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.
+lemma sem_test_char_multi :
+ ∀alpha,test,n,i.i ≤ n →
+ inject_TM ? (test_char ? test) n i ⊨
+ [ tc_true : Rtc_multi_true alpha test n i, Rtc_multi_false alpha test n i ].
+#alpha #test #n #i #Hin #int
+cases (acc_sem_inject … Hin (sem_test_char alpha test) int)
+#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
+[ @Hloop
+| #Hqtrue lapply (Htrue Hqtrue) * * * #c *
+ #Hcur #Htestc #Hnth_i #Hnth_j %
+ [ %{c} % //
+ | @(eq_vec … (niltape ?)) #i0 #Hi0
+ cases (decidable_eq_nat i0 i) #Hi0i
+ [ >Hi0i @Hnth_i
+ | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
+| #Hqfalse lapply (Hfalse Hqfalse) * * #Htestc #Hnth_i #Hnth_j %
+ [ @Htestc
+ | @(eq_vec … (niltape ?)) #i0 #Hi0
+ cases (decidable_eq_nat i0 i) #Hi0i
+ [ >Hi0i @Hnth_i
+ | @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 R_match_step_false ≝
λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
∀ls,x,xs,end,rs.
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).
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) →
- (â\88\83cj,rs1.rs0 = cj::rs1 → ci ≠ cj →
+ (â\88\80cj,rs1.rs0 = cj::rs1 → ci ≠ cj →
(outt = change_vec ?? int
- (tape_move â\80¦ (nth dst ? int (niltape ?)) (Some ? â\8c©x,Râ\8cª)) dst â\88§ is_endc ci = false)) â\88¨
+ (tape_move â\80¦ (nth dst ? int (niltape ?)) (Some ? â\8c©x,Râ\8cª)) dst â\88§ is_endc ci = false)) â\88§
(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_test_char_multi :
- ∀alpha,test,n,i.i ≤ n →
- inject_TM ? (test_char ? test) n i ⊨
- [ tc_true : Rtc_multi_true alpha test n i, Rtc_multi_false alpha test n i ].
-#alpha #test #n #i #Hin #int
-cases (acc_sem_inject … Hin (sem_test_char alpha test) int)
-#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
-[ @Hloop
-| #Hqtrue lapply (Htrue Hqtrue) * * * #c *
- #Hcur #Htestc #Hnth_i #Hnth_j %
- [ %{c} % //
- | @(eq_vec … (niltape ?)) #i0 #Hi0
- cases (decidable_eq_nat i0 i) #Hi0i
- [ >Hi0i @Hnth_i
- | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
-| #Hqfalse lapply (Hfalse Hqfalse) * * #Htestc #Hnth_i #Hnth_j %
- [ @Htestc
- | @(eq_vec … (niltape ?)) #i0 #Hi0
- cases (decidable_eq_nat i0 i) #Hi0i
- [ >Hi0i @Hnth_i
- | @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.
-
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 ⊨
]
|#ls #x #xs #ci #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc
cases rs00 in Htadst_mid;
- [(* case rs empty *) #Htadst_mid %2 #_
- cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) -Hcomp2
+ [(* 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)
[ >Hidst >nth_change_vec // <Htbdst // cases (reverse sig ?) //
|@sym_eq @Htbelse @sym_not_eq //
]
- |#cj #rs0 #Htadst_mid % %{cj} %{rs0} #_ #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
+ |#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 // ] //
- ]
+ [ >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) // ]
+ ]
+ ]
+|#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_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) // ]
+ >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
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) →
- is_startc x = true →
(∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
- ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) →
- current sig (nth dst (tape sig) outt (niltape sig)) = None ?)
- (* outt = int) *) ∧
+ (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 ∧
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).
+ ∀l,l1.x0::rs0 ≠ l@x::xs@l1)).
(*
definition R_match_m ≝
#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 #Hfalse #ls #x #xs #end #rs #Hmid_src #Hstart #Hnotend #Hend
+[ #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 //
- |#ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
+ |#Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
normalize in ⊢ (%→?); #H destruct (H)
]
|* #ls0 * #rs0 * #Hmid_dst #HFalse %
[ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
- |#ls1 #x1 #rs1 >Hmid_dst #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 #Hstart #Hnotend #Hend
+ #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)
- [#H @H //
- |
-
- |#cur_dst #Hcur_dst %2 #ls0 #x0 #rs0 #Hmid_dst
- whd in Htrue; >Hmid_src in Htrue; #Htrue
- cases (Htrue x (refl …) Hstart ?) -Htrue
+ [#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==cur_dst)) #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 Hxs e notendx1 *)]
- #ci -tl1 #tl1 #Hxs #H cases (H … (refl … ))
+ 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 *)
- |#Htrue #_ cases(Htrue cur_dst Hcur_dst (\Pf eqx)) -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
- ]
+ #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
+ ]
]
]
]
projects / helm.git / commitdiff