--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A||
+ \ / This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ V_____________________________________________________________*)
+
+include "turing/turing.ma".
+include "turing/inject.ma".
+include "turing/while_multi.ma".
+
+definition parmove_states ≝ initN 3.
+
+definition parmove0 : parmove_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition parmove1 : parmove_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+definition parmove2 : parmove_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+
+(*
+
+src: a b c ... z # ---→ a b c ... z #
+ ^ ^
+
+dst: _ _ _ ... _ d ---→ a b c ... z d
+ ^ ^
+
+0) (x ≠ sep,_) → (x,x)(R,R) → 1
+ (sep,d) → None 2
+1) (_,_) → None 1
+2) (_,_) → None 2
+
+*)
+
+definition trans_parmove_step ≝
+ λsrc,dst,sig,n,D,is_sep.
+ λp:parmove_states × (Vector (option sig) (S n)).
+ let 〈q,a〉 ≝ p in
+ match pi1 … q with
+ [ O ⇒ match nth src ? a (None ?) with
+ [ None ⇒ 〈parmove2,null_action ? n〉
+ | Some a0 ⇒
+ if is_sep a0 then 〈parmove2,null_action ? n〉
+ else match nth dst ? a (None ?) with
+ [ 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〉 ] ]
+ | S q ⇒ match q with
+ [ O ⇒ (* 1 *) 〈parmove1,null_action ? n〉
+ | S _ ⇒ (* 2 *) 〈parmove2,null_action ? n〉 ] ].
+
+definition parmove_step ≝
+ λsrc,dst,sig,n,D,is_sep.
+ mk_mTM sig n parmove_states (trans_parmove_step src dst sig n D is_sep)
+ parmove0 (λq.q == parmove1 ∨ q == parmove2).
+
+definition R_parmove_step_true ≝
+ λsrc,dst,sig,n,D,is_sep.λint,outt: Vector (tape sig) (S n).
+ ∃x1,x2.
+ current ? (nth src ? int (niltape ?)) = Some ? x1 ∧
+ current ? (nth dst ? int (niltape ?)) = Some ? x2 ∧
+ 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.
+
+definition R_parmove_step_false ≝
+ λsrc,dst:nat.λsig,n,is_sep.λint,outt: Vector (tape sig) (S n).
+ ((∃x1.
+ current ? (nth src ? int (niltape ?)) = Some ? x1 ∧
+ is_sep x1 = true) ∨
+ current ? (nth src ? int (niltape ?)) = None ? ∨
+ current ? (nth dst ? int (niltape ?)) = None ?) ∧
+ outt = int.
+
+lemma parmove_q0_q2_null_src :
+ ∀src,dst,sig,n,D,is_sep,v.src < S n → dst < S n →
+ nth src ? (current_chars ?? v) (None ?) = None ? →
+ step sig n (parmove_step src dst sig n D is_sep)
+ (mk_mconfig ??? parmove0 v) =
+ mk_mconfig ??? parmove2 v.
+#src #dst #sig #n #D #is_sep #v #Hsrc #Hdst #Hcurrent
+whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?);
+@eq_f2
+[ whd in ⊢ (??(???%)?); >Hcurrent %
+| whd in ⊢ (??(???????(???%))?); >Hcurrent @tape_move_null_action ]
+qed.
+
+lemma parmove_q0_q2_sep :
+ ∀src,dst,sig,n,D,is_sep,v,s.src < S n → dst < S n →
+ nth src ? (current_chars ?? v) (None ?) = Some ? s → is_sep s = true →
+ step sig n (parmove_step src dst sig n D is_sep)
+ (mk_mconfig ??? parmove0 v) =
+ mk_mconfig ??? parmove2 v.
+#src #dst #sig #n #D #is_sep #v #s #Hsrc #Hdst #Hcurrent #Hsep
+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 ]
+qed.
+
+lemma parmove_q0_q2_null_dst :
+ ∀src,dst,sig,n,D,is_sep,v,s.src < S n → dst < S n →
+ nth src ? (current_chars ?? v) (None ?) = Some ? s → is_sep s = false →
+ nth dst ? (current_chars ?? v) (None ?) = None ? →
+ step sig n (parmove_step src dst sig n D is_sep)
+ (mk_mconfig ??? parmove0 v) =
+ mk_mconfig ??? parmove2 v.
+#src #dst #sig #n #D #is_sep #v #s #Hsrc #Hdst #Hcursrc #Hsep #Hcurdst
+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 ]
+qed.
+
+lemma parmove_q0_q1 :
+ ∀src,dst,sig,n,D,is_sep,v.src ≠ dst → src < S n → dst < S n →
+ ∀a1,a2.
+ nth src ? (current_chars ?? v) (None ?) = Some ? a1 →
+ nth dst ? (current_chars ?? v) (None ?) = Some ? a2 →
+ is_sep a1 = false →
+ step sig n (parmove_step src dst sig n D is_sep)
+ (mk_mconfig ??? parmove0 v) =
+ 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).
+#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
+ @eq_f2 // @eq_f2 // >nth_change_vec_neq //
+]
+qed.
+
+lemma sem_parmove_step :
+ ∀src,dst,sig,n,D,is_sep.src ≠ dst → src < S n → dst < S n →
+ parmove_step src dst sig n D is_sep ⊨
+ [ parmove1: R_parmove_step_true src dst sig n D is_sep,
+ R_parmove_step_false src dst sig n is_sep ].
+#src #dst #sig #n #D #is_sep #Hneq #Hsrc #Hdst #int
+lapply (refl ? (current ? (nth src ? int (niltape ?))))
+cases (current ? (nth src ? int (niltape ?))) in ⊢ (???%→?);
+[ #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 ?))))
+ cases (current ? (nth dst ? int (niltape ?))) in ⊢ (???%→?);
+ [ #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 % ]
+]]]]
+qed.
+
+definition parmove ≝ λsrc,dst,sig,n,D,is_sep.
+ whileTM … (parmove_step src dst sig n D is_sep) parmove1.
+
+definition R_parmoveL ≝
+ λsrc,dst,sig,n,is_sep.λint,outt: Vector (tape sig) (S n).
+ (∀ls,x,xs,rs,sep.
+ nth src ? int (niltape ?) = midtape sig (xs@sep::ls) x rs →
+ (∀c.memb ? c (x::xs) = true → is_sep c = false) → is_sep sep = true →
+ ∀ls0,x0,target,c,rs0.|xs| = |target| →
+ nth dst ? int (niltape ?) = midtape sig (target@c::ls0) x0 rs0 →
+ 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).
+
+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
+ 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 // ]
+ | #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) //
+ >change_vec_change_vec
+ >(change_vec_commute … ?? tc ?? 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) //
+ | >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)
+] ]
+qed.
+
+lemma terminate_copy : ∀src,dst,sig,n,is_sep,t.
+ src ≠ dst → src < S n → dst < S n →
+ copy src dst sig n is_sep ↓ t.
+#src #dst #sig #n #is_sep #t #Hneq #Hsrc #Hdst
+@(terminate_while … (sem_copy_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 //]
+ >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
+ >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH
+ ]
+]
+qed.
+
+lemma sem_copy : ∀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/
+qed.
\ No newline at end of file
projects / helm.git / commitdiff