else nop
*)
-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)
- (single_finalTM ??
- (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst)))
- (nop …)
- tc_true).
-
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.
λalpha,test,n,i.λt1,t2:Vector ? (S n).
(∀c. current alpha (nth i ? t1 (niltape ?)) = Some ? c → test c = false) ∧ t2 = t1.
-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) ∨
- (∃ls0,rs0.
- nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
- ∀rsj,end,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_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 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.
- 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 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).
-*)
-
-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 →
- 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).
-
lemma sem_test_char_multi :
∀alpha,test,n,i.i ≤ n →
inject_TM ? (test_char ? test) n i ⊨
axiom daemon : ∀X:Prop.X.
+(*
+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) ∨
+ (∃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 ? → outt = int) ∧
+ (∀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)) ∧
+ (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)).
+
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 ⊨
(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 %
- [ #s1 #Hcurta_dst #Hneqss1
+ #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 // whd in ⊢ (%→?); * * #_ #Htbdst #Htbelse %
+ #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 #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 ??)) //
- [| >Hcomp2 >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 //
+ ]
+ |#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 // ] //
+ ]
+ ]
+ | >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) // ]
]
- | >Hcomp2 >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 //]
- (* STOP. *)
- 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 // ]
- >Hcomp2 >nth_change_vec_neq [|@sym_not_eq // ]
- >nth_change_vec_neq [|@sym_not_eq // ] //
- ]
- ]
- | >Hcomp2 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) // ]
]
|#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb
whd in ⊢ (%→?); #Hout >Hout >Htb whd
[#_ #Hmid_dst cases (Hmid_dst c_dst (refl …)) -Hmid_dst
#ls_dst * #rs_dst #Hmid_dst %2
cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * *
- #Hrs_src #Hrs_dst #Hnotendc #Hneq
- %{ls_dst} %{rsj} %
- [<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
- 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_dst]
- ]
- |#_ #Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls * #rs #Hsrc
- %1 %
- [% % %{c_src} % // lapply (Hc c_src) -Hc >Hcomp1
- [| %2 % % @(not_to_not ??? (\Pf Hceq)) #H destruct (H) // ]
- cases (is_endc c_src) //
- >Hsrc #Hc lapply (Hc (refl ??)) normalize #H destruct (H)
- |@Hcomp1 %2 %1 %1 @(not_to_not ??? (\Pf Hceq)) #H destruct (H) //
- ]
- ]
- ]
- ]
-qed.
+ #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq %{ls_dst} %{rsj} >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 // ]
+ ]
+ ]
+ | #Hxsxs1 >Hmid_dst >Hxsxs1 % //
+ #rsj0 #c #Hcrsj destruct (Hxsxs1 Hrs2 Hcrsj) @eq_f3 //
+ @eq_f3 // lapply (append_l2_injective ?????? Hrs_src) //
+ #Hendr1 destruct (Hendr1) % ]
+ ]
+ ]
+ (* 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)
+ ]
+ ]
+]
+qed.
+*)
-#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb
- 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 %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 % % % #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 #Hcomp1
- #Hmid_src cases (Hmid_src c_src (refl …)) -Hmid_src
- #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} %
- [% [% // <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
- 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_dst]
- ]
- |#_ #Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls * #rs #Hsrc
- %1 %
- [% % %{c_src} % // lapply (Hc c_src) -Hc >Hcomp1
- [| %2 % % @(not_to_not ??? (\Pf Hceq)) #H destruct (H) // ]
- cases (is_endc c_src) //
- >Hsrc #Hc lapply (Hc (refl ??)) normalize #H destruct (H)
- |@Hcomp1 %2 %1 %1 @(not_to_not ??? (\Pf Hceq)) #H destruct (H) //
- ]
+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)
+ (nop …)
+ tc_true).
+
+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) ∨
+ (∃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
+ (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)) ∧
+ (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)).
+
+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)))) :
+ 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 …
+ (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 // ] //
+ ]
+ ]
+ | >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) // ]
]
- ]
-qed.
+ ]
+|#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 %2
+ 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
+ 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 // ]
+ ]
+ ]
+ | #Hxsxs1 >Hmid_dst >Hxsxs1 % //
+ #rsj0 #c #Hcrsj destruct (Hxsxs1 Hrs2 Hcrsj) @eq_f3 //
+ @eq_f3 // lapply (append_l2_injective ?????? Hrs_src) //
+ #Hendr1 destruct (Hendr1) % ]
+ ]
+ ]
+ (* 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)
+ ]
+ ]
+]
+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 ≝
+ 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 →
+ (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) ∨
(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.
#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 #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 #Hdiff #Hstartc #Hendc #Hnotend
- #Hnthi >Hnthi in Hci; normalize in ⊢ (%→?); #H destruct (H) ] ]
- | #Hcj #Houtc %
- [ #_ @Houtc
- | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hdiff #Hstartc #Hendc #_ #_ #Hnthj >Hnthj in Hcj;
- normalize in ⊢ (%→?); #H destruct (H) ]
+[ #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)
]
- |* #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) //
- ]
+ |* #ls0 * #rs0 * #Hmid_dst #HFalse %
+ [ >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)
+ ]
+ ]
+|#ta #tb #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
+ #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend
+ 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 … ) 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 ⊢ (%→?);
+ #H destruct (H)
+ ]
+ | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ]
+ #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
+ [2: #z #membz @daemon (*aggiungere l'ipotesi*)]
+ cases (true_or_false (x==c)) #eqx
+ [ #_ #Htrue cases (comp_list ? (xs@end::rs) rs0 is_endc)
+ #x1 * #tl1 * #tl2 * * * #Hxs #Hrs0 #Hnotendx1
+ cases tl1 in Hxs;
+ [>append_nil #Hx1 @daemon (* absurd by Hx1 e notendx1 *)]
+ #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 //
+
+ >Hrs0 in Hmid_dst; #Hmid_dst
+ cases(Htrue ???????? Hmid_dst) -Htrue #Htb #Hendx
+ whd in IH;
+ cases(IH ls x xs end rs ? Hstart Hnotend Hend)
+ [* #H1 #H2 >Htb in H1; >nth_change_vec //
+ >Hmid_dst cases rs0 [2: #a #tl normalize in ⊢ (%→?); #H destruct (H)]
+ #_ %2 @daemon (* si dimostra *)
+ |@daemon
+ |>Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src
+ ]
]
]
-|#tc #td #te #Hd #Hstar #IH #He lapply (IH He) -IH *
- #IH1 #IH2 % [@IH1]
-
-
- cases (comp_list ? (x1::xs1@ci::rsi) (x2::rs2) is_endc)
- #l * #tl1 * #tl2 * * * #H1 #H2 #H3 #H4
+]
+qed.
projects / helm.git / blobdiff