(current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
current ? (nth i ? int (niltape ?)) = None ? ∨
current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
- (∀ls,x,xs,ci,rs,ls0,cj,rs0.
+ (∀ls,x,xs,ci,rs,ls0,rs0.
nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
- nth j ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) →
+ nth j ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
(∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
+ (rs0 = [ ] →
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
+ (mk_tape sig (reverse ? xs@x::ls0) (None ?) []) j) ∨
+ ∀cj,rs1.rs0 = cj::rs1 →
(is_endc ci = true ∨ ci ≠ cj) →
outt = change_vec ??
(change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
definition Rtc_multi_false ≝
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 ? (nth src ? int (niltape ?)) ≠ current ? (nth dst ? int (niltape ?)) ∨ *)
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. ∀rsi,rsj,end,c.
+ (∃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 src ? int (niltape ?) = midtape sig ls x (xs@rs) ∧
- nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
+ 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.
+ (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).
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 #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.
+
+#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;
[ % % [ % % %{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}
+ %{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
| @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_src]
- |@Hmid_dst] ]
+ % // % [
+ >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
(((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
current ? (nth i ? int (niltape ?)) = None ? ∨
current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
- (∀ls,x,xs,ci,rs,ls0,x0,rs0.
+ (∀ls,x,xs,ci,rs,ls0,x0,rs0.
+ (∀x. is_startc x ≠ is_endc x) →
is_startc x = true → is_endc ci = true →
+ (∀z. memb ? z (x::xs) = true → is_endc x = false) →
nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
- ∃l,cj,l1.x0::rs0 = l@x::xs@cj::l1 ∧
- outt = change_vec ??
- (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
- (midtape sig ((reverse ? (l@x::xs))@ls0) cj l1) j).
+ (∃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) 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.
+ ∃l1,l2. l = l1@ls@l2 ∨ ∀l1,l2. l ≠ l1@ls@l2.
+*)
lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
src ≠ dst → src < S n → dst < S n →
#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 ⊢ (%→?); *
+[ #tc whd in ⊢ (%→%); *
[ * * [ *
[ * #cur_src * #H1 #H2 #Houtc %
[ #_ @Houtc
- | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #_ #Hnthi #Hnthj
- >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H %
+ | #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 #Hnthi >Hnthi in Hci;
- normalize in ⊢ (%→?); #H destruct (H) ] ]
- | #Hcj #Houtc %
- [ #_ @Houtc
- | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #Hnthj >Hnthj in Hcj;
- normalize in ⊢ (%→?); #H destruct (H) ] ]
-
-
-
-[ #tc whd in ⊢ (%→?); * * [ *
+ | #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) ]
+ ]
+ |* #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) //
+ ]
+ ]
+ ]
+|#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
projects / helm.git / blobdiff