((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨
(∃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 ?) ∨
+ outt = change_vec ??
+ (change_vec ?? int (mk_tape sig (reverse ? rs0@x::ls) (option_hd ? xs0) (tail ? xs0)) src)
+ (mk_tape ? (reverse ? rs0@x::ls0) (None ?) [ ]) dst) ∨
(∃ls0,rs0.
nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
(* ∀rsj,c.
∀s,rs.nth src ? int (niltape ?) = midtape ? [ ] s rs →
outt = change_vec ?? int
(tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst ∧
- (current sig (nth dst (tape sig) int (niltape sig)) = Some ? s →
+ (∃s0.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s0 ∧
+ (s0 = s →
∃xs,ci,rs',ls0,cj,rs0.
rs = xs@ci::rs' ∧
nth dst ? int (niltape ?) = midtape sig ls0 s (xs@cj::rs0) ∧
- ci ≠ cj).
+ ci ≠ cj)).
-axiom daemon : ∀X:Prop.X.
-
lemma sem_match_step :
∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
match_step src dst sig n ⊨
#_ #Hcomp cases (Hcomp ????? (refl ??) (refl ??)) -Hcomp [ *
[ * #rs' * #_ #Hcurtc_dst * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
>nth_current_chars >nth_current_chars >Hcurtc_dst
- cases (current ? (nth src …)) [| #x ]
- normalize in ⊢ (%→?); #H destruct (H)
+ cases (current ? (nth src …))
+ [normalize in ⊢ (%→?); #H destruct (H)
+ | #x >nth_change_vec // cases (reverse ? rs0)
+ [ normalize in ⊢ (%→?); #H destruct (H)
+ | #r1 #rs1 normalize in ⊢ (%→?); #H destruct (H) ] ]
| * #rs0' * #_ #Hcurtc_src * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
>(?:nth src ? (current_chars ?? tc) (None ?) = None ?)
[|>nth_current_chars >Hcurtc_src >nth_change_vec_neq
| <Htb2 [|@(\Pf Hdsti)] >nth_change_vec_neq [| @(\Pf Hdsti)]
>Hrs0 >reverse_reverse >nth_change_vec_neq in ⊢ (???%);
<Hrs <Hmidta_src [|@(\Pf Hdsti)] >change_vec_same % ]
- | #_ >Hmidta_dst >Hrs0
- %{xs} %{ci} %{rs''} %{ls0} %{cj} %{rs0'} % // % //
+ | >Hmidta_dst %{s'} % [%] #_
+ >Hrs0 %{xs} %{ci} %{rs''} %{ls0} %{cj} %{rs0'} % // % //
]
]
| lapply (\Pf Hss0) -Hss0 #Hss0 #Htc cut (tc = ta)
[ <(\P Hdsti) >(Htb1 … Hmidta_dst) >nth_change_vec // >Hmidta_dst
cases rs0 [|#r2 #rs2] %
| <Htb2 [|@(\Pf Hdsti)] >nth_change_vec_neq [| @(\Pf Hdsti)] % ]
- | >Hs0 #H destruct (H) @False_ind cases (Hss0) /2/ ]
+ | >Hs0 %{s0} % // #H destruct (H) @False_ind cases (Hss0) /2/ ]
]
]
]
>nth_current_chars >Hta_src >nth_current_chars >Hta_dst
whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ] -Hcomp1
cases (Hcomp2 … Hta_src Hta_dst) [ *
- [ * #rs' * #Hxs #Hcurtc % %2 %{ls0} %{rs0} %{rs'} % // % //
+ [ * #rs' * #Hxs #Hcurtc % %2 %{ls0} %{rs0} %{rs'} %
+ [ % // | >Hcurtc % ]
| * #rs0' * #Hxs #Htc %2 >Htc %{ls0} %{rs0'} % // ]
| * #xs0 * #ci * #cj * #rs' * #rs0' * * *
#Hci #Hxs #Hrs0 #Htc @False_ind
λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
∀x,rs.
nth src ? int (niltape ?) = midtape sig [ ] x rs →
- (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧
+ (current sig (nth dst (tape sig) int (niltape sig)) = None ? →
+ right ? (nth dst (tape sig) int (niltape sig)) = [ ] → outt = int) ∧
(∀ls0,x0,rs0.
nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
(∃l,l1.x0::rs0 = l@x::rs@l1 ∧
| * #ls0 * #rs0 * #xs0 * * #Htc_dst #Hrs0 #HNone %
[ >Htc_dst normalize in ⊢ (%→?); #H destruct (H)
| #ls1 #x1 #rs1 >Htc_dst #H destruct (H)
- >Hrs0 cases xs0
+ >Hrs0 >HNone cases xs0
[ % %{[ ]} %{[ ]} % [ >append_nil >append_nil %]
- (* change false case
- #cj #ls2 #H destruct (H) *) @daemon
+ @eq_f3 //
+ [ >reverse_append %
+ | >reverse_append >reverse_cons >reverse_append
+ >associative_append >associative_append % ]
| #x2 #xs2 %2 #l #l1 % #Habs lapply (eq_f ?? (length ?) ?? Habs)
>length_append whd in ⊢ (??%(??%)→?); >length_append
>length_append normalize >commutative_plus whd in ⊢ (???%→?);
lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
[#Hcurta_dst %
- [#_ whd in Htrue; >Hmidta_src in Htrue; #Htrue
- cases (Htrue ?? (refl ??)) -Htrue >Hcurta_dst
- (* dovremmo sapere che ta.dst è sul margine destro, da cui la move non
- ha effetto *) #_ cut (tc = ta) [@daemon] #Htc destruct (Htc) #_
+ [#Hcurta_dst #Hrightta_dst whd in Htrue; >Hmidta_src in Htrue; #Htrue
+ cases (Htrue ?? (refl ??)) -Htrue #Htc
+ cut (tc = ta)
+ [ >Htc whd in match (tape_move_mono ???); whd in match (tape_write ???);
+ <(change_vec_same … ta dst (niltape ?)) in ⊢ (???%);
+ lapply Hrightta_dst lapply Hcurta_dst -Hrightta_dst -Hcurta_dst
+ cases (nth dst ? ta (niltape ?))
+ [ #_ #_ %
+ | #r0 #rs0 #_ normalize in ⊢ (%→?); #H destruct (H)
+ | #l0 #ls0 #_ #_ %
+ | #ls #x0 #rs normalize in ⊢ (%→?); #H destruct (H) ] ]
+ -Htc #Htc destruct (Htc) #_
cases (IH … Hmidta_src) #Houtc #_ @Houtc //
|#ls0 #x0 #rs0 #Hmidta_dst >Hmidta_dst in Hcurta_dst;
normalize in ⊢ (%→?); #H destruct (H)
#H destruct (H) whd in Htrue; >Hmidta_src in Htrue; #Htrue
cases (Htrue ?? (refl …)) -Htrue >Hmidta_dst #Htc
cases (true_or_false (x==c)) #eqx
- [ lapply (\P eqx) -eqx #eqx destruct (eqx)
- #Htrue cases (Htrue (refl ??)) -Htrue
+ [ lapply (\P eqx) -eqx #eqx destruct (eqx) * #s0 * whd in ⊢ (??%?→?); #Hs0
+ destruct (Hs0) #Htrue cases (Htrue (refl ??)) -Htrue
#xs0 * #ci * #rs' * #ls1 * #cj * #rs1 * * #Hxs #H destruct (H) #Hcicj
>Htc in IH; whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
#IH cases (IH … Hmidta_src) -IH #_ >nth_change_vec //
#Hxs1 >Hxs1 #IH cases (IH … (refl ??)) -IH
[ * #l * #l1 * #Hxs1'
>change_vec_commute // >change_vec_change_vec
- #Houtc % %{(c::l)} %{l1} %
+ #Houtc % %{(s0::l)} %{l1} %
[ normalize <Hxs1' %
| >reverse_cons >associative_append >change_vec_commute // @Houtc ]
- | #H %2 #l #l1 >(?:l@c::xs@l1 = l@(c::xs)@l1) [|%]
+ | #H %2 #l #l1 >(?:l@s0::xs@l1 = l@(s0::xs)@l1) [|%]
@not_sub_list_merge
[ #l2 >Hxs <Hxs1 % normalize #H1 lapply (cons_injective_r ????? H1)
>associative_append #H2 lapply (append_l2_injective ????? (refl ??) H2)
-H1 #H1 cases (H l2 l3) #H2 @H2 @H1
]
]
- | (* in match_step_true manca il caso di fallimento immediato
- (con i due current diversi) *)
- @daemon
- (*
- #_ lapply (\Pf eqx) -eqx #eqx >Hmidta_dst
- cases (Htrue ? (refl ??) eqx) -Htrue #Htb #Hendcx #_
- cases rs0 in Htb;
- [ #_ %2 #l #l1 cases l
+ | #_ cases (IH x xs ?) -IH
+ [| >Htc >nth_change_vec_neq [|@sym_not_eq //] @Hmidta_src ]
+ >Htc >nth_change_vec // cases rs0
+ [ #_ #_ %2 #l #l1 cases l
[ normalize cases xs
[ cases l1
- [ normalize % #H destruct (H) cases eqx /2/
+ [ normalize % #H destruct (H) cases (\Pf eqx) /2/
| #tmp1 #l2 normalize % #H destruct (H) ]
| #tmp1 #l2 normalize % #H destruct (H) ]
| #tmp1 #l2 normalize % #H destruct (H)cases l2 in e0;
[ normalize #H1 destruct (H1)
- | #tmp2 #l3 normalize #H1 destruct (H1) ]
- ]
- | #r1 #rs1 normalize in ⊢ (???(????%?)→?); #Htb >Htb in IH; #IH
- cases (IH ls x xs end rs ? Hnotend Hend Hnotstart)
- [| >Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src ] -IH
- #_ #IH cases (IH Hstart (c::ls0) r1 rs1 ?)
- [|| >nth_change_vec // ] -IH
- [ * #l * #l1 * #Hll1 #Hout % %{(c::l)} %{l1} % >Hll1 //
- >reverse_cons >associative_append #cj0 #ls #Hl1 >(Hout ?? Hl1)
- >change_vec_commute in ⊢ (??(???%??)?); // @sym_not_eq //
- | #IH %2 @(not_sub_list_merge_2 ?? (x::xs)) normalize [|@IH]
- #l1 % #H destruct (H) cases eqx /2/
- ] *)
+ | #tmp2 #l3 normalize #H1 destruct (H1) ] ]
+ | #r1 #rs1 #_ #IH cases (IH … (refl ??)) -IH
+ [ * #l * #l1 * #Hll1 #Houtc % %{(c::l)} %{l1} % [ >Hll1 % ]
+ >Houtc >change_vec_commute // >change_vec_change_vec
+ >change_vec_commute [|@sym_not_eq //]
+ >reverse_cons >associative_append %
+ | #Hll1 %2 @(not_sub_list_merge_2 ?? (x::xs)) normalize [|@Hll1]
+ #l1 % #H destruct (H) cases (\Pf eqx) /2/
+ ]
+ ]
]
]
]
qed.
definition Pre_match_m ≝
- λsrc,sig,n,is_startc,is_endc.λt: Vector (tape sig) (S n).
- ∃start,xs,end.
- nth src (tape sig) t (niltape sig) = midtape ? [] start (xs@[end]) ∧
- is_startc start = true ∧
- (∀c.c ∈ (xs@[end]) = true → is_startc c = false) ∧
- (∀c.c ∈ (start::xs) = true → is_endc c = false) ∧
- is_endc end = true.
+ λsrc,sig,n.λt: Vector (tape sig) (S n).
+ ∃x,xs.
+ nth src (tape sig) t (niltape sig) = midtape ? [] x xs.
lemma terminate_match_m :
- ∀src,dst,sig,n,is_startc,is_endc,t.
+ ∀src,dst,sig,n,t.
src ≠ dst → src < S n → dst < S n →
- Pre_match_m src sig n is_startc is_endc t →
- match_m src dst sig n is_startc is_endc ↓ t.
-#src #dst #sig #n #is_startc #is_endc #t #Hneq #Hsrc #Hdst * #start * #xs * #end
-* * * * #Hmid_src #Hstart #Hnotstart #Hnotend #Hend
-@(terminate_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst)) //
+ Pre_match_m src sig n t →
+ match_m src dst sig n ↓ t.
+#src #dst #sig #n #t #Hneq #Hsrc #Hdst * #start * #xs
+#Hmid_src
+@(terminate_while … (sem_match_step src dst sig n Hneq Hsrc Hdst)) //
<(change_vec_same … t dst (niltape ?))
lapply (refl ? (nth dst (tape sig) t (niltape ?)))
cases (nth dst (tape sig) t (niltape ?)) in ⊢ (???%→?);
[ #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
- >Hmid_src #HR cases (HR ? (refl ??)) -HR
- >nth_change_vec // >Htape_dst normalize in ⊢ (%→?);
- * #H @False_ind @H %
+ >Hmid_src #HR cases (HR ?? (refl ??)) -HR
+ >nth_change_vec // >Htape_dst #_ * #s0 * normalize in ⊢ (%→?); #H destruct (H)
| #x0 #xs0 #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
- >Hmid_src #HR cases (HR ? (refl ??)) -HR
- >nth_change_vec // >Htape_dst normalize in ⊢ (%→?);
- * #H @False_ind @H %
+ >Hmid_src #HR cases (HR ?? (refl ??)) -HR
+ >nth_change_vec // >Htape_dst #_ normalize in ⊢ (%→?);
+ * #s0 * #H destruct (H)
| #x0 #xs0 #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
- >Hmid_src #HR cases (HR ? (refl ??)) -HR
- >nth_change_vec // >Htape_dst normalize in ⊢ (%→?);
- * #H @False_ind @H %
+ >Hmid_src #HR cases (HR ?? (refl ??)) -HR
+ >nth_change_vec // >Htape_dst #_ normalize in ⊢ (%→?);
+ * #s0 * #H destruct (H)
| #ls #s #rs lapply s -s lapply ls -ls lapply Hmid_src lapply t -t elim rs
[#t #Hmid_src #ls #s #Hmid_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
- >Hmid_src >nth_change_vec // >Hmid_dst #HR cases (HR ? (refl ??)) -HR #_
- #HR cases (HR Hstart Hnotstart)
- cases (true_or_false (start == s)) #Hs
- [ lapply (\P Hs) -Hs #Hs <Hs #_ #Htrue
- cut (∃ci,xs1.xs@[end] = ci::xs1)
- [ cases xs
- [ %{end} %{[]} %
- | #x1 #xs1 %{x1} %{(xs1@[end])} % ] ] * #ci * #xs1 #Hxs
- >Hxs in Htrue; #Htrue
- cases (Htrue [ ] start [ ] ? xs1 ? [ ] (refl ??) (refl ??) ?)
- [ * #_ * #H @False_ind @H % ]
- #c0 #Hc0 @Hnotend >(memb_single … Hc0) @memb_hd
- | lapply (\Pf Hs) -Hs #Hs #Htrue #_
- cases (Htrue ? (refl ??) Hs) -Htrue #Ht1 #_ %
- #t2 whd in ⊢ (%→?); #HR cases (HR start ?)
- [ >Ht1 >nth_change_vec // normalize in ⊢ (%→?); * #H @False_ind @H %
- | >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec_neq [|@sym_not_eq //] >Hmid_src % ]
- ]
+ >Hmid_src >nth_change_vec // >Hmid_dst #HR cases (HR ?? (refl ??)) -HR
+ >change_vec_change_vec #Ht1 #_ % #t2 whd in ⊢ (%→?);
+ >Ht1 >nth_change_vec_neq [|@sym_not_eq //] >Hmid_src #HR
+ cases (HR ?? (refl ??)) -HR #_
+ >nth_change_vec // * #s1 * normalize in ⊢ (%→?); #H destruct (H)
|#r0 #rs0 #IH #t #Hmid_src #ls #s #Hmid_dst % #t1 whd in ⊢ (%→?);
>nth_change_vec_neq [|@sym_not_eq //] >Hmid_src
- #Htrue cases (Htrue ? (refl ??)) -Htrue #_ #Htrue
- <(change_vec_same … t1 dst (niltape ?))
- cases (Htrue Hstart Hnotstart) -Htrue
- cases (true_or_false (start == s)) #Hs
- [ lapply (\P Hs) -Hs #Hs <Hs #_ #Htrue
- cut (∃ls0,xs0,ci,rs,rs0.
- nth src ? t (niltape ?) = midtape sig [ ] start (xs0@ci::rs) ∧
- nth dst ? t (niltape ?) = midtape sig ls0 s (xs0@rs0) ∧
- (is_endc ci = true ∨ (is_endc ci = false ∧ (∀b,tlb.rs0 = b::tlb → ci ≠ b))))
- [cases (comp_list ? (xs@[end]) (r0::rs0) is_endc) #xs0 * #xs1 * #xs2
- * * * #Hxs #Hrs #Hxs0notend #Hcomp >Hrs
- cut (∃y,ys. xs1 = y::ys)
- [ lapply Hxs0notend lapply Hxs lapply xs0 elim xs
- [ *
- [ normalize #Hxs1 <Hxs1 #_ %{end} %{[]} %
- | #z #zs normalize in ⊢ (%→?); #H destruct (H) #H
- lapply (H ? (memb_hd …)) -H >Hend #H1 destruct (H1)
- ]
- | #y #ys #IH0 *
- [ normalize in ⊢ (%→?); #Hxs1 <Hxs1 #_ %{y} %{(ys@[end])} %
- | #z #zs normalize in ⊢ (%→?); #H destruct (H) #Hmemb
- @(IH0 ? e0 ?) #c #Hc @Hmemb @memb_cons // ] ] ] * #y * #ys #Hxs1
- >Hxs1 in Hxs; #Hxs >Hmid_src >Hmid_dst >Hxs >Hrs
- %{ls} %{xs0} %{y} %{ys} %{xs2}
- % [ % // | @Hcomp // ] ]
- * #ls0 * #xs0 * #ci * #rs * #rs0 * * #Hmid_src' #Hmid_dst' #Hcomp
- <Hmid_src in Htrue; >nth_change_vec // >Hs #Htrue destruct (Hs)
- lapply (Htrue ??????? Hmid_src' Hmid_dst' ?) -Htrue
- [ #c0 #Hc0 @Hnotend cases (orb_true_l … Hc0) -Hc0 #Hc0
- [ whd in ⊢ (??%?); >Hc0 %
- | @memb_cons >Hmid_src in Hmid_src'; #Hmid_src' destruct (Hmid_src')
- lapply e0 -e0 @(list_elim_left … rs)
- [ #e0 destruct (e0) lapply (append_l1_injective_r ?????? e0) //
- | #x1 #xs1 #_ >append_cons in ⊢ (???%→?);
- <associative_append #e0 lapply (append_l1_injective_r ?????? e0) //
- #e1 >e1 @memb_append_l1 @memb_append_l1 // ] ]
- | * * #Hciendc cases rs0 in Hcomp;
- [ #_ * #H @False_ind @H %
- | #r1 #rs1 * [ >Hciendc #H destruct (H) ]
- * #_ #Hcomp lapply (Hcomp ?? (refl ??)) -Hcomp #Hcomp #_ #Htrue
- cases (Htrue ?? (refl ??) Hcomp) #Ht1 #_ >Ht1 @(IH ?? (s::ls) r0)
- [ >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src
- | >nth_change_vec // >Hmid_dst % ] ] ]
- | >Hmid_dst >nth_change_vec // lapply (\Pf Hs) -Hs #Hs #Htrue #_
- cases (Htrue ? (refl ??) Hs) #Ht1 #_ >Ht1 @(IH ?? (s::ls) r0)
- [ >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src
- | >nth_change_vec // ] ] ] ]
+ #Htrue cases (Htrue ?? (refl ??)) -Htrue >change_vec_change_vec
+ >nth_change_vec // >Hmid_dst whd in match (tape_move_mono ???); #Ht1
+ * #s0 * whd in ⊢ (??%?→?); #H destruct (H) #_ >Ht1
+ lapply (IH t1 ? (s0::ls) r0 ?)
+ [ >Ht1 >nth_change_vec //
+ | >Ht1 >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src
+ | >Ht1 >nth_change_vec // ]
+ ]
+]
qed.
\ No newline at end of file
projects / helm.git / blobdiff