--- /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: _ _ _ ... _ ---→ 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.
+
+axiom 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 #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 whd in ⊢ (??(????(???%))?);
+ change with (pmap_vec ???????) in ⊢ (??%?);
+ whd in match (vec_map ?????);
+ >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.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/
+ <(nth_vec_map ?? (current …) src ? int (niltape ?)) //
+ | 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/
+ [ <(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 ??)
+ [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) ∧
+ ((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)
+ | #Hcur_dst #x #xs #rs #Hsrctb #ls0 #x0 #target
+ #rs0 #Hlen #Hdsttb >Hdsttb in Hcur_dst; normalize in ⊢ (%→?); #H destruct (H)
+ ]
+| #td #te * #c0 * #c1 * * #Hc0 #Hc1 #Hd #Hstar #IH #He
+ lapply (IH He) -IH * #IH1 #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
+ >(IH1 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 // ]
+ ]
+ | >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.
\ No newline at end of file
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