(* *)
(**************************************************************************)
+include "turing/simple_machines.ma".
include "turing/multi_universal/compare.ma".
include "turing/multi_universal/par_test.ma".
include "turing/multi_universal/moves_2.ma".
-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.
-
-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.
-
-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.
-
-definition Rm_test_null_true ≝
- λalpha,n,i.λt1,t2:Vector ? (S n).
- current alpha (nth i ? t1 (niltape ?)) ≠ None ? ∧ t2 = t1.
-
-definition Rm_test_null_false ≝
- λalpha,n,i.λt1,t2:Vector ? (S n).
- current alpha (nth i ? t1 (niltape ?)) = None ? ∧ t2 = t1.
-
-lemma sem_test_null_multi : ∀alpha,n,i.i ≤ n →
- inject_TM ? (test_null ?) n i ⊨
- [ tc_true : Rm_test_null_true alpha n i, Rm_test_null_false alpha n i ].
-#alpha #n #i #Hin #int
-cases (acc_sem_inject … Hin (sem_test_null alpha) int)
-#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
-[ @Hloop
-| #Hqtrue lapply (Htrue Hqtrue) * * #Hcur #Hnth_i #Hnth_j % //
- @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) #Hi0i
- [ >Hi0i @sym_eq @Hnth_i | @sym_eq @Hnth_j @sym_not_eq // ] ]
-| #Hqfalse lapply (Hfalse Hqfalse) * * #Hcur #Hnth_i #Hnth_j %
- [ @Hcur
- | @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) //
- #Hi0i @sym_eq @Hnth_j @sym_not_eq // ] ]
-qed.
-
definition match_test ≝ λsrc,dst.λsig:DeqSet.λn.λv:Vector ? n.
match (nth src (option sig) v (None ?)) with
[ None ⇒ false
| Some x ⇒ notb (nth dst (DeqOption sig) v (None ?) == None ?) ].
-
-definition mmove_states ≝ initN 2.
-
-definition mmove0 : mmove_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
-definition mmove1 : mmove_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
-
-definition trans_mmove ≝
- λi,sig,n,D.
- λp:mmove_states × (Vector (option sig) (S n)).
- let 〈q,a〉 ≝ p in match (pi1 … q) with
- [ O ⇒ 〈mmove1,change_vec ? (S n) (null_action ? n) (〈None ?,D〉) i〉
- | S _ ⇒ 〈mmove1,null_action sig n〉 ].
-
-definition mmove ≝
- λi,sig,n,D.
- mk_mTM sig n mmove_states (trans_mmove i sig n D)
- mmove0 (λq.q == mmove1).
-
-definition Rm_multi ≝
- λalpha,n,i,D.λt1,t2:Vector ? (S n).
- t2 = change_vec ? (S n) t1 (tape_move alpha (nth i ? t1 (niltape ?)) D) i.
-
-lemma sem_move_multi :
- ∀alpha,n,i,D.i ≤ n →
- mmove i alpha n D ⊨ Rm_multi alpha n i D.
-#alpha #n #i #D #Hin #int %{2}
-%{(mk_mconfig ? mmove_states n mmove1 ?)}
-[| %
- [ whd in ⊢ (??%?); @eq_f whd in ⊢ (??%?); @eq_f %
- | whd >tape_move_multi_def
- <(change_vec_same … (ctapes …) i (niltape ?))
- >pmap_change <tape_move_multi_def >tape_move_null_action % ] ]
- qed.
definition rewind ≝ λsrc,dst,sig,n.
parmove src dst sig n L · mmove src sig n R · mmove dst sig n R.
+definition R_rewind_strong ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ (∀x,x0,xs,rs.
+ nth src ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
+ ∀ls0,y,y0,target,rs0.|xs| = |target| →
+ nth dst ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) src)
+ (midtape sig ls0 y0 (reverse ? target@y::rs0)) dst) ∧
+ (∀x,x0,xs,rs.
+ nth dst ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
+ ∀ls0,y,y0,target,rs0.|xs| = |target| →
+ nth src ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) dst)
+ (midtape sig ls0 y0 (reverse ? target@y::rs0)) src) ∧
+ (∀x,rs.nth src ? int (niltape ?) = midtape sig [] x rs →
+ ∀ls0,y,rs0.nth dst ? int (niltape ?) = midtape sig ls0 y rs0 →
+ outt = int) ∧
+ (∀x,rs.nth dst ? int (niltape ?) = midtape sig [] x rs →
+ ∀ls0,y,rs0.nth src ? int (niltape ?) = midtape sig ls0 y rs0 →
+ outt = int).
+
definition R_rewind ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
(∀x,x0,xs,rs.
nth src ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
qed.
*)
-lemma sem_rewind : ∀src,dst,sig,n.
+lemma sem_rewind_strong : ∀src,dst,sig,n.
src ≠ dst → src < S n → dst < S n →
- rewind src dst sig n ⊨ R_rewind src dst sig n.
+ rewind src dst sig n ⊨ R_rewind_strong src dst sig n.
#src #dst #sig #n #Hneq #Hsrc #Hdst
@(sem_seq_app sig n ????? (sem_parmoveL src dst sig n Hneq Hsrc Hdst) ?)
[| @(sem_seq_app sig n ????? (sem_move_multi … R ?) (sem_move_multi … R ?)) //
@le_S_S_to_le // ]
-#ta #tb * #tc * * #Htc #_ * #td * whd in ⊢ (%→%→?); #Htd #Htb %
+#ta #tb * #tc * * * #Htc1 #Htc2 #_ * #td * whd in ⊢ (%→%→?); #Htd #Htb % [ % [ %
[ #x #x0 #xs #rs #Hmidta_src #ls0 #y #y0 #target #rs0 #Hlen #Hmidta_dst
- >(Htc ??? Hmidta_src ls0 y (target@[y0]) rs0 ??) in Htd;
+ >(Htc1 ??? Hmidta_src ls0 y (target@[y0]) rs0 ??) in Htd;
[|>Hmidta_dst //
|>length_append >length_append >Hlen % ]
>change_vec_commute [|@sym_not_eq //]
>rev_append_def >append_nil #Htd >Htd in Htb;
>change_vec_change_vec >nth_change_vec //
cases ls0 [|#l1 #ls1] normalize in match (tape_move ???); //
+| #x #x0 #xs #rs #Hmidta_dst #ls0 #y #y0 #target #rs0 #Hlen #Hmidta_src
+ >(Htc2 ??? Hmidta_dst ls0 y (target@[y0]) rs0 ??) in Htd;
+ [|>Hmidta_src //
+ |>length_append >length_append >Hlen % ]
+ >change_vec_change_vec
+ >change_vec_commute [|@sym_not_eq //]
+ >nth_change_vec //
+ >reverse_append >reverse_single
+ >reverse_append >reverse_single
+ cases ls0 [|#l1 #ls1] normalize in match (tape_move ???);
+ #Htd >Htd in Htb; >change_vec_change_vec >nth_change_vec //
+ >rev_append_def >change_vec_commute // normalize in match (tape_move ???); // ]
| #x #rs #Hmidta_src #ls0 #y #rs0 #Hmidta_dst
- lapply (Htc … Hmidta_src … (refl ??) Hmidta_dst) -Htc #Htc >Htc in Htd;
+ lapply (Htc1 … Hmidta_src … (refl ??) Hmidta_dst) -Htc1 #Htc >Htc in Htd;
>change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
>nth_change_vec_neq [|@sym_not_eq //]
>nth_change_vec // lapply (refl ? ls0) cases ls0 in ⊢ (???%→%);
>nth_change_vec // >change_vec_change_vec
whd in match (tape_move ???);whd in match (tape_move ???); <Hmidta_src
<Hls0 <Hmidta_dst >change_vec_same >change_vec_same //
-]]
+ ] ]
+| #x #rs #Hmidta_dst #ls0 #y #rs0 #Hmidta_src
+ lapply (Htc2 … Hmidta_dst … (refl ??) Hmidta_src) -Htc2 #Htc >Htc in Htd;
+ >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
+ >nth_change_vec // lapply (refl ? ls0) cases ls0 in ⊢ (???%→%);
+ [ #Hls0 destruct (Hls0) #Htd >Htd in Htb;
+ >nth_change_vec // >change_vec_change_vec
+ whd in match (tape_move ???);whd in match (tape_move ???);
+ <Hmidta_src <Hmidta_dst >change_vec_same >change_vec_same //
+ | #l1 #ls1 #Hls0 destruct (Hls0) #Htd >Htd in Htb;
+ >nth_change_vec // >change_vec_change_vec
+ whd in match (tape_move ???); whd in match (tape_move ???); <Hmidta_src
+ <Hmidta_dst >change_vec_same >change_vec_same //
+ ]
+]
+qed.
+
+lemma sem_rewind : ∀src,dst,sig,n.
+ src ≠ dst → src < S n → dst < S n →
+ rewind src dst sig n ⊨ R_rewind src dst sig n.
+#src #dst #sig #n #Hneq #Hsrc #Hdst @(Realize_to_Realize … (sem_rewind_strong …)) //
+#ta #tb * * * #H1 #H2 #H3 #H4 % /2 by /
qed.
definition match_step ≝ λsrc,dst,sig,n.
∀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 ⊨
| <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/ ]
]
]
]
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 ∧
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)
]
qed.
-definition Pre_match_m ≝
- λsrc,sig,n.λt: Vector (tape sig) (S n).
- ∃x,xs.
- nth src (tape sig) t (niltape sig) = midtape ? [] x xs.
+axiom daemon : ∀P:Prop.P.
+
+(* XXX: move to turing (or mono) *)
+definition option_cons ≝ λsig.λc:option sig.λl.
+ match c with [ None ⇒ l | Some c0 ⇒ c0::l ].
+
+definition R_match_step_true_naive ≝
+ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ |left ? (nth src ? outt (niltape ?))| +
+ |option_cons ? (current ? (nth dst ? outt (niltape ?))) (right ? (nth dst ? outt (niltape ?)))| <
+ |left ? (nth src ? int (niltape ?))| +
+ |option_cons ? (current ? (nth dst ? int (niltape ?))) (right ? (nth dst ? int (niltape ?)))|.
+
+axiom right_mk_tape : ∀sig,ls,c,rs.right ? (mk_tape sig ls c rs) = rs.
+axiom left_mk_tape : ∀sig,ls,c,rs.left ? (mk_tape sig ls c rs) = ls.
+axiom current_mk_tape : ∀sig,ls,c,rs.current ? (mk_tape sig ls c rs) = c.
+axiom length_tail : ∀A,l.0 < |l| → |tail A l| < |l|.
+axiom lists_length_split :
+ ∀A.∀l1,l2:list A.(∃la,lb.(|la| = |l1| ∧ l2 = la@lb) ∨ (|la| = |l2| ∧ l1 = la@lb)).
+axiom opt_cons_tail_expand : ∀A,l.l = option_cons A (option_hd ? l) (tail ? l).
-lemma terminate_match_m :
+lemma sem_match_step_termination :
+ ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
+ match_step src dst sig n ⊨
+ [ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
+ R_match_step_true_naive src dst sig n,
+ R_match_step_false src dst sig n ].
+#src #dst #sig #n #Hneq #Hsrc #Hdst
+@(acc_sem_seq_app sig n … (sem_compare src dst sig n Hneq Hsrc Hdst)
+ (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?))
+ (sem_seq …
+ (sem_rewind_strong ???? Hneq Hsrc Hdst)
+ (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
+ (sem_nop …)))
+[ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?))))
+ cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%);
+ [ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %]
+ whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
+ >nth_current_chars >Hcurta_src normalize in ⊢ (%→?); #H destruct (H)
+ | #s #Hs lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
+ cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
+ [ #Hcurta_dst #Hcomp #_ * #td * >Hcomp [| %2 %]
+ whd in ⊢ (%→?); * whd in ⊢ (??%?→?);
+ >nth_current_chars >nth_current_chars >Hs >Hcurta_dst
+ normalize in ⊢ (%→?); #H destruct (H)
+ | #s0 #Hs0
+ cases (current_to_midtape … Hs) #ls * #rs #Hmidta_src >Hmidta_src
+ cases (current_to_midtape … Hs0) #ls0 * #rs0 #Hmidta_dst >Hmidta_dst
+ cases (true_or_false (s == s0)) #Hss0
+ [ lapply (\P Hss0) -Hss0 #Hss0 destruct (Hss0)
+ #_ #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 …))
+ [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
+ [>nth_change_vec [cases (append ???) // | @Hsrc]
+ |@(not_to_not … Hneq) //
+ ]]
+ normalize in ⊢ (%→?); #H destruct (H) ]
+ | * #xs * #ci * #cj * #rs'' * #rs0' * * * #Hcicj #Hrs #Hrs0
+ #Htc * #td * * #Hmatch #Htd destruct (Htd) * #te * * *
+ >Htc >change_vec_commute // >nth_change_vec //
+ >change_vec_commute [|@sym_not_eq //] >nth_change_vec //
+ cases (lists_length_split ? ls ls0) #lsa * #lsb * * #Hlen #Hlsalsb
+ destruct (Hlsalsb) *
+ [ #Hte #_ #_ <(reverse_reverse … ls) in Hte; <(reverse_reverse … lsa)
+ cut (|reverse ? lsa| = |reverse ? ls|) [ // ] #Hlen'
+ @(list_cases2 … Hlen')
+ [ #H1 #H2 >H1 >H2 -H1 -H2 normalize in match (reverse ? [ ]); #Hte #_
+ lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
+ >change_vec_commute // >change_vec_change_vec
+ >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
+ >Hte * * #_ >nth_change_vec // >reverse_reverse
+ #H lapply (H … (refl ??)) -H #Htb1 #Htb2
+ cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta (midtape sig [] s0 (xs@ci::rs'')) src) (mk_tape sig (s0::lsb) (option_hd sig (xs@cj::rs0')) (tail sig (xs@cj::rs0'))) dst)
+ [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
+ >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
+ >right_mk_tape normalize in match (left ??);
+ >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand
+ whd in match (option_cons ???); >Hrs0
+ normalize in ⊢ (?(?%)%); //
+ | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
+ >reverse_cons >reverse_cons #Hte
+ lapply (Hte ci hdb (reverse ? xs@s0::reverse ? tlb) rs'' ?
+ lsb cj hda (reverse ? xs@s0::reverse ? tla) rs0' ??)
+ [ /2 by cons_injective_l, nil/
+ | >length_append >length_append @eq_f @(eq_f ?? S)
+ >H1 in Hlen'; >H2 whd in ⊢ (??%%→?); #Hlen'
+ >length_reverse >length_reverse destruct (Hlen') //
+ | /2 by refl, trans_eq/ ] -Hte
+ #Hte #_ * * #_ >Hte >nth_change_vec // #Htb1 #Htb2
+ cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
+ (mk_tape sig (hda::lsb) (option_hd ? (reverse sig (reverse sig xs@s0::reverse sig tla)@cj::rs0')) (tail ? (reverse sig (reverse sig xs@s0::reverse sig tla)@cj::rs0'))) dst)
+ (midtape ? [ ] hdb (reverse sig (reverse sig xs@s0::reverse sig tlb)@ci::rs'')) src)
+ [@daemon] -Htb1 -Htb2 #Htb >Htb whd
+ >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
+ >right_mk_tape >Hmidta_src >Hmidta_dst
+ whd in match (left ??); whd in match (left ??); whd in match (right ??);
+ >current_mk_tape <opt_cons_tail_expand whd in match (option_cons ???);
+ >Hrs0 >length_append whd in ⊢ (??(??%)); >length_append >length_reverse
+ >length_append >commutative_plus in match (|reverse ??| + ?);
+ whd in match (|?::?|); >length_reverse >length_reverse
+ <(length_reverse ? ls) <Hlen' >H1 normalize // ]
+ | #_ #Hte #_ <(reverse_reverse … ls0) in Hte; <(reverse_reverse … lsa)
+ cut (|reverse ? lsa| = |reverse ? ls0|) [ // ] #Hlen'
+ @(list_cases2 … Hlen')
+ [ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #Hte
+ lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte
+ >change_vec_change_vec >change_vec_commute [|@sym_not_eq //]
+ >change_vec_change_vec #Hte #_
+ >Hte * * #_ >nth_change_vec // >reverse_reverse
+ #H lapply (H … (refl ??)) -H #Htb1 #Htb2
+ cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta (mk_tape ? [s0] (option_hd ? (xs@cj::rs0')) (tail ? (xs@cj::rs0'))) dst)
+ (midtape ? lsb s0 (xs@ci::rs'')) src)
+ [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
+ >nth_change_vec_neq // >nth_change_vec //
+ >right_mk_tape normalize in match (left ??);
+ >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand >Hrs0
+ >length_append normalize >length_append >length_append
+ <(reverse_reverse ? lsa) >H1 normalize //
+ | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
+ >reverse_cons >reverse_cons #Hte
+ lapply (Hte cj hdb (reverse ? xs@s0::reverse ? tlb) rs0' ?
+ lsb ci hda (reverse ? xs@s0::reverse ? tla) rs'' ??)
+ [ /2 by cons_injective_l, nil/
+ | >length_append >length_append @eq_f @(eq_f ?? S)
+ >H1 in Hlen'; >H2 whd in ⊢ (??%%→?); #Hlen'
+ >length_reverse >length_reverse destruct (Hlen') //
+ | /2 by refl, trans_eq/ ] -Hte
+ #Hte #_ * * #_ >Hte >nth_change_vec_neq // >nth_change_vec // #Htb1 #Htb2
+ cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
+ (mk_tape sig [hdb] (option_hd ? (reverse sig (reverse sig xs@s0::reverse sig tlb)@cj::rs0')) (tail ? (reverse sig (reverse sig xs@s0::reverse sig tlb)@cj::rs0'))) dst)
+ (midtape ? lsb hda (reverse sig (reverse sig xs@s0::reverse sig tla)@ci::rs'')) src)
+ [@daemon] -Htb1 -Htb2 #Htb >Htb whd
+ >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
+ >right_mk_tape >Hmidta_src >Hmidta_dst
+ whd in match (left ??); whd in match (left ??); whd in match (right ??);
+ >current_mk_tape <opt_cons_tail_expand
+ whd in match (option_cons ???);
+ >Hrs0 >length_append whd in ⊢ (??(??%)); >length_append >length_reverse
+ >length_append >commutative_plus in match (|reverse ??| + ?);
+ whd in match (|?::?|); >length_reverse >length_reverse
+ <(length_reverse ? lsa) >Hlen' >H2 >length_append
+ normalize //
+ ]
+ ]
+ ]
+ | lapply (\Pf Hss0) -Hss0 #Hss0 #Htc cut (tc = ta)
+ [@Htc % % @(not_to_not ??? Hss0) #H destruct (H) %]
+ -Htc #Htc destruct (Htc) #_ * #td * whd in ⊢ (%→?); * #_
+ #Htd destruct (Htd) * #te * * * * >Hmidta_src >Hmidta_dst
+ cases (lists_length_split ? ls ls0) #lsa * #lsb * * #Hlen #Hlsalsb
+ destruct (Hlsalsb)
+ [ <(reverse_reverse … ls) <(reverse_reverse … lsa)
+ cut (|reverse ? lsa| = |reverse ? ls|) [ // ] #Hlen'
+ @(list_cases2 … Hlen')
+ [ #H1 #H2 >H1 >H2 -H1 -H2 #_ #_ normalize in match (reverse ? [ ]); #Hte #_
+ lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte) * * #_
+ >Hmidta_dst #Htb1 lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2
+ cut (tb = change_vec ?? ta (mk_tape ? (s0::lsa@lsb) (option_hd ? rs0) (tail ? rs0)) dst)
+ [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
+ >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst
+ >right_mk_tape normalize in match (left ??); normalize in match (right ??);
+ >Hmidta_src >Hmidta_dst >current_mk_tape <opt_cons_tail_expand
+ normalize //
+ | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
+ >reverse_cons >reverse_cons >associative_append #Hte
+ lapply (Hte ???? (refl ??) ? s0 ? (reverse ? tla) ?? (refl ??))
+ [ >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) //
+ normalize #H destruct (H) // ] #Hte #_ #_ #_
+ * * #_ >Hte >nth_change_vec // #Htb1 #Htb2
+ cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
+ (mk_tape sig (hda::lsb) (option_hd ? (reverse sig (reverse sig tla)@s0::rs0)) (tail ? (reverse sig (reverse sig tla)@s0::rs0))) dst)
+ (midtape ? [ ] hdb (reverse sig (reverse sig tlb)@s::rs)) src)
+ [@daemon] -Htb1 -Htb2 #Htb >Htb whd
+ >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
+ >right_mk_tape >Hmidta_src >Hmidta_dst
+ whd in match (left ??); whd in match (left ??); whd in match (right ??);
+ >current_mk_tape <opt_cons_tail_expand >length_append
+ >length_reverse >length_reverse <(length_reverse ? ls) <Hlen'
+ >H1 normalize // ]
+ | #_ <(reverse_reverse … ls0) <(reverse_reverse … lsa)
+ cut (|reverse ? lsa| = |reverse ? ls0|) [ // ] #Hlen'
+ @(list_cases2 … Hlen')
+ [ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #_ #_ #Hte
+ lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte)
+ * * #_ >Hmidta_dst #Htb1 lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2
+ cut (tb = change_vec (tape sig) (S n) ta (mk_tape ? (s0::ls0) (option_hd ? rs0) (tail ? rs0)) dst)
+ [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec //
+ >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst
+ >current_mk_tape >right_mk_tape normalize in ⊢ (??%); <opt_cons_tail_expand
+ normalize //
+ | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2
+ >reverse_cons >reverse_cons #Hte #_ #_
+ lapply (Hte s0 hdb (reverse ? tlb) rs0 ?
+ lsb s hda (reverse ? tla) rs ??)
+ [ /2 by cons_injective_l, nil/
+ | >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) //
+ normalize #H destruct (H) //
+ | /2 by refl, trans_eq/ ] -Hte
+ #Hte * * #_ >Hte >nth_change_vec_neq // >nth_change_vec // #Htb1 #Htb2
+ cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta
+ (mk_tape sig [hdb] (option_hd ? (reverse sig (reverse sig tlb)@s0::rs0)) (tail ? (reverse sig (reverse sig tlb)@s0::rs0))) dst)
+ (midtape ? lsb hda (reverse sig (reverse sig tla)@s::rs)) src)
+ [@daemon] -Htb1 -Htb2 #Htb >Htb whd
+ >nth_change_vec // >nth_change_vec_neq // >nth_change_vec //
+ >right_mk_tape >Hmidta_src >Hmidta_dst
+ whd in match (left ??); whd in match (left ??); whd in match (right ??);
+ >current_mk_tape <opt_cons_tail_expand >length_append
+ normalize in ⊢ (??%); >length_append >reverse_reverse
+ <(length_reverse ? lsa) >Hlen' >H2 normalize //
+ ]
+ ]
+ ]
+ ]
+ ]
+| #ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd destruct (Htd)
+ whd in ⊢ (%→?); #Htb destruct (Htb) #ls #x #xs #Hta_src
+ lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
+ cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
+ [ #Hcurta_dst % % % // @Hcomp1 %2 //
+ | #x0 #Hcurta_dst cases (current_to_midtape … Hcurta_dst) -Hcurta_dst
+ #ls0 * #rs0 #Hta_dst cases (true_or_false (x == x0)) #Hxx0
+ [ lapply (\P Hxx0) -Hxx0 #Hxx0 destruct (Hxx0)
+ | >(?:tc=ta) in Htest;
+ [|@Hcomp1 % % >Hta_src >Hta_dst @(not_to_not ??? (\Pf Hxx0)) normalize
+ #Hxx0' destruct (Hxx0') % ]
+ whd in ⊢ (??%?→?);
+ >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'} %
+ [ % // | >Hcurtc % ]
+ | * #rs0' * #Hxs #Htc %2 >Htc %{ls0} %{rs0'} % // ]
+ | * #xs0 * #ci * #cj * #rs' * #rs0' * * *
+ #Hci #Hxs #Hrs0 #Htc @False_ind
+ whd in Htest:(??%?);
+ >(?:nth src ? (current_chars ?? tc) (None ?) = Some ? ci) in Htest;
+ [|>nth_current_chars >Htc >nth_change_vec_neq [|@(not_to_not … Hneq) //]
+ >nth_change_vec //]
+ >(?:nth dst ? (current_chars ?? tc) (None ?) = Some ? cj)
+ [|>nth_current_chars >Htc >nth_change_vec //]
+ normalize #H destruct (H) ] ] ]
+qed.
+
+axiom terminate_match_m :
∀src,dst,sig,n,t.
src ≠ dst → src < S n → dst < S n →
- 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)) //
+(*#src #dst #sig #n #ta #Hneq #Hsrc #Hdst
+@(terminate_while … (sem_match_step_termination src dst sig n Hneq Hsrc Hdst)) // % #tb
+letin f ≝ (λt0:Vector (tape sig) (S n).|left ? (nth src (tape ?) t0 (niltape ?))|
+ +|option_cons ? (current ? (nth dst (tape ?) t0 (niltape ?)))
+ (right ? (nth dst (tape ?) t0 (niltape ?)))|)
+change with (f tb < f ta) in ⊢ (%→?); @(nat_elim1 (f tb))
+#x lapply (refl ? x) cases x in ⊢ (???%→%);
+[ #Hx
+*
+#x #IH #Hx cases
+
+ @IH % #tc change with (f tc < f tb) in ⊢ (%→?);
+
+ )(|left @(nat_elim1 (|left ? (nth ? (tape ?) t (niltape ?))|
+ +|option_cons sig (current ? (nth dst (tape ?) t (niltape ?)))
+ (right ? (nth dst (tape ?) t (niltape ?)))|))
<(change_vec_same … t dst (niltape ?))
+<(change_vec_same … t src (niltape ?)) in ⊢ (???(???%??));
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
+ >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 // ] ] ] ]
-qed.
\ No newline at end of file
+ #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. *)
+
+lemma sem_match_m : ∀src,dst,sig,n.
+src ≠ dst → src < S n → dst < S n →
+ match_m src dst sig n \vDash R_match_m src dst sig n.
+#src #dst #sig #n #Hneq #Hsrc #Hdst @WRealize_to_Realize [/2/| @wsem_match_m // ]
+qed.
projects / helm.git / blobdiff