+++ /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".
-include "turing/while_machine.ma".
-include "turing/simple_machines.ma".
-include "turing/if_machine.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: _ _ _ ... _ ---→ a b c ... z
- ^ ^
-
-0) (x,_) → (x,_)(R,R) → 1
- (None,_) → None 2
-1) (_,_) → None 1
-2) (_,_) → None 2
-
-*)
-
-definition trans_parmove_step ≝
- λsrc,dst,sig,n,D.
- 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 sig n〉
- | Some a0 ⇒ 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) (〈None ?,D〉) src)
- (〈None ?,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.
- mk_mTM sig n parmove_states (trans_parmove_step src dst sig n D)
- parmove0 (λq.q == parmove1 ∨ q == parmove2).
-
-definition R_parmove_step_true ≝
- λsrc,dst,sig,n,D.λint,outt: Vector (tape sig) (S n).
- ∃x1,x2.
- current ? (nth src ? int (niltape ?)) = Some ? x1 ∧
- current ? (nth dst ? int (niltape ?)) = Some ? x2 ∧
- outt = change_vec ??
- (change_vec ?? int
- (tape_move ? (nth src ? int (niltape ?)) D) src)
- (tape_move ? (nth dst ? int (niltape ?)) D) dst.
-
-definition R_parmove_step_false ≝
- λsrc,dst:nat.λsig,n.λint,outt: Vector (tape sig) (S n).
- (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,v.src < S n → dst < S n →
- nth src ? (current_chars ?? v) (None ?) = None ? →
- step sig n (parmove_step src dst sig n D)
- (mk_mconfig ??? parmove0 v) =
- mk_mconfig ??? parmove2 v.
-#src #dst #sig #n #D #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_null_dst :
- ∀src,dst,sig,n,D,v,s.src < S n → dst < S n →
- nth src ? (current_chars ?? v) (None ?) = Some ? s →
- nth dst ? (current_chars ?? v) (None ?) = None ? →
- step sig n (parmove_step src dst sig n D)
- (mk_mconfig ??? parmove0 v) =
- mk_mconfig ??? parmove2 v.
-#src #dst #sig #n #D #v #s #Hsrc #Hdst #Hcursrc #Hcurdst
-whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?);
-@eq_f2
-[ whd in ⊢ (??(???%)?); >Hcursrc whd in ⊢ (??(???%)?); >Hcurdst %
-| whd in ⊢ (??(????(???%))?); >Hcursrc
- whd in ⊢ (??(????(???%))?); >Hcurdst @tape_move_null_action ]
-qed.
-
-lemma parmove_q0_q1 :
- ∀src,dst,sig,n,D,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 →
- step sig n (parmove_step src dst sig n D)
- (mk_mconfig ??? parmove0 v) =
- mk_mconfig ??? parmove1
- (change_vec ? (S n)
- (change_vec ?? v
- (tape_move ? (nth src ? v (niltape ?)) D) src)
- (tape_move ? (nth dst ? v (niltape ?)) D) dst).
-#src #dst #sig #n #D #v #Hneq #Hsrc #Hdst
-#a1 #a2 #Hcursrc #Hcurdst
-whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
-[ whd in match (trans ????);
- >Hcursrc >Hcurdst %
-| whd in match (trans ????);
- >Hcursrc >Hcurdst whd in ⊢ (??(????(???%))?);
- >tape_move_multi_def <(change_vec_same ?? v dst (niltape ?)) in ⊢ (??%?);
- >pmap_change <(change_vec_same ?? v src (niltape ?)) in ⊢(??%?);
- >pmap_change <tape_move_multi_def >tape_move_null_action
- @eq_f2 // >nth_change_vec_neq //
-]
-qed.
-
-lemma sem_parmove_step :
- ∀src,dst,sig,n,D.src ≠ dst → src < S n → dst < S n →
- parmove_step src dst sig n D ⊨
- [ parmove1: R_parmove_step_true src dst sig n D,
- R_parmove_step_false src dst sig n ].
-#src #dst #sig #n #D #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/
- | normalize in ⊢ (%→?); #H destruct (H) ]
- | #_ % // % // ] ]
-| #a #Ha lapply (refl ? (current ? (nth dst ? int (niltape ?))))
- cases (current ? (nth dst ? int (niltape ?))) in ⊢ (???%→?);
- [ #Hcurdst %{2} %
- [| % [ %
- [ whd in ⊢ (??%?); >(parmove_q0_q2_null_dst …) /2/
- | normalize in ⊢ (%→?); #H destruct (H) ]
- | #_ % // %2 // ] ]
- | #b #Hb %{2} %
- [| % [ %
- [whd in ⊢ (??%?); >(parmove_q0_q1 … Hneq Hsrc Hdst ? b ??)
- [2: <(nth_vec_map ?? (current …) dst ? int (niltape ?)) //
- |3: <(nth_vec_map ?? (current …) src ? int (niltape ?)) //
- | // ]
- | #_ %{a} %{b} % // % // ]
- | * #H @False_ind @H % ]
-]]]
-qed.
-
-definition parmove ≝ λsrc,dst,sig,n,D.
- whileTM … (parmove_step src dst sig n D) parmove1.
-
-definition R_parmoveL ≝
- λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
- (∀x,xs,rs.
- nth src ? int (niltape ?) = midtape sig xs x rs →
- ∀ls0,x0,target,rs0.|xs| = |target| →
- nth dst ? int (niltape ?) = midtape sig (target@ls0) x0 rs0 →
- outt = change_vec ??
- (change_vec ?? int (mk_tape sig [] (None ?) (reverse ? xs@x::rs)) src)
- (mk_tape sig (tail ? ls0) (option_hd ? ls0) (reverse ? target@x0::rs0)) dst) ∧
- (∀x,xs,rs.
- nth dst ? int (niltape ?) = midtape sig xs x rs →
- ∀ls0,x0,target,rs0.|xs| = |target| →
- nth src ? int (niltape ?) = midtape sig (target@ls0) x0 rs0 →
- outt = change_vec ??
- (change_vec ?? int (mk_tape sig [] (None ?) (reverse ? xs@x::rs)) dst)
- (mk_tape sig (tail ? ls0) (option_hd ? ls0) (reverse ? target@x0::rs0)) src) ∧
- ((current ? (nth src ? int (niltape ?)) = None ? ∨
- current ? (nth dst ? int (niltape ?)) = None ?) →
- outt = int).
-
-lemma wsem_parmoveL : ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
- parmove src dst sig n L ⊫ R_parmoveL src dst sig n.
-#src #dst #sig #n #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
-lapply (sem_while … (sem_parmove_step src dst sig n L Hneq Hsrc Hdst) … Hloop) //
--Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ whd in ⊢ (%→?); * #H #Houtc % [2: #_ @Houtc ] cases H #Hcurtb
- [ %
- [ #x #xs #rs #Hsrctb >Hsrctb in Hcurtb; normalize in ⊢ (%→?);
- #Hfalse destruct (Hfalse)
- | #x #xs #rs #Hdsttb #ls0 #x0 #target #rs0 #Hlen #Hsrctb >Hsrctb in Hcurtb;
- normalize in ⊢ (%→?); #H destruct (H)
- ]
- | %
- [ #x #xs #rs #Hsrctb #ls0 #x0 #target
- #rs0 #Hlen #Hdsttb >Hdsttb in Hcurtb; normalize in ⊢ (%→?); #H destruct (H)
- | #x #xs #rs #Hdsttb >Hdsttb in Hcurtb; normalize in ⊢ (%→?);
- #Hfalse destruct (Hfalse)
- ]
- ]
-| #td #te * #c0 * #c1 * * #Hc0 #Hc1 #Hd #Hstar #IH #He
- lapply (IH He) -IH * * #IH1a #IH1b #IH2 % [ %
- [ #x #xs #rs #Hsrc_td #ls0 #x0 #target
- #rs0 #Hlen #Hdst_td
- >Hsrc_td in Hc0; normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
- >Hdst_td in Hd; >Hsrc_td @(list_cases2 … Hlen)
- [ #Hxsnil #Htargetnil >Hxsnil >Htargetnil #Hd >IH2
- [2: %1 >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 //
- >(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_td in Hc1; >Htargetnil
- normalize in ⊢ (%→?); #Hc1 destruct (Hc1) cases ls0 //
- | >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)] %
- ]
- ]
- | #hd1 #hd2 #tl1 #tl2 #Hxs #Htarget >Hxs >Htarget #Hd
- >(IH1a hd1 tl1 (c0::rs) ? ls0 hd2 tl2 (x0::rs0))
- [ >Hd >(change_vec_commute … ?? td ?? src dst) //
- >change_vec_change_vec
- >(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 //
- | >Hxs in Hlen; >Htarget normalize #Hlen destruct (Hlen) //
- | >Hd >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec // ]
- ]
- | #x #xs #rs #Hdst_td #ls0 #x0 #target
- #rs0 #Hlen #Hsrc_td
- >Hdst_td in Hc0; normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
- >Hsrc_td in Hd; >Hdst_td @(list_cases2 … Hlen)
- [ #Hxsnil #Htargetnil >Hxsnil >Htargetnil #Hd >IH2
- [2: %2 >Hd >nth_change_vec //]
- >Hd -Hd @(eq_vec … (niltape ?))
- #i #Hi cases (decidable_eq_nat i dst) #Hidst
- [ >Hidst >(nth_change_vec_neq … dst src) //
- >nth_change_vec // >nth_change_vec //
- | cases (decidable_eq_nat i src) #Hisrc
- [ >Hisrc >nth_change_vec // >(nth_change_vec_neq …) [|@sym_not_eq //]
- >Hsrc_td in Hc1; >Htargetnil
- normalize in ⊢ (%→?); #Hc1 destruct (Hc1) >nth_change_vec //
- cases ls0 //
- | >nth_change_vec_neq [|@(sym_not_eq … Hidst)]
- >nth_change_vec_neq [|@(sym_not_eq … Hisrc)]
- >nth_change_vec_neq [|@(sym_not_eq … Hisrc)]
- >nth_change_vec_neq [|@(sym_not_eq … Hidst)] %
- ]
- ]
- | #hd1 #hd2 #tl1 #tl2 #Hxs #Htarget >Hxs >Htarget #Hd
- >(IH1b hd1 tl1 (x::rs) ? ls0 hd2 tl2 (x0::rs0))
- [ >Hd >(change_vec_commute … ?? td ?? dst src) [|@sym_not_eq //]
- >change_vec_change_vec
- >(change_vec_commute … ?? td ?? src dst) //
- >change_vec_change_vec
- >reverse_cons >associative_append
- >reverse_cons >associative_append
- >change_vec_commute [|@sym_not_eq //] %
- | >Hd >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
- | >Hxs in Hlen; >Htarget normalize #Hlen destruct (Hlen) //
- | >Hd >nth_change_vec // ]
- ]
- ]
-| >Hc0 >Hc1 * [ #Hc0 destruct (Hc0) | #Hc1 destruct (Hc1) ]
-] ]
-qed.
-
-lemma terminate_parmoveL : ∀src,dst,sig,n,t.
- src ≠ dst → src < S n → dst < S n →
- parmove src dst sig n L ↓ t.
-#src #dst #sig #n #t #Hneq #Hsrc #Hdst
-@(terminate_while … (sem_parmove_step …)) //
-<(change_vec_same … t src (niltape ?))
-cases (nth src (tape sig) t (niltape ?))
-[ % #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 >change_vec_change_vec #Ht1 %
- #t2 * #y1 * #y2 * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H)
- |#l0 #ls0 #IH #t #c #rs % #t1 * #x1 * #x2 * * >nth_change_vec //
- normalize in ⊢ (%→?); #H destruct (H) #Hcurdst
- >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH
- ]
-]
-qed.
-
-lemma sem_parmoveL : ∀src,dst,sig,n.
- src ≠ dst → src < S n → dst < S n →
- parmove src dst sig n L ⊨ R_parmoveL src dst sig n.
-#src #dst #sig #n #Hneq #Hsrc #Hdst @WRealize_to_Realize
-[/2/ | @wsem_parmoveL //]
-qed.
-
-(* while {
- if current != null
- then move_r
- else nop
- }
- *)
-
-definition mte_step ≝ λalpha,D.
-ifTM ? (test_null alpha) (single_finalTM ? (move alpha D)) (nop ?) tc_true.
-
-definition R_mte_step_true ≝ λalpha,D,t1,t2.
- ∃ls,c,rs.
- t1 = midtape alpha ls c rs ∧ t2 = tape_move ? t1 D.
-
-definition R_mte_step_false ≝ λalpha.λt1,t2:tape alpha.
- current ? t1 = None ? ∧ t1 = t2.
-
-definition mte_acc : ∀alpha,D.states ? (mte_step alpha D) ≝
-λalpha,D.(inr … (inl … (inr … start_nop))).
-
-lemma sem_mte_step :
- ∀alpha,D.mte_step alpha D ⊨
- [ mte_acc … : R_mte_step_true alpha D, R_mte_step_false alpha ] .
-#alpha #D #ta
-@(acc_sem_if_app ??????????? (sem_test_null …)
- (sem_move_single …) (sem_nop alpha) ??)
-[ #tb #tc #td * #Hcurtb
- lapply (refl ? (current ? tb)) cases (current ? tb) in ⊢ (???%→?);
- [ #H @False_ind >H in Hcurtb; * /2/ ]
- -Hcurtb #c #Hcurtb #Htb whd in ⊢ (%→?); #Htc whd
- cases (current_to_midtape … Hcurtb) #ls * #rs #Hmidtb
- %{ls} %{c} %{rs} % //
-| #tb #tc #td * #Hcurtb #Htb whd in ⊢ (%→?); #Htc whd % // ]
-qed.
-
-definition move_to_end ≝ λsig,D.whileTM sig (mte_step sig D) (mte_acc …).
-
-definition R_move_to_end_r ≝
- λsig,int,outt.
- (current ? int = None ? → outt = int) ∧
- ∀ls,c,rs.int = midtape sig ls c rs → outt = mk_tape ? (reverse ? rs@c::ls) (None ?) [ ].
-
-lemma wsem_move_to_end_r : ∀sig. move_to_end sig R ⊫ R_move_to_end_r sig.
-#sig #ta #k #outc #Hloop
-lapply (sem_while … (sem_mte_step sig R) … Hloop) //
--Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ * #Hcurtb #Houtc % /2/ #ls #c #rs #Htb >Htb in Hcurtb; normalize in ⊢ (%→?); #H destruct (H)
-| #tc #td * #ls * #c * #rs * #Htc >Htc cases rs
- [ normalize in ⊢ (%→?); #Htd >Htd #Hstar #IH whd in ⊢ (%→?); #Hfalse
- lapply (IH Hfalse) -IH * #Htd1 #_ %
- [ normalize in ⊢ (%→?); #H destruct (H)
- | #ls0 #c0 #rs0 #H destruct (H) >Htd1 // ]
- | #r0 #rs0 whd in ⊢ (???%→?); #Htd >Htd #Hstar #IH whd in ⊢ (%→?); #Hfalse
- lapply (IH Hfalse) -IH * #_ #IH %
- [ normalize in ⊢ (%→?); #H destruct (H)
- | #ls1 #c1 #rs1 #H destruct (H) >reverse_cons >associative_append @IH % ] ] ]
-qed.
-
-lemma terminate_move_to_end_r : ∀sig,t.move_to_end sig R ↓ t.
-#sig #t @(terminate_while … (sem_mte_step sig R …)) //
-cases t
-[ % #t1 * #ls * #c * #rs * #H destruct
-|2,3: #a0 #al0 % #t1 * #ls * #c * #rs * #H destruct
-| #ls #c #rs lapply c -c lapply ls -ls elim rs
- [ #ls #c % #t1 * #ls0 * #c0 * #rs0 * #Hmid #Ht1 destruct %
- #t2 * #ls1 * #c1 * #rs1 * normalize in ⊢ (%→?); #H destruct
- | #r0 #rs0 #IH #ls #c % #t1 * #ls1 * #c1 * #rs1 * #Hmid #Ht1 destruct @IH
- ]
-]
-qed.
-
-lemma sem_move_to_end_r : ∀sig. move_to_end sig R ⊨ R_move_to_end_r sig.
-#sig @WRealize_to_Realize //
-qed.
-
-definition R_move_to_end_l ≝
- λsig,int,outt.
- (current ? int = None ? → outt = int) ∧
- ∀ls,c,rs.int = midtape sig ls c rs → outt = mk_tape ? [ ] (None ?) (reverse ? ls@c::rs).
-
-lemma wsem_move_to_end_l : ∀sig. move_to_end sig L ⊫ R_move_to_end_l sig.
-#sig #ta #k #outc #Hloop
-lapply (sem_while … (sem_mte_step sig L) … Hloop) //
--Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ * #Hcurtb #Houtc % /2/ #ls #c #rs #Htb >Htb in Hcurtb; normalize in ⊢ (%→?); #H destruct (H)
-| #tc #td * #ls * #c * #rs * #Htc >Htc cases ls
- [ normalize in ⊢ (%→?); #Htd >Htd #Hstar #IH whd in ⊢ (%→?); #Hfalse
- lapply (IH Hfalse) -IH * #Htd1 #_ %
- [ normalize in ⊢ (%→?); #H destruct (H)
- | #ls0 #c0 #rs0 #H destruct (H) >Htd1 // ]
- | #l0 #ls0 whd in ⊢ (???%→?); #Htd >Htd #Hstar #IH whd in ⊢ (%→?); #Hfalse
- lapply (IH Hfalse) -IH * #_ #IH %
- [ normalize in ⊢ (%→?); #H destruct (H)
- | #ls1 #c1 #rs1 #H destruct (H) >reverse_cons >associative_append @IH % ] ] ]
-qed.
-
-lemma terminate_move_to_end_l : ∀sig,t.move_to_end sig L ↓ t.
-#sig #t @(terminate_while … (sem_mte_step sig L …)) //
-cases t
-[ % #t1 * #ls * #c * #rs * #H destruct
-|2,3: #a0 #al0 % #t1 * #ls * #c * #rs * #H destruct
-| #ls elim ls
- [ #c #rs % #t1 * #ls0 * #c0 * #rs0 * #Hmid #Ht1 destruct %
- #t2 * #ls1 * #c1 * #rs1 * normalize in ⊢ (%→?); #H destruct
- | #l0 #ls0 #IH #c #rs % #t1 * #ls1 * #c1 * #rs1 * #Hmid #Ht1 destruct @IH
- ]
-]
-qed.
-
-lemma sem_move_to_end_l : ∀sig. move_to_end sig L ⊨ R_move_to_end_l sig.
-#sig @WRealize_to_Realize //
-qed.
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