*)
definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc.
- compare src dst sig n ·
+ compare src dst sig n is_endc ·
(ifTM ?? (inject_TM ? (test_char ? (λa.is_endc a == false)) n src)
(single_finalTM ??
(parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst)))
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.
+
+axiom 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 ∨ (∀b,tlb.tl2 = b::tlb → a≠b).
+
+
+definition R_match_step_false ≝
+ λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
+ (((∃x.current ? (nth src ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
+ (* current ? (nth src ? int (niltape ?)) ≠ current ? (nth dst ? int (niltape ?)) ∨ *)
+ current sig (nth src (tape sig) int (niltape sig)) = None ? ∨
+ current sig (nth dst (tape sig) int (niltape sig)) = None ? ) → outt = int) ∧
+ (∀ls,ls0,x,x0,rs,rs0.
+ nth src ? int (niltape ?) = midtape sig ls x rs →
+ nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
+ x ≠ x0 ∨
+ (x = x0 ∧
+ ∀xs,end,rs',rs0'.rs = xs@end::rs' → rs0 = xs@rs0' →
+ (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
+ is_endc end = false ∨
+ (is_endc end = true ∧
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
+ (mk_tape sig (reverse ? xs@x::ls0) (option_hd ? rs0) (tail ? rs0)) dst))).
+ ∀ls,ls0,rs,rs0,x,xs,end.
+ (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
+ nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
+ nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
+ is_endc end = false ∨
+ (is_endc end = true ∧
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
+ (mk_tape sig (reverse ? xs@x::ls0) (option_hd ? rs0) (tail ? rs0)) dst)).
+
+(*
definition R_match_step_false ≝
λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
(((∃x.current ? (nth src ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
current sig (nth dst (tape sig) int (niltape sig)) = None ? ) ∧ outt = int) ∨
∃ls,ls0,rs,rs0,x,xs. ∀rsi,rsj,end,c.
rs = end::rsi → rs0 = c::rsj →
- is_endc x = false ∧ is_endc end = true ∧
+ (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) ∧ is_endc end = true ∧
nth src ? int (niltape ?) = midtape sig ls x (xs@rs) ∧
nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
outt = change_vec ??
(change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rsi) src)
(midtape sig (reverse ? xs@x::ls0) c rsj) dst.
+*)
+
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 →
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 →
+ (∀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,rs,ls0,cj,rs0.
nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
nth dst ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) → ci ≠ cj →
+ (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
outt = change_vec ?? int
(tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false).
| @sym_eq @Hnth_j @sym_not_eq // ] ] ]
qed.
-axiom comp_list: ∀S:DeqSet. ∀l1,l2:list S. ∃l,tl1,tl2.
- l1 = l@tl1 ∧ l2 = l@tl2 ∧ ∀a,b,tla,tlb. tl1 = a::tla → tl2 = b::tlb → a≠b.
+axiom 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 ∨ (∀b,tlb.tl2 = b::tlb → a≠b).
axiom daemon : ∀X:Prop.X.
match_step src dst sig n is_startc is_endc ⊨
[ 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 ].
+ R_match_step_false src dst sig n is_endc ].
+@daemon
+(*
#src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst
-@(acc_sem_seq_app sig n … (sem_compare src dst sig n 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_test_char_multi sig (λa.is_endc a == false) n src (le_S_S_to_le … Hsrc))
(sem_seq …
(sem_parmoveL ???? is_startc Hneq Hsrc Hdst)
#Htb #s #Hcurta_src #Hstart #Hnotstart %
[ #s1 #Hcurta_dst #Hneqss1
lapply Htb lapply Hcurtc -Htb -Hcurtc >(?:tc=ta)
- [|@Hcomp1 % % >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) % ]
+ [|@Hcomp1 %2 % % >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) % ]
#Hcurtc * #te * * #_ #Hte >Hte // whd in ⊢ (%→?); * * #_ #Htbdst #Htbelse %
[ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
[ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
| >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Htbelse @sym_not_eq // ]
| >Hcurtc in Hcurta_src; #H destruct (H) cases (is_endc s) in Hcend;
normalize #H destruct (H) // ]
- |#ls #x #xs #ci #rs #ls0 #cj #rs0 #Htasrc_mid #Htadst_mid #Hcicj
- lapply (Hcomp2 … Htasrc_mid Htadst_mid Hcicj) -Hcomp2 #Hcomp2
+ |#ls #x #xs #ci #rs #ls0 #cj #rs0 #Htasrc_mid #Htadst_mid #Hcicj #Hnotendc
+ lapply (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc (or_intror ?? Hcicj))
+ -Hcomp2 #Hcomp2
cases Htb #td * * #Htd #_ >Htasrc_mid in Hcurta_src; normalize in ⊢ (%→?);
#H destruct (H)
>(Htd ls ci (reverse ? xs) rs s ??? ls0 cj (reverse ? xs) s rs0 (refl ??)) //
whd in ⊢ (%→?); #Hout >Hout >Htb whd
lapply (current_to_midtape sig (nth src ? intape (niltape ?)))
cases (current … (nth src ? intape (niltape ?))) in Hcomp1;
- [#Hcomp1 #_ %1 % [%1 %2 // | @Hcomp1 %1 %2 %]
+ [#Hcomp1 #_ %1 % [%1 %2 // | @Hcomp1 %2 %1 %2 %]
|#c_src lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
cases (current … (nth dst ? intape (niltape ?)))
- [#_ #Hcomp1 #_ %1 % [%2 % | @Hcomp1 %2 %]
+ [#_ #Hcomp1 #_ %1 % [%2 % | @Hcomp1 %2 % % % #H destruct (H)]
|#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 #_
+ #ls_dst * #rs_dst #Hmid_dst #Hcomp1
#Hmid_src cases (Hmid_src c_src (refl …)) -Hmid_src
- #ls_src * #rs_src #Hmid_src %2
- cases (comp_list … rs_src rs_dst) #xs * #rsi * #rsj * *
- #Hrs_src #Hrs_dst #Hneq
- %{ls_src} %{ls_dst} %{rsi} %{rsj} %{c_src} %{xs}
- #rsi0 #rsj0 #end #c #Hend #Hc_dst
- >Hrs_src in Hmid_src; >Hend #Hmid_src
- >Hrs_dst in Hmid_dst; >Hc_dst <(\P Hceq) #Hmid_dst
- lapply(Hcomp2 … Hmid_src Hmid_dst ?)
- [@(Hneq … Hend Hc_dst)]
- -Hcomp2 #Hcomp2 <Hcomp2
- % // % [ %
- [>Hcomp2 in Hc; >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec // #H lapply (H ? (refl …))
- cases (is_endc end) normalize //
- |@Hmid_src]
- |@Hmid_dst]
+ #ls_src * #rs_src #Hmid_src
+ cases (true_or_false (is_endc c_src)) #Hc_src
+ [ % % [ % % %{c_src} % // | @Hcomp1 % %{c_src} % // ]
+ | %2 cases (comp_list … rs_src rs_dst is_endc) #xs * #rsi * #rsj * * *
+ #Hrs_src #Hrs_dst #Hnotendc #Hneq
+ %{ls_src} %{ls_dst} %{rsi} %{rsj} %{c_src} %{xs}
+ #rsi0 #rsj0 #end #c #Hend #Hc_dst
+ >Hrs_src in Hmid_src; >Hend #Hmid_src
+ >Hrs_dst in Hmid_dst; >Hc_dst <(\P Hceq) #Hmid_dst
+ cut (is_endc end = true ∨ end ≠ c)
+ [cases (Hneq … Hend) /2/ -Hneq #Hneq %2 @(Hneq … Hc_dst) ] #Hneq
+ lapply (Hcomp2 … Hmid_src Hmid_dst ? Hneq)
+ [#c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
+ [ >(\P Hc0) //
+ | @Hnotendc // ]
+ ]
+ -Hcomp2 #Hcomp2 <Hcomp2
+ % // % [ %
+ [>Hcomp2 in Hc; >nth_change_vec_neq [|@sym_not_eq //]
+ >nth_change_vec // #H lapply (H ? (refl …))
+ cases (is_endc end) [|normalize #H destruct (H) ]
+ #_ % // #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
+ [ >(\P Hc0) // | @Hnotendc // ]
+ |@Hmid_src]
+ |@Hmid_dst] ]
|#_ #Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls * #rs #Hsrc
%1 %
[% % %{c_src} % // lapply (Hc c_src) -Hc >Hcomp1
- [| % % @(not_to_not ??? (\Pf Hceq)) #H destruct (H) // ]
+ [| %2 % % @(not_to_not ??? (\Pf Hceq)) #H destruct (H) // ]
cases (is_endc c_src) //
>Hsrc #Hc lapply (Hc (refl ??)) normalize #H destruct (H)
- |@Hcomp1 %1 %1 @(not_to_not ??? (\Pf Hceq)) #H destruct (H) //
+ |@Hcomp1 %2 %1 %1 @(not_to_not ??? (\Pf Hceq)) #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)
(inr ?? (inr ?? (inl … (inr ?? start_nop)))).
+(*
definition R_match_m ≝
λi,j,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
+ ∀ls,x,rs,ls0,x0,rs0.
+ nth i ? int (niltape ?) = midtape sig ls x rs →
+ nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
+
+ ,xs,ci,rs,ls0,x0,rs0.
+ is_startc x = true → is_endc ci = true →
+ (∀c0.c0 ∈ (x::xs) = true → is_endc c0 = false) →
+ nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
+ nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
+
(((∃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.
is_startc x = true → is_endc ci = true →
+ (∀c0.c0 ∈ (x::xs) = true → is_endc c0 = false) →
nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
∃l,cj,l1.x0::rs0 = l@x::xs@cj::l1 ∧
outt = change_vec ??
(change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
(midtape sig ((reverse ? (l@x::xs))@ls0) cj l1) j).
+
+lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
+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) //
+-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
+[ #tc whd in ⊢ (%→?); *
+ [ * * [ *
+ [ * #cur_src * #H1 #H2 #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #_ #Hnotendc #Hnthsrc
+ @False_ind >Hnthsrc in H1;normalize
+ #H1 destruct (H1) >(Hnotendc ? (memb_hd …)) in H2; #H2 destruct (H2)
+ ]
+ | #Hci #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hstart #Hend_ci #Hnotend
+ #Hnthi >Hnthi in Hci; normalize in ⊢ (%→?); #H destruct (H) ] ]
+ | #Hcj #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #_ #_ #_ #Hnthj >Hnthj in Hcj;
+ normalize in ⊢ (%→?); #H destruct (H) ] ]
+ | * #ls * #ls0 * #rs * #rs0 * #x * #xs #Houtc %
+ [ Houtc ?? x x (refl ??) (refl ??))
+ #Hcases
+ cut (∃end,rsi.rs = end::rsi ∧ nth src ? tc (niltape ?) = midtape ? ls x (xs@rs))
+ [ cases (nth src ? tc (niltape ?)) in
+
+
+lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
+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) //
+-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
+[ #tc whd in ⊢ (%→?); *
+ [ * * [ *
+ [ * #cur_src * #H1 #H2 #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #_ #Hnthi #Hnthj
+ >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H %
+ ]
+ | #Hci #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi >Hnthi in Hci;
+ normalize in ⊢ (%→?); #H destruct (H) ] ]
+ | #Hcj #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #Hnthj >Hnthj in Hcj;
+ normalize in ⊢ (%→?); #H destruct (H) ] ]
+
+
+
+[ #tc whd in ⊢ (%→?); * * [ *
+
+*)
+
+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.
+ is_startc x = true → is_endc ci = true →
+ (∀c0.c0 ∈ (x::xs) = true → is_endc c0 = false) →
+ nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
+ nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
+ (∃x1. is_endc x1 = false ∧ current ? (nth i ? outt (niltape ?)) = Some ? x1) ∨
+ (∃l,cj,l1.x0::rs0 = l@x::xs@cj::l1 ∧
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
+ (midtape sig ((reverse ? (l@x::xs))@ls0) cj l1) j)).
+
+lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
+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) //
+-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
+[ #tc whd in ⊢ (%→?); * #HR1 #HR2 % [ @HR1 ]
+ #ls #x #xs #ci #rs #ls0 #x0 #rs0 #Hstartc #Hendc #Hnotendc #Hsrctc #Hdsttc
+ cases (comp_list ? (x::xs@ci::rs) (x0::rs0) is_endc)
+ #l0 * #l1 * #l2 * * * #Heqsrc #Heqdst #Hnotendsrc #Hor
+ cut (∃x1,l1'.l1 = x1::l1') [@daemon] * #x1 * #l1' #Hl1
+ cases (Hor ?? Hl1) -Hor
+ [
+ cases HR2 -HR2 #HR2 [% @HR2]
+ |cut (is_endc x1 = false) [@daemon] #Hx1
+
+
+ [ * * [ *
+ [ * #cur_src * #H1 #H2 #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #_ #Hnotendc #Hnthsrc
+ @False_ind >Hnthsrc in H1;normalize
+ #H1 destruct (H1) >(Hnotendc ? (memb_hd …)) in H2; #H2 destruct (H2)
+ ]
+ | #Hci #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hstart #Hend_ci #Hnotend
+ #Hnthi >Hnthi in Hci; normalize in ⊢ (%→?); #H destruct (H) ] ]
+ | #Hcj #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #_ #_ #_ #Hnthj >Hnthj in Hcj;
+ normalize in ⊢ (%→?); #H destruct (H) ] ]
+ | * #ls * #ls0 * #rs * #rs0 * #x * #xs #Houtc %
+ [ Houtc ?? x x (refl ??) (refl ??))
+ #Hcases
+ cut (∃end,rsi.rs = end::rsi ∧ nth src ? tc (niltape ?) = midtape ? ls x (xs@rs))
+ [ cases (nth src ? tc (niltape ?)) in Hcases;
+ [
+
lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
src ≠ dst → src < S n → dst < S n →
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