From c460dc2c764c678778095d3ff210c7915eb07ef1 Mon Sep 17 00:00:00 2001 From: Wilmer Ricciotti Date: Tue, 8 Jan 2013 10:26:00 +0000 Subject: [PATCH] parmove based on end of the tape --- .../lib/turing/multi_universal/moves_2.ma | 260 ++++++++++++++++++ 1 file changed, 260 insertions(+) create mode 100644 matita/matita/lib/turing/multi_universal/moves_2.ma diff --git a/matita/matita/lib/turing/multi_universal/moves_2.ma b/matita/matita/lib/turing/multi_universal/moves_2.ma new file mode 100644 index 000000000..2455ec582 --- /dev/null +++ b/matita/matita/lib/turing/multi_universal/moves_2.ma @@ -0,0 +1,260 @@ +(* + ||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 -- 2.39.2