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) ∨
- (current sig (nth dst (tape sig) outt (niltape sig)) = None ?) ∨
- (∃ls0,rs0.
- nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
- ∀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).
+ (∃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 ?) ∨
+ (∃ls0,rs0.
+ nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
+ ∀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).
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) →
current sig (nth dst (tape sig) int (niltape sig)) ≠ None ? ∧
- (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 → s ≠ s1 →
- outt = change_vec ?? int
+ (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)).
+ (∀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 ??
(sem_nop …)))
[#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd >Htd -Htd
* #te * #Hte #Htb whd
- #s #Hcurta_src #Hstart #Hnotstart % [ %
- [ @daemon
- | #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 //
- ]
- | >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) %
+ #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 //
+ ]
+ | >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) %
+ ]
+ ]
]
- ]
|#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb
whd in ⊢ (%→?); #Hout >Hout >Htb whd
#ls #c_src #xs #end #rs #Hmid_src #Hnotend #Hend
[ #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
[ >(\P Hc0) @Hnotend @memb_hd | @Hnotendxs1 //] ]
*
- [ * #Hrsj >Hrsj #Hta % %2 >Hta >nth_change_vec // cases (reverse ? xs1) //
+ [ * #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 ⊢ (??%?→?);
]
]
]
-qed.
+qed.
definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc.
whileTM … (match_step src dst sig n is_startc is_endc)
-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
[ #tc #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend
cases (Hfalse … Hmid_src Hnotend Hend) -Hfalse
- [(* current dest = None *) * #Hcur_dst #Houtc %
- [#_ >Houtc //
- |#Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
- normalize in ⊢ (%→?); #H destruct (H)
+ [(* current dest = None *) *
+ [ * #Hcur_dst #Houtc %
+ [#_ >Houtc //
+ |#Hstart #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
+ [ % %{[ ]} %{[ ]} % [ >append_nil >append_nil %]
+ #cj #ls2 #H destruct (H)
+ | #x2 #xs2 %2 #l #l1 % #Habs lapply (eq_f ?? (length ?) ?? Habs)
+ >length_append whd in ⊢ (??%(??%)→?); >length_append
+ >length_append normalize >commutative_plus whd in ⊢ (???%→?);
+ #H destruct (H) lapply e0 >(plus_n_O (|rs1|)) in ⊢ (??%?→?);
+ >associative_plus >associative_plus
+ #e1 lapply (injective_plus_r ??? e1) whd in ⊢ (???%→?);
+ #e2 destruct (e2)
+ ]
+ ]
]
|* #ls0 * #rs0 * #Hmid_dst #HFalse %
[ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
[#Hmid_dst %
[#_ whd in Htrue; >Hmid_src in Htrue; #Htrue
- cases (Htrue x (refl … ) Hstart ?) -Htrue [2: @daemon]
- * #Htb #_ #_ >Htb in IH; // #IH
- cases (IH ls x xs end rs Hmid_src Hstart Hnotend Hend)
- #Hcur_outc #_ @Hcur_outc //
- |#ls0 #x0 #rs0 #Hmid_dst2 >Hmid_dst2 in Hmid_dst; normalize in ⊢ (%→?);
+ 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)
]
| #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ]
- #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcurta_dst; normalize in ⊢ (%→?);
+ #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 …) Hstart ?) -Htrue
+ cases (Htrue x (refl …)) -Htrue #_ #Htrue cases (Htrue Hstart ?) -Htrue
[2: #z #membz @daemon (*aggiungere l'ipotesi*)]
cases (true_or_false (x==c)) #eqx
[ #_ #Htrue cases (comp_list ? (xs@end::rs) rs0 is_endc)
#ci -tl1 #tl1 #Hxs #H cases (H … (refl … ))
[(* this is absurd, since Htrue conlcudes is_endc ci =false *)
#Hend_ci @daemon (* lapply(Htrue … (refl …)) -Htrue *)
- |#Hcomp lapply (Htrue ls x x1 ci tl1 ls0 tl2 ???)
- [ #c0 #Hc0 cases (orb_true_l … Hc0) #Hc0
- [ @Hnotend >(\P Hc0) @memb_hd
- | @Hnotendx1 // ]
- | >Hmid_dst >Hrs0 >(\P eqx) %
- | >Hxs %
- | * cases tl2 in Hrs0;
- [ >append_nil #Hrs0 #_ #Htb whd in IH;
- lapply (IH ls x x1 ci tl1 ? Hstart ??)
- [
- |
- | >Htb // >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
+ |cases tl2 in Hrs0;
+ [
+ | #cj #tl2' #Hrs0 #Hcomp lapply (Htrue ls x 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 >(\P eqx) %
+ | >Hxs %
+ | * #Htb >Htb #Hendci %2 >Hrs0 >Hxs
+ cases (IH ls x xs end rs ? Hnotend Hend) [|
+ STOP
+
+
+
+ >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
>Hrs0 in Hmid_dst; #Hmid_dst
cases(Htrue ???????? Hmid_dst) -Htrue #Htb #Hendx
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