #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
- #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend #Hnotstart
+|-ta #ta #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
+ #x #xs #Hmidta_src
lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
- [#Hmid_dst %
- [#_ whd in Htrue; >Hmid_src in Htrue; #Htrue
- cases (Htrue x (refl … )) -Htrue * #Htaneq #_
- @False_ind >Hmid_dst in Htaneq; /2/
- |#Hstart #ls0 #x0 #rs0 #Hmid_dst2 >Hmid_dst2 in Hmid_dst; normalize in ⊢ (%→?);
- #H destruct (H)
+ [#Hcurta_dst %
+ [#Hcurta_dst #Hrightta_dst whd in Htrue; >Hmidta_src in Htrue; #Htrue
+ cases (Htrue ?? (refl ??)) -Htrue #Htc
+ cut (tc = ta)
+ [ >Htc whd in match (tape_move_mono ???); whd in match (tape_write ???);
+ <(change_vec_same … ta dst (niltape ?)) in ⊢ (???%);
+ lapply Hrightta_dst lapply Hcurta_dst -Hrightta_dst -Hcurta_dst
+ cases (nth dst ? ta (niltape ?))
+ [ #_ #_ %
+ | #r0 #rs0 #_ normalize in ⊢ (%→?); #H destruct (H)
+ | #l0 #ls0 #_ #_ %
+ | #ls #x0 #rs normalize in ⊢ (%→?); #H destruct (H) ] ]
+ -Htc #Htc destruct (Htc) #_
+ cases (IH … Hmidta_src) #Houtc #_ @Houtc //
+ |#ls0 #x0 #rs0 #Hmidta_dst >Hmidta_dst in Hcurta_dst;
+ normalize in ⊢ (%→?); #H destruct (H)
]
| #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ]
- #Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcurta_dst; normalize in ⊢ (%→?);
- #H destruct (H) whd in Htrue; >Hmid_src in Htrue; #Htrue
- cases (Htrue x (refl …)) -Htrue #_ #Htrue cases (Htrue Hstart Hnotstart) -Htrue
+ #ls0 #x0 #rs0 #Hmidta_dst >Hmidta_dst in Hcurta_dst; normalize in ⊢ (%→?);
+ #H destruct (H) whd in Htrue; >Hmidta_src in Htrue; #Htrue
+ cases (Htrue ?? (refl …)) -Htrue >Hmidta_dst #Htc
cases (true_or_false (x==c)) #eqx
- [ lapply (\P eqx) -eqx #eqx destruct (eqx)
- #_ #Htrue cases (comp_list ? (xs@end::rs) rs0 is_endc)
- #x1 * #tl1 * #tl2 * * * #Hxs #Hrs0 #Hnotendx1
- cases tl1 in Hxs;
- [>append_nil #Hx1 <Hx1 in Hnotendx1; #Hnotendx1
- lapply (Hnotendx1 end ?) [ @memb_append_l2 @memb_hd ]
- >Hend #H destruct (H) ]
- #ci -tl1 #tl1 #Hxs #H cases (H … (refl … ))
- [ #Hendci % cases (IH ????? Hmid_src Hnotend Hend Hnotstart)
- (* this is absurd, since Htrue conlcudes is_endc ci =false *)
- (* no, è più complicato
- #Hend_ci lapply (Htrue ls c xi
- *)
- @daemon (* lapply(Htrue … (refl …)) -Htrue *)
- |cases tl2 in Hrs0;
- [ >append_nil #Hrs0 destruct (Hrs0) * #Hcifalse#_ %2
- cut (∃l.xs = x1@ci::l)
- [lapply Hxs lapply Hnotendx1 lapply Hnotend lapply xs
- -Hxs -xs -Hnotendx1 elim x1
- [ *
- [ #_ #_ normalize #H1 destruct (H1) >Hend in Hcifalse;
- #H1 destruct (H1)
- | #x2 #xs2 #_ #_ normalize #H >(cons_injective_l ????? H) %{xs2} % ]
- | #x2 #xs2 #IHin *
- [ #_ #Hnotendxs2 normalize #H destruct (H)
- >(Hnotendxs2 ? (memb_hd …)) in Hend; #H destruct (H)
- | #x3 #xs3 #Hnotendxs3 #Hnotendxs2 normalize #H destruct (H)
- cases (IHin ??? e0)
- [ #xs4 #Hxs4 >Hxs4 %{xs4} %
- | #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
- [ >(\P Hc0) @Hnotendxs3 @memb_hd
- | @Hnotendxs3 @memb_cons @memb_cons @Hc0 ]
- | #c0 #Hc0 @Hnotendxs2 @memb_cons @Hc0 ]
- ]
- ]
- ] * #l #Hxs' >Hxs'
- #l0 #l1 % #H lapply (eq_f ?? (length ?) ?? H) -H
- >length_append normalize >length_append >length_append
- normalize >commutative_plus normalize #H destruct (H) -H
- >associative_plus in e0; >associative_plus
- >(plus_n_O (|x1|)) in ⊢(??%?→?); #H lapply (injective_plus_r … H)
- -H normalize #H destruct (H)
- | #cj #tl2' #Hrs0 * #Hcifalse #Hcomp lapply (Htrue ls c x1 ci cj tl1 ls0 tl2' ????)
- [ @(Hcomp ?? (refl ??))
- | #c0 #Hc0 cases (orb_true_l … Hc0) #Hc0
- [ @Hnotend >(\P Hc0) @memb_hd
- | @Hnotendx1 // ]
- | >Hmid_dst >Hrs0 %
- | >Hxs %
- | * #Htb >Htb #Hendci >Hrs0 >Hxs
- cases (IH ls c xs end rs ? Hnotend Hend Hnotstart) -IH
- [| >Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src ]
- #_ #IH lapply Hxs lapply Hnotendx1 -Hxs -Hnotendx1 cases x1 in Hrs0;
- [ #Hrs0 #_ whd in ⊢ (???%→?); #Hxs
- cases (IH Hstart (c::ls0) cj tl2' ?)
- [ -IH * #l * #l1 * #Hll1 #IH % %{(c::l)} %{l1}
- % [ @eq_f @Hll1 ]
- #cj0 #l2 #Hcj0 >(IH … Hcj0) >Htb
- >change_vec_commute // >change_vec_change_vec
- >change_vec_commute [|@sym_not_eq // ] @eq_f3 //
- >reverse_cons >associative_append %
- | #IH %2 #l #l1 >(?:l@c::xs@l1 = l@(c::xs)@l1) [|%]
- @not_sub_list_merge
- [ #l2 cut (∃xs'.xs = ci::xs')
- [ cases xs in Hxs;
- [ normalize #H destruct (H) >Hend in Hendci; #H destruct (H)
- | #ci' #xs' normalize #H lapply (cons_injective_l ????? H)
- #H1 >H1 %{xs'} % ]
- ]
- * #xs' #Hxs' >Hxs' normalize % #H destruct (H)
- lapply (Hcomp … (refl ??)) * /2/
- |#t #l2 #l3 % normalize #H lapply (cons_injective_r ????? H)
- -H #H >H in IH; #IH cases (IH l2 l3) -IH #IH @IH % ]
- | >Htb >nth_change_vec // >Hmid_dst >Hrs0 % ]
- | #x2 #xs2 normalize in ⊢ (%→?); #Hrs0 #Hnotendxs2 normalize in ⊢ (%→?);
- #Hxs cases (IH Hstart (c::ls0) x2 (xs2@cj::tl2') ?)
- [ -IH * #l * #l1 * #Hll1 #IH % %{(c::l)} %{l1}
- % [ @eq_f @Hll1 ]
- #cj0 #l2 #Hcj0 >(IH … Hcj0) >Htb
- >change_vec_commute // >change_vec_change_vec
- >change_vec_commute [|@sym_not_eq // ] @eq_f3 //
- >reverse_cons >associative_append %
- | -IH #IH %2 #l #l1 >(?:l@c::xs@l1 = l@(c::xs)@l1) [|%]
- @not_sub_list_merge_2 [| @IH]
- cut (∃l2.xs = (x2::xs2)@ci::l2)
- [lapply Hnotendxs2
- lapply Hnotend -Hnotend lapply Hxs
- >(?:x2::xs2@ci::tl1 = (x2::xs2)@ci::tl1) [|%]
- lapply (x2::xs2) elim xs
- [ *
- [ normalize in ⊢ (%→?); #H1 destruct (H1)
- >Hendci in Hend; #Hend destruct (Hend)
- | #x3 #xs3 normalize in ⊢ (%→?); #H1 destruct (H1)
- #_ #Hnotendx3 >(Hnotendx3 ? (memb_hd …)) in Hend;
- #Hend destruct (Hend)
- ]
- | #x3 #xs3 #IHin *
- [ normalize in ⊢ (%→?); #Hxs3 destruct (Hxs3) #_ #_
- %{xs3} %
- | #x4 #xs4 normalize in ⊢ (%→?); #Hxs3xs4 #Hnotend
- #Hnotendxs4 destruct (Hxs3xs4) cases (IHin ? e0 ??)
- [ #l0 #Hxs3 >Hxs3 %{l0} %
- | #c0 #Hc0 @Hnotend cases (orb_true_l … Hc0) -Hc0 #Hc0
- [ >(\P Hc0) @memb_hd
- | @memb_cons @memb_cons @Hc0 ]
- | #c0 #Hc0 @Hnotendxs4 @memb_cons //
- ]
- ]
- ]
- ] * #l2 #Hxs'
- >Hxs' #l3 normalize >associative_append normalize % #H
- destruct (H) lapply (append_l2_injective ?????? e1) //
- #H1 destruct (H1) cases (Hcomp ?? (refl ??)) /2/
- | >Htb >nth_change_vec // >Hmid_dst >Hrs0 % ]
- ]
- ]
+ [ lapply (\P eqx) -eqx #eqx destruct (eqx) * #s0 * whd in ⊢ (??%?→?); #Hs0
+ destruct (Hs0) #Htrue cases (Htrue (refl ??)) -Htrue
+ #xs0 * #ci * #rs' * #ls1 * #cj * #rs1 * * #Hxs #H destruct (H) #Hcicj
+ >Htc in IH; whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
+ #IH cases (IH … Hmidta_src) -IH #_ >nth_change_vec //
+ cut (∃x1,xs1.xs0@cj::rs1 = x1::xs1)
+ [ cases xs0 [ %{cj} %{rs1} % | #x1 #xs1 %{x1} %{(xs1@cj::rs1)} % ] ] * #x1 * #xs1
+ #Hxs1 >Hxs1 #IH cases (IH … (refl ??)) -IH
+ [ * #l * #l1 * #Hxs1'
+ >change_vec_commute // >change_vec_change_vec
+ #Houtc % %{(s0::l)} %{l1} %
+ [ normalize <Hxs1' %
+ | >reverse_cons >associative_append >change_vec_commute // @Houtc ]
+ | #H %2 #l #l1 >(?:l@s0::xs@l1 = l@(s0::xs)@l1) [|%]
+ @not_sub_list_merge
+ [ #l2 >Hxs <Hxs1 % normalize #H1 lapply (cons_injective_r ????? H1)
+ >associative_append #H2 lapply (append_l2_injective ????? (refl ??) H2)
+ #H3 lapply (cons_injective_l ????? H3) #H3 >H3 in Hcicj; * /2/
+ |#t #l2 #l3 % normalize #H1 lapply (cons_injective_r ????? H1)
+ -H1 #H1 cases (H l2 l3) #H2 @H2 @H1
]
]
- |lapply (\Pf eqx) -eqx #eqx >Hmid_dst #Htrue
- cases (Htrue ? (refl ??) eqx) -Htrue #Htb #Hendcx #_
- cases rs0 in Htb;
- [ #_ %2 #l #l1 cases l
+ | #_ cases (IH x xs ?) -IH
+ [| >Htc >nth_change_vec_neq [|@sym_not_eq //] @Hmidta_src ]
+ >Htc >nth_change_vec // cases rs0
+ [ #_ #_ %2 #l #l1 cases l
[ normalize cases xs
[ cases l1
- [ normalize % #H destruct (H) cases eqx /2/
+ [ normalize % #H destruct (H) cases (\Pf eqx) /2/
| #tmp1 #l2 normalize % #H destruct (H) ]
| #tmp1 #l2 normalize % #H destruct (H) ]
| #tmp1 #l2 normalize % #H destruct (H)cases l2 in e0;
[ normalize #H1 destruct (H1)
- | #tmp2 #l3 normalize #H1 destruct (H1) ]
- ]
- | #r1 #rs1 normalize in ⊢ (???(????%?)→?); #Htb >Htb in IH; #IH
- cases (IH ls x xs end rs ? Hnotend Hend Hnotstart)
- [| >Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src ] -IH
- #_ #IH cases (IH Hstart (c::ls0) r1 rs1 ?)
- [|| >nth_change_vec // ] -IH
- [ * #l * #l1 * #Hll1 #Hout % %{(c::l)} %{l1} % >Hll1 //
- >reverse_cons >associative_append #cj0 #ls #Hl1 >(Hout ?? Hl1)
- >change_vec_commute in ⊢ (??(???%??)?); // @sym_not_eq //
- | #IH %2 @(not_sub_list_merge_2 ?? (x::xs)) normalize [|@IH]
- #l1 % #H destruct (H) cases eqx /2/
- ]
+ | #tmp2 #l3 normalize #H1 destruct (H1) ] ]
+ | #r1 #rs1 #_ #IH cases (IH … (refl ??)) -IH
+ [ * #l * #l1 * #Hll1 #Houtc % %{(c::l)} %{l1} % [ >Hll1 % ]
+ >Houtc >change_vec_commute // >change_vec_change_vec
+ >change_vec_commute [|@sym_not_eq //]
+ >reverse_cons >associative_append %
+ | #Hll1 %2 @(not_sub_list_merge_2 ?? (x::xs)) normalize [|@Hll1]
+ #l1 % #H destruct (H) cases (\Pf eqx) /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 ?))| +
+ |option_cons ? (current ? (nth dst ? outt (niltape ?))) (right ? (nth dst ? 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 ⊨
+ [ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
+ R_match_step_true_naive src dst sig n,
+ R_match_step_false src dst sig n ].
+#src #dst #sig #n #Hneq #Hsrc #Hdst
+@(acc_sem_seq_app sig n … (sem_compare src dst sig n Hneq Hsrc Hdst)
+ (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 …)))
+[ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?))))
+ cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%);
+ [ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %]
+ whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
+ >nth_current_chars >Hcurta_src normalize in ⊢ (%→?); #H destruct (H)
+ | #s #Hs lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
+ cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
+ [ #Hcurta_dst #Hcomp #_ * #td * >Hcomp [| %2 %]
+ whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
+ >nth_current_chars >nth_current_chars >Hs >Hcurta_dst
+ normalize in ⊢ (%→?); #H destruct (H)
+ | #s0 #Hs0
+ cases (current_to_midtape … Hs) #ls * #rs #Hmidta_src >Hmidta_src
+ cases (current_to_midtape … Hs0) #ls0 * #rs0 #Hmidta_dst >Hmidta_dst
+ cases (true_or_false (s == s0)) #Hss0
+ [ lapply (\P Hss0) -Hss0 #Hss0 destruct (Hss0)
+ #_ #Hcomp cases (Hcomp ????? (refl ??) (refl ??)) -Hcomp [ *
+ [ * #rs' * #_ #Hcurtc_dst * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
+ >nth_current_chars >nth_current_chars >Hcurtc_dst
+ cases (current ? (nth src …))
+ [normalize in ⊢ (%→?); #H destruct (H)
+ | #x >nth_change_vec // cases (reverse ? rs0)
+ [ normalize in ⊢ (%→?); #H destruct (H)
+ | #r1 #rs1 normalize in ⊢ (%→?); #H destruct (H) ] ]
+ | * #rs0' * #_ #Hcurtc_src * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
+ >(?:nth src ? (current_chars ?? tc) (None ?) = None ?)
+ [|>nth_current_chars >Hcurtc_src >nth_change_vec_neq
+ [>nth_change_vec [cases (append ???) // | @Hsrc]
+ |@(not_to_not … Hneq) //
+ ]]
+ 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 //
+ 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 #_
+ 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
+ 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 ??);
+ >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand
+ whd in match (option_cons ???); >Hrs0
+ normalize in ⊢ (?(?%)%); //
+ | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
+ >reverse_cons >reverse_cons #Hte
+ lapply (Hte ci hdb (reverse ? xs@s0::reverse ? tlb) rs'' ?
+ 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
+ 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
+ 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
+ >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 (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 >current_mk_tape <opt_cons_tail_expand >Hrs0
+ >length_append normalize >length_append >length_append
+ <(reverse_reverse ? lsa) >H1 normalize //
+ | #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 ??);
+ >current_mk_tape <opt_cons_tail_expand
+ whd in match (option_cons ???);
+ >Hrs0 >length_append whd in ⊢ (??(??%)); >length_append >length_reverse
+ >length_append >commutative_plus in match (|reverse ??| + ?);
+ whd in match (|?::?|); >length_reverse >length_reverse
+ <(length_reverse ? lsa) >Hlen' >H2 >length_append
+ normalize //
+ ]
+ ]
+ ]
+ | 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
+ 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
+ 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 //
+ >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst
+ >right_mk_tape 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 ??))
+ [ >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) //
+ normalize #H destruct (H) // ] #Hte #_ #_ #_
+ * * #_ >Hte >nth_change_vec // #Htb1 #Htb2
+ 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)
+ [@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 ??);
+ >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)
+ * * #_ >Hmidta_dst #Htb1 lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2
+ cut (tb = change_vec (tape sig) (S n) ta (mk_tape ? (s0::ls0) (option_hd ? rs0) (tail ? rs0)) dst)
+ [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
+ >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst
+ >current_mk_tape >right_mk_tape 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 * * #_ >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 tlb)@s0::rs0)) (tail ? (reverse sig (reverse sig tlb)@s0::rs0))) dst)
+ (midtape ? lsb hda (reverse sig (reverse sig tla)@s::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 ??);
+ >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 ?))))
+ cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
+ [ #Hcurta_dst % % % // @Hcomp1 %2 //
+ | #x0 #Hcurta_dst cases (current_to_midtape … Hcurta_dst) -Hcurta_dst
+ #ls0 * #rs0 #Hta_dst cases (true_or_false (x == x0)) #Hxx0
+ [ lapply (\P Hxx0) -Hxx0 #Hxx0 destruct (Hxx0)
+ | >(?:tc=ta) in Htest;
+ [|@Hcomp1 % % >Hta_src >Hta_dst @(not_to_not ??? (\Pf Hxx0)) normalize
+ #Hxx0' destruct (Hxx0') % ]
+ whd in ⊢ (??%?→?);
+ >nth_current_chars >Hta_src >nth_current_chars >Hta_dst
+ whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ] -Hcomp1
+ cases (Hcomp2 … Hta_src Hta_dst) [ *
+ [ * #rs' * #Hxs #Hcurtc % %2 %{ls0} %{rs0} %{rs'} %
+ [ % // | >Hcurtc % ]
+ | * #rs0' * #Hxs #Htc %2 >Htc %{ls0} %{rs0'} % // ]
+ | * #xs0 * #ci * #cj * #rs' * #rs0' * * *
+ #Hci #Hxs #Hrs0 #Htc @False_ind
+ whd in Htest:(??%?);
+ >(?:nth src ? (current_chars ?? tc) (None ?) = Some ? ci) in Htest;
+ [|>nth_current_chars >Htc >nth_change_vec_neq [|@(not_to_not … Hneq) //]
+ >nth_change_vec //]
+ >(?:nth dst ? (current_chars ?? tc) (None ?) = Some ? cj)
+ [|>nth_current_chars >Htc >nth_change_vec //]
+ normalize #H destruct (H) ] ] ]
+qed.
+
+axiom terminate_match_m :
+ ∀src,dst,sig,n,t.
+ src ≠ dst → src < S n → dst < S n →
+ match_m src dst sig n ↓ t.
+(*#src #dst #sig #n #ta #Hneq #Hsrc #Hdst
+@(terminate_while … (sem_match_step_termination src dst sig n Hneq Hsrc Hdst)) // % #tb
+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 (f tb < f ta) in ⊢ (%→?); @(nat_elim1 (f tb))
+#x lapply (refl ? x) cases x in ⊢ (???%→%);
+[ #Hx
+*
+#x #IH #Hx cases
+
+ @IH % #tc change with (f tc < f tb) in ⊢ (%→?);
+
+ )(|left @(nat_elim1 (|left ? (nth ? (tape ?) t (niltape ?))|
+ +|option_cons sig (current ? (nth dst (tape ?) t (niltape ?)))
+ (right ? (nth dst (tape ?) t (niltape ?)))|))
+<(change_vec_same … t dst (niltape ?))
+<(change_vec_same … t src (niltape ?)) in ⊢ (???(???%??));
+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 // ]
+ ]
+]
+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.
projects / helm.git / blobdiff