∀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) ∨
+ ((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.
∀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,cj,rs0.
+ (∀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@cj::rs0) → ci ≠ cj →
+ nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
(∀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).
-
+ (∃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_test_char_multi :
∀alpha,test,n,i.i ≤ n →
inject_TM ? (test_char ? test) n i ⊨
(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
- 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 //
+ ]
+ |#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
+ [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 //
+ ]
+ |#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
+ 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 i) [|@sym_not_eq // ]
- >Htc >nth_change_vec_neq [|@sym_not_eq // ]
- >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) // ]
+ | >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
]
]
(* STOP *)
- | #Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls0 * #rs0 #Hsrc %2
- %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) //
- ]
- ]
+ |#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) //
- ]
- ]
- ]
- ]
-qed.
+]
+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).
+ ∀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) *) ∧
+ (∀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 #Hstart #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;
+ 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)
+ %1 %{[ ]} %{rs0} % [%] #cj #l2 #Hnotnil
+ >reverse_cons >associative_append @(HFalse ?? Hnotnil)
]
- |* #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) //
+ ]
+|#ta #tb #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
+ #ls #x #xs #end #rs #Hmid_src #Hstart #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
+ [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 … ))
+ [(* 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
]
]
]
-|#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.
outt = change_vec ??
(change_vec ?? int (midtape sig ls sep (reverse ? xs@x::rs)) src)
(midtape sig ls0 c (reverse ? target@x0::rs0)) dst) ∧
- (∀s.current ? (nth src ? int (niltape ?)) = Some ? s → is_sep s = true →
- outt = int).
+ (((∃s.current ? (nth src ? int (niltape ?)) = Some ? s ∧ is_sep s = true) ∨
+ current ? (nth src ? int (niltape ?)) = None ? ∨
+ current ? (nth dst ? int (niltape ?)) = None ?) →
+ outt = int).
lemma wsem_parmoveL : ∀src,dst,sig,n,is_sep.src ≠ dst → src < S n → dst < S n →
parmove src dst sig n L is_sep ⊫ R_parmoveL src dst sig n is_sep.
#src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
lapply (sem_while … (sem_parmove_step src dst sig n L is_sep Hneq Hsrc Hdst) … Hloop) //
-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ #tc whd in ⊢ (%→?); * * [ *
- [ * #x * #Hx #Hsep #Houtc %
- [ #ls #x0 #xs #rs #sep #Hsrctc #Hnosep >Hsrctc in Hx; normalize in ⊢ (%→?);
- #Hx0 destruct (Hx0) lapply (Hnosep ? (memb_hd …)) >Hsep
- #Hfalse destruct (Hfalse)
- | #s #Hs #Hseps @Houtc ]
- | #Hcur #Houtc %
- [ #ls #x0 #xs #rs #sep #Hsrctc >Hsrctc in Hcur; normalize in ⊢ (%→?);
- #Hcur destruct (Hcur)
- | >Hcur #s #Hs destruct (Hs) ] ]
- | #Hcur #Houtc %
- [ #ls #x0 #xs #rs #sep #Hsrctc #Hnosep #Hsep #ls0 #x1 #target #c #rs0 #Hlen
- #Hdsttc >Hdsttc in Hcur; normalize in ⊢ (%→?); #Hcur destruct (Hcur)
- | #s #Hs #Hseps @Houtc ]
- ]
+[ #tc whd in ⊢ (%→?); * #H #Houtc % [2: #_ @Houtc ] cases H
+ [ * [ * #x * #Hx #Hsep #ls #x0 #xs #rs #sep #Hsrctc #Hnosep >Hsrctc in Hx; normalize in ⊢ (%→?);
+ #Hx0 destruct (Hx0) lapply (Hnosep ? (memb_hd …)) >Hsep
+ #Hfalse destruct (Hfalse)
+ | #Hcur_src #ls #x0 #xs #rs #sep #Hsrctc >Hsrctc in Hcur_src;
+ normalize in ⊢ (%→?); #H destruct (H)]
+ |#Hcur_dst #ls #x0 #xs #rs #sep #Hsrctc #Hnosep #Hsep #ls0 #x1 #target
+ #c #rs0 #Hlen #Hdsttc >Hdsttc in Hcur_dst; normalize in ⊢ (%→?); #H destruct (H)
+ ]
| #tc #td #te * #c0 * #c1 * * * #Hc0 #Hc1 #Hc0nosep #Hd #Hstar #IH #He
lapply (IH He) -IH * #IH1 #IH2 %
[ #ls #x #xs #rs #sep #Hsrc_tc #Hnosep #Hsep #ls0 #x0 #target
#c #rs0 #Hlen #Hdst_tc
- >Hsrc_tc in Hc0; normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
-(* <(change_vec_same … tc src (niltape ?)) in Hd:(???(???(???%??)??));
- <(change_vec_same … tc dst (niltape ?)) in ⊢(???(???(???%??)??)→?); *)
- >Hdst_tc in Hd; >Hsrc_tc
-(* >change_vec_change_vec >change_vec_change_vec
- >(change_vec_commute ?? tc ?? dst src) [|@(sym_not_eq … Hneq)]
- >change_vec_change_vec *) @(list_cases2 … Hlen)
- [ #Hxsnil #Htargetnil >Hxsnil >Htargetnil #Hd >(IH2 … Hsep)
- [ >Hd -Hd @(eq_vec … (niltape ?))
- #i #Hi cases (decidable_eq_nat i src) #Hisrc
- [ >Hisrc >(nth_change_vec_neq … src dst) [|@(sym_not_eq … Hneq)]
- >nth_change_vec //
- >(nth_change_vec_neq … src dst) [|@(sym_not_eq … Hneq)]
- >nth_change_vec //
- | cases (decidable_eq_nat i dst) #Hidst
- [ >Hidst >nth_change_vec // >nth_change_vec //
- >Hdst_tc in Hc1; >Htargetnil
- normalize in ⊢ (%→?); #Hc1 destruct (Hc1) %
- | >nth_change_vec_neq [|@(sym_not_eq … Hidst)]
- >nth_change_vec_neq [|@(sym_not_eq … Hisrc)]
- >nth_change_vec_neq [|@(sym_not_eq … Hidst)]
- >nth_change_vec_neq [|@(sym_not_eq … Hisrc)] % ]
- ]
- | >Hd >nth_change_vec_neq [|@(sym_not_eq … Hneq)]
- >nth_change_vec // ]
+ >Hsrc_tc in Hc0; normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
+ >Hdst_tc in Hd; >Hsrc_tc @(list_cases2 … Hlen)
+ [ #Hxsnil #Htargetnil >Hxsnil >Htargetnil #Hd >IH2
+ [2: %1 %1 %{sep} % // >Hd >nth_change_vec_neq [|@(sym_not_eq … Hneq)]
+ >nth_change_vec //]
+ >Hd -Hd @(eq_vec … (niltape ?))
+ #i #Hi cases (decidable_eq_nat i src) #Hisrc
+ [ >Hisrc >(nth_change_vec_neq … src dst) [|@(sym_not_eq … Hneq)]
+ >nth_change_vec //
+ >(nth_change_vec_neq … src dst) [|@(sym_not_eq … Hneq)]
+ >nth_change_vec //
+ | cases (decidable_eq_nat i dst) #Hidst
+ [ >Hidst >nth_change_vec // >nth_change_vec //
+ >Hdst_tc in Hc1; >Htargetnil
+ normalize in ⊢ (%→?); #Hc1 destruct (Hc1) %
+ | >nth_change_vec_neq [|@(sym_not_eq … Hidst)]
+ >nth_change_vec_neq [|@(sym_not_eq … Hisrc)]
+ >nth_change_vec_neq [|@(sym_not_eq … Hidst)]
+ >nth_change_vec_neq [|@(sym_not_eq … Hisrc)] %
+ ]
+ ]
| #hd1 #hd2 #tl1 #tl2 #Hxs #Htarget >Hxs >Htarget #Hd
>(IH1 ls hd1 tl1 (c0::rs) sep ?? Hsep ls0 hd2 tl2 c (x0::rs0))
[ >Hd >(change_vec_commute … ?? tc ?? src dst) //
| >Hd >nth_change_vec_neq [|@sym_not_eq //]
>nth_change_vec // ]
]
- | #c #Hc #Hsepc >Hc in Hc0; #Hcc0 destruct (Hcc0) >Hc0nosep in Hsepc;
- #H destruct (H)
-] ]
+ | >Hc0 >Hc1 * [* [ * #c * #Hc destruct (Hc) >Hc0nosep]] #Habs destruct (Habs)
+ ] ]
qed.
lemma terminate_parmoveL : ∀src,dst,sig,n,is_sep,t.
lemma sem_parmoveL : ∀src,dst,sig,n,is_sep.
src ≠ dst → src < S n → dst < S n →
parmove src dst sig n L is_sep ⊨ R_parmoveL src dst sig n is_sep.
-#src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst @WRealize_to_Realize /2/
+#src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst @WRealize_to_Realize
+[/2/ | @wsem_parmoveL //]
qed.
\ No newline at end of file
projects / helm.git / commitdiff