[ None ⇒ 〈parmove2,null_action ? n〉
| Some a1 ⇒ 〈parmove1,change_vec ? (S n)
(change_vec ?(S n)
- (null_action ? n) (Some ? 〈a0,D〉) src)
- (Some ? 〈a1,D〉) dst〉 ] ]
+ (null_action ? n) (〈None sig,D〉) src)
+ (〈None ?,D〉) dst〉 ] ]
| S q ⇒ match q with
[ O ⇒ (* 1 *) 〈parmove1,null_action ? n〉
| S _ ⇒ (* 2 *) 〈parmove2,null_action ? n〉 ] ].
is_sep x1 = false ∧
outt = change_vec ??
(change_vec ?? int
- (tape_move ? (nth src ? int (niltape ?)) (Some ? 〈x1,D〉)) src)
- (tape_move ? (nth dst ? int (niltape ?)) (Some ? 〈x2,D〉)) dst.
+ (tape_move_mono ? (nth src ? int (niltape ?)) (〈None ?,D〉)) src)
+ (tape_move_mono ? (nth dst ? int (niltape ?)) (〈None ?,D〉)) dst.
definition R_parmove_step_false ≝
λsrc,dst:nat.λsig,n,is_sep.λint,outt: Vector (tape sig) (S n).
whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?);
@eq_f2
[ whd in ⊢ (??(???%)?); >Hcurrent %
-| whd in ⊢ (??(???????(???%))?); >Hcurrent @tape_move_null_action ]
+| whd in ⊢ (??(????(???%))?); >Hcurrent @tape_move_null_action ]
qed.
lemma parmove_q0_q2_sep :
whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?);
@eq_f2
[ whd in ⊢ (??(???%)?); >Hcurrent whd in ⊢ (??(???%)?); >Hsep %
-| whd in ⊢ (??(???????(???%))?); >Hcurrent
- whd in ⊢ (??(???????(???%))?); >Hsep @tape_move_null_action ]
+| whd in ⊢ (??(????(???%))?); >Hcurrent
+ whd in ⊢ (??(????(???%))?); >Hsep @tape_move_null_action ]
qed.
lemma parmove_q0_q2_null_dst :
whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?);
@eq_f2
[ whd in ⊢ (??(???%)?); >Hcursrc whd in ⊢ (??(???%)?); >Hsep >Hcurdst %
-| whd in ⊢ (??(???????(???%))?); >Hcursrc
- whd in ⊢ (??(???????(???%))?); >Hsep >Hcurdst @tape_move_null_action ]
+| whd in ⊢ (??(????(???%))?); >Hcursrc
+ whd in ⊢ (??(????(???%))?); >Hsep >Hcurdst @tape_move_null_action ]
qed.
lemma parmove_q0_q1 :
mk_mconfig ??? parmove1
(change_vec ? (S n)
(change_vec ?? v
- (tape_move ? (nth src ? v (niltape ?)) (Some ? 〈a1,D〉)) src)
- (tape_move ? (nth dst ? v (niltape ?)) (Some ? 〈a2,D〉)) dst).
+ (tape_move_mono ? (nth src ? v (niltape ?)) (〈None ?, D〉)) src)
+ (tape_move_mono ? (nth dst ? v (niltape ?)) (〈None ?, D〉)) dst).
#src #dst #sig #n #D #is_sep #v #Hneq #Hsrc #Hdst
#a1 #a2 #Hcursrc #Hcurdst #Hsep
whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
[ whd in match (trans ????);
>Hcursrc >Hcurdst whd in ⊢ (??(???%)?); >Hsep //
| whd in match (trans ????);
- >Hcursrc >Hcurdst whd in ⊢ (??(???????(???%))?); >Hsep
- change with (change_vec ?????) in ⊢ (??(???????%)?);
- <(change_vec_same … v dst (niltape ?)) in ⊢ (??%?);
- <(change_vec_same … v src (niltape ?)) in ⊢ (??%?);
- >pmap_change >pmap_change >tape_move_null_action
+ >Hcursrc >Hcurdst whd in ⊢ (??(????(???%))?); >Hsep whd in ⊢ (??(????(???%))?);
+ <(change_vec_same ?? v dst (niltape ?)) in ⊢ (??%?);
+ >tape_move_multi_def >pmap_change
+ <(change_vec_same ?? v src (niltape ?)) in ⊢ (??%?);
+ >pmap_change <tape_move_multi_def >tape_move_null_action
@eq_f2 // @eq_f2 // >nth_change_vec_neq //
]
qed.
[ #Hcursrc %{2} %
[| % [ %
[ whd in ⊢ (??%?); >parmove_q0_q2_null_src /2/
- <(nth_vec_map ?? (current …) src ? int (niltape ?)) //
| normalize in ⊢ (%→?); #H destruct (H) ]
| #_ % // % %2 // ] ]
| #a #Ha cases (true_or_false (is_sep a)) #Hsep
[ %{2} %
[| % [ %
[ whd in ⊢ (??%?); >(parmove_q0_q2_sep … Hsep) /2/
- <(nth_vec_map ?? (current …) src ? int (niltape ?)) //
| normalize in ⊢ (%→?); #H destruct (H) ]
| #_ % // % % %{a} % // ] ]
| lapply (refl ? (current ? (nth dst ? int (niltape ?))))
[ #Hcurdst %{2} %
[| % [ %
[ whd in ⊢ (??%?); >(parmove_q0_q2_null_dst … Hsep) /2/
- [ <(nth_vec_map ?? (current …) dst ? int (niltape ?)) //
- | <(nth_vec_map ?? (current …) src ? int (niltape ?)) // ]
| normalize in ⊢ (%→?); #H destruct (H) ]
| #_ % // %2 // ] ]
| #b #Hb %{2} %
[| % [ %
[whd in ⊢ (??%?); >(parmove_q0_q1 … Hneq Hsrc Hdst ? b ?? Hsep) //
- [ <(nth_vec_map ?? (current …) dst ? int (niltape ?)) //
- | <(nth_vec_map ?? (current …) src ? int (niltape ?)) // ]
| #_ %{a} %{b} % // % // % // ]
| * #H @False_ind @H % ]
]]]]
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 #td #te * #c0 * #c1 * * * #Hc0 #Hc1 #Hc0nosep #Hd #Hstar #IH #He
+[ 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)
+ ]
+| #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 //
+ | cases (decidable_eq_nat i dst) #Hidst
+ [ >Hidst >nth_change_vec //
+ | >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 >(change_vec_commute … ?? td ?? src dst) //
>change_vec_change_vec
- >(change_vec_commute … ?? tc ?? dst src) [|@sym_not_eq //]
+ >(change_vec_commute … ?? td ?? dst src) [|@sym_not_eq //]
>change_vec_change_vec
>reverse_cons >associative_append
>reverse_cons >associative_append %
- | >Hd >nth_change_vec // >Hdst_tc >Htarget >Hdst_tc in Hc1;
- normalize in ⊢ (%→?); #H destruct (H) //
+ | >Hd >nth_change_vec //
| >Hxs in Hlen; >Htarget normalize #Hlen destruct (Hlen) //
| <Hxs #c1 #Hc1 @Hnosep @memb_cons //
| >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_copy : ∀src,dst,sig,n,is_sep,t.
+lemma terminate_parmoveL : ∀src,dst,sig,n,is_sep,t.
src ≠ dst → src < S n → dst < S n →
- copy src dst sig n is_sep ↓ t.
+ parmove src dst sig n L is_sep ↓ t.
#src #dst #sig #n #is_sep #t #Hneq #Hsrc #Hdst
-@(terminate_while … (sem_copy_step …)) //
+@(terminate_while … (sem_parmove_step …)) //
<(change_vec_same … t src (niltape ?))
cases (nth src (tape sig) t (niltape ?))
-[ % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
-|2,3: #a0 #al0 % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
-| #ls #c #rs lapply c -c lapply ls -ls lapply t -t elim rs
- [#t #ls #c % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?);
- #H1 destruct (H1) #Hxsep >change_vec_change_vec #Ht1 %
- #t2 * #x0 * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
+[ % #t1 * #x1 * #x2 * * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
+|2,3: #a0 #al0 % #t1 * #x1 * #x2 * * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
+| #ls lapply t -t elim ls
+ [#t #c #rs % #t1 * #x1 * #x2 * * * >nth_change_vec // normalize in ⊢ (%→?);
+ #H1 destruct (H1) #Hcurdst #Hxsep >change_vec_change_vec #Ht1 %
+ #t2 * #y1 * #y2 * * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
>nth_change_vec // normalize in ⊢ (%→?); #H destruct (H)
- |#r0 #rs0 #IH #t #ls #c % #t1 * #x * * >nth_change_vec //
- normalize in ⊢ (%→?); #H destruct (H) #Hxsep
+ |#l0 #ls0 #IH #t #c #rs % #t1 * #x1 * #x2 * * * >nth_change_vec //
+ normalize in ⊢ (%→?); #H destruct (H) #Hcurdst #Hxsep
>change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH
]
]
qed.
-lemma sem_copy : ∀src,dst,sig,n,is_sep.
+lemma sem_parmoveL : ∀src,dst,sig,n,is_sep.
src ≠ dst → src < S n → dst < S n →
- copy src dst sig n is_sep ⊨ R_copy src dst sig n is_sep.
-#src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst @WRealize_to_Realize /2/
+ 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/ | @wsem_parmoveL //]
qed.
\ No newline at end of file
projects / helm.git / blobdiff