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 ? (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,rs,rs0,x,xs. ∀rsi,rsj,end,c.
+ (∃ls,ls0,rs,rs0,x,xs.
+ nth src ? int (niltape ?) = midtape sig ls x (xs@rs) ∧ is_endc x = false ∧
+ nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
+ ∀rsi,rsj,end,c.
rs = end::rsi → rs0 = c::rsj →
(∀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) ∧
+ nth dst ? int (niltape ?) = midtape sig ls0 x (xs@c::rsj) ∧
outt = change_vec ??
(change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rsi) src)
- (midtape sig (reverse ? xs@x::ls0) c rsj) dst.
-
-*)
+ (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).
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 ].
-@daemon
-(*
+ 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_test_char_multi sig (λa.is_endc a == false) n src (le_S_S_to_le … Hsrc))
[ % % [ % % %{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}
+ %{ls_src} %{ls_dst} %{rsi} %{rsj} %{c_src} %{xs} %
+ [% [% // <Hrs_src //|<Hrs_dst >(\P Hceq) // ]]
#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
| @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] ]
+ % // % [
+ >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_dst]
+ ]
|#_ #Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls * #rs #Hsrc
%1 %
[% % %{c_src} % // lapply (Hc c_src) -Hc >Hcomp1
]
]
]
-*)
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.
+ (∀ls,x,xs,ci,rs,ls0,x0,rs0.
+ (∀x. is_startc x ≠ is_endc x) →
is_startc x = true → is_endc ci = true →
- (∀c0.c0 ∈ (x::xs) = true → is_endc c0 = false) →
+ (∀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,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
+ (∃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.
lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
src ≠ dst → src < S n → dst < S n →
#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 ⊢ (%→?); *
+[ #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)
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hdiff #Hstartc #Hendc #Hnotend #Hnthi
+ @False_ind
+ >Hnthi in H1; whd in ⊢ (??%?→?); #H destruct (H) cases (Hdiff cur_src)
+ #Habs @Habs //
]
| #Hci #Houtc %
[ #_ @Houtc
- | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hstart #Hend_ci #Hnotend
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hdiff #Hstartc #Hendc #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 →
- 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 ⊢ (%→?); * * [ *
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hdiff #Hstartc #Hendc #_ #_ #Hnthj >Hnthj in Hcj;
+ normalize in ⊢ (%→?); #H destruct (H) ]
+ ]
+ |* #ls * #ls0 * #rs * #rs0 * #x0 * #xs * * * #Hsrc #Hx0 #Hdst #H %
+ [>Hsrc *
+ [* [* #x * whd in ⊢ (??%?→?); #Habs destruct (Habs) >Hx0 #Habs destruct (Habs)
+ |whd in ⊢ (??%?→?); #Habs destruct (Habs) ]
+ |>Hdst whd in ⊢ (??%?→?); #Habs destruct (Habs) ]
+ |#ls1 #x1 #xs1 #ci #rsi #ls2 #x2 #rs2
+ #Hdiff #Hstart #Hend #Hnotend
+ >Hsrc #Hsrc1 destruct (Hsrc1) >Hdst #Hdst1 destruct (Hdst1)
+ %1 %{[ ]} %{rs0} normalize in ⊢ (%→?); #Heq #cj #l2 #Hl1
+ cut (xs=xs1)
+ [@(append_l1_injective_r … rs0 rs0 (refl …)) @(cons_injective_r …Heq)]
+ #eqxs <eqxs
+ whd in match (append ? [ ] (x2::xs)); >reverse_cons >associative_append
+ normalize in match (append ? [x2] ls2);
+ cases (H rsi l2 ci cj ? Hl1)
+ [* #_ #_ #H3 @H3
+ |>eqxs in e0; #e0 @(append_l2_injective … e0) //
+ ]
+ ]
+ ]
+|
+
+
+ cases (comp_list ? (x1::xs1@ci::rsi) (x2::rs2) is_endc)
+ #l * #tl1 * #tl2 * * * #H1 #H2 #H3 #H4
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