(**************************************************************************)
include "turing/multi_universal/compare.ma".
+include "turing/multi_universal/par_test.ma".
+
definition Rtc_multi_true ≝
λalpha,test,n,i.λt1,t2:Vector ? (S n).
| @sym_eq @Hnth_j @sym_not_eq // ] ] ]
qed.
+definition Rm_test_null_true ≝
+ λalpha,n,i.λt1,t2:Vector ? (S n).
+ current alpha (nth i ? t1 (niltape ?)) ≠ None ? ∧ t2 = t1.
+
+definition Rm_test_null_false ≝
+ λalpha,n,i.λt1,t2:Vector ? (S n).
+ current alpha (nth i ? t1 (niltape ?)) = None ? ∧ t2 = t1.
+
+lemma sem_test_null_multi : ∀alpha,n,i.i ≤ n →
+ inject_TM ? (test_null ?) n i ⊨
+ [ tc_true : Rm_test_null_true alpha n i, Rm_test_null_false alpha n i ].
+#alpha #n #i #Hin #int
+cases (acc_sem_inject … Hin (sem_test_null alpha) int)
+#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
+[ @Hloop
+| #Hqtrue lapply (Htrue Hqtrue) * * #Hcur #Hnth_i #Hnth_j % //
+ @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) #Hi0i
+ [ >Hi0i @sym_eq @Hnth_i | @sym_eq @Hnth_j @sym_not_eq // ] ]
+| #Hqfalse lapply (Hfalse Hqfalse) * * #Hcur #Hnth_i #Hnth_j %
+ [ @Hcur
+ | @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) //
+ #Hi0i @sym_eq @Hnth_j @sym_not_eq // ] ]
+qed.
+
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.
-
+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 ?))].
definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc.
compare src dst sig n is_endc ·
- (ifTM ?? (inject_TM ? (test_char ? (λa.is_endc a == false)) n src)
- (ifTM ?? (inject_TM ? (test_null ?) n src)
- (single_finalTM ??
- (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst)))
- (nop …) tc_true)
+ (ifTM ?? (partest sig n (match_test src dst sig ? is_endc))
+ (single_finalTM ??
+ (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst)))
(nop …)
- tc_true).
+ partest1).
definition R_match_step_false ≝
λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
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) ∨
- (∃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,rs,ls0,rs0.
- nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
- nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
- (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
- (∀cj,rs1.rs0 = cj::rs1 → 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 ??
(change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) src)
- (mk_tape sig (reverse ? xs@x::ls0) (None ?) [ ]) dst)).
+ (mk_tape sig (reverse ? xs@x::ls0) (None ?) [ ]) dst)). *)
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 ⊨
- [ inr ?? (inr ?? (inl … (inr ?? (inr ?? start_nop)))) :
+ [ 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
-(* test_null versione multi? *)
@(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))
- (acc_sem_if ? n … (sem_test_null sig (λa.is_endc a == false) n src (le_S_S_to_le … Hsrc))
-
- sem_seq …
+ (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ? is_endc))
+ (sem_seq …
(sem_parmoveL ???? is_startc Hneq Hsrc Hdst)
(sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
(sem_nop …)))
-[#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * * #c * #Hcurtc #Hcend #Htd >Htd -Htd
- #Htb #s #Hcurta_src #Hstart #Hnotstart % [ %
- [#Hdst_none @daemon
- | #s1 #Hcurta_dst #Hneqss1
- lapply Htb lapply Hcurtc -Htb -Hcurtc >(?:tc=ta)
- [|@Hcomp1 %2 % % >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) % ]
- #Hcurtc * #te * * #_ #Hte >Hte [2: %1 %1 %{s} % //]
- 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)
- #ls * #rs #Hta_mid >(Htbdst … Hta_mid) >Hta_mid cases rs //
- | >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 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc
- cases rs00 in Htadst_mid;
- [(* case rs empty *) #Htadst_mid % [ #cj #rs1 #H destruct (H) ]
- #_ cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) -Hcomp2
- [2: * #x0 * #rs1 * #H destruct (H) ]
- * #_ #Htc cases Htb #td * * #_ #Htd >Htasrc_mid in Hcurta_src;
- normalize in ⊢ (%→?); #H destruct (H)
- >Htd [2: %2 >Htc >nth_change_vec // cases (reverse sig ?) //]
- >Htc * * >nth_change_vec // #Htbdst #_ #Htbelse
- @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
- [ >Hidst >nth_change_vec // <Htbdst // cases (reverse sig ?) //
- |@sym_eq @Htbelse @sym_not_eq //
- ]
- |#cj0 #rs0 #Htadst_mid % [| #H destruct (H) ]
- #cj #rs1 #H destruct (H) #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 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 ??)) //
- [| >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 >nth_change_vec_neq [|@sym_not_eq // ] @nth_change_vec // ]
- * * #_ #Htbdst #Htbelse %
- [ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
- [ >Hidst >nth_change_vec // >Htadst_mid >(Htbdst ls0 s (xs@cj'::rs0'))
- [ cases xs //
- | >nth_change_vec // ]
- | >nth_change_vec_neq [|@sym_not_eq //]
- <Htbelse [|@sym_not_eq // ]
- >nth_change_vec_neq [|@sym_not_eq //]
- cases (decidable_eq_nat i src) #Hisrc
- [ >Hisrc >nth_change_vec // >Htasrc_mid //
- | >nth_change_vec_neq [|@sym_not_eq //]
- <(Htbelse i) [|@sym_not_eq // ]
- >Htc >nth_change_vec_neq [|@sym_not_eq // ]
- >nth_change_vec_neq [|@sym_not_eq // ] //
+[#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 //
]
- ]
- | >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec // whd in ⊢ (??%?→?);
- #H destruct (H) cases (is_endc c) in Hcend;
- normalize #H destruct (H) // ]
+ | >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
lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
cases (current … (nth dst ? intape (niltape ?))) in Hcomp1;
- [#Hcomp1 #_ %1 % [% | @Hcomp1 %2 %2 % ]
+ [#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 %2
+ #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 %{ls_dst} %{rsj} >Hrs_dst in Hmid_dst; #Hmid_dst
+ #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 #Hta %
- [ >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
- #Hc lapply (Hc ? (refl ??)) #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 >Hmid_dst >Hxsxs1 % ]
- | #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0) ]
- | * #cj * #rs2 * #Hrs2 #Hta lapply (Hta ?)
- [ cases (Hneq … Hrs1) /2/ #H %2 @(H ?? Hrs2) ]
- -Hta #Hta >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec // #Hc lapply (Hc ? (refl ??)) #Hendr1
- (* lemmatize this proof *) 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 // ]
- ]
+ [ >(\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 >Hmid_dst >Hxsxs1 % //
- #rsj0 #c #Hcrsj destruct (Hxsxs1 Hrs2 Hcrsj) @eq_f3 //
- @eq_f3 // lapply (append_l2_injective ?????? Hrs_src) //
- #Hendr1 destruct (Hendr1) % ]
+ ]
+ | #Hxsxs1 destruct (Hxsxs1) >Hmid_dst %{ls_dst} %{rsj} % //
+ #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0)
+ lapply (append_l2_injective … Hrs_src) // #Hrs' destruct (Hrs') %
]
]
- (* STOP *)
- |#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; >Hmid_src #Hc lapply (Hc ? (refl …)) -Hc
- >(Hnotend c_src) // normalize #H destruct (H)
+ |#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)
]
]
]
-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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