From d036ed7ebc9b6ecdefc773b0913980066cc778a2 Mon Sep 17 00:00:00 2001 From: Wilmer Ricciotti Date: Wed, 14 Nov 2012 17:10:33 +0000 Subject: [PATCH] copy machine (multi-tape) completed --- .../matita/lib/turing/multi_universal/copy.ma | 208 +++++++++++++++--- 1 file changed, 178 insertions(+), 30 deletions(-) diff --git a/matita/matita/lib/turing/multi_universal/copy.ma b/matita/matita/lib/turing/multi_universal/copy.ma index a68c22624..e49312026 100644 --- a/matita/matita/lib/turing/multi_universal/copy.ma +++ b/matita/matita/lib/turing/multi_universal/copy.ma @@ -14,6 +14,7 @@ include "turing/turing.ma". include "turing/inject.ma". +include "turing/while_multi.ma". definition copy_states ≝ initN 3. @@ -57,17 +58,23 @@ definition copy_step ≝ mk_mTM sig n copy_states (trans_copy_step src dst sig n is_sep) copy0 (λq.q == copy1 ∨ q == copy2). -definition R_copy_step ≝ +definition R_copy_step_true ≝ λsrc,dst,sig,n,is_sep.λint,outt: Vector (tape sig) (S n). - (∀x1,x2,xls,xrs. - nth src ? int (niltape ?) = midtape sig xls x1 (x2::xrs) → - (is_sep x1 = true ∧ outt = int) ∨ - (is_sep x1 = false ∧ - outt = change_vec ?? - (change_vec ?? int (midtape sig (x1::xls) x2 xrs) src) - (tape_move ? (nth dst ? int (niltape ?)) (Some ? 〈x1,R〉)) dst)) ∧ - (current ? (nth src ? int (niltape ?)) = None ? → - outt = int). + ∃x1. + current ? (nth src ? int (niltape ?)) = Some ? x1 ∧ + is_sep x1 = false ∧ + outt = change_vec ?? + (change_vec ?? int + (tape_move ? (nth src ? int (niltape ?)) (Some ? 〈x1,R〉)) src) + (tape_move ? (nth dst ? int (niltape ?)) (Some ? 〈x1,R〉)) dst. + +definition R_copy_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 ∧ outt = int) ∨ + current ? (nth src ? int (niltape ?)) = None ? ∧ + outt = int. lemma copy_q0_q2_null : ∀src,dst,sig,n,is_sep,v,t.src < S n → dst < S n → @@ -100,39 +107,180 @@ whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2 ] qed. +lemma change_vec_commute : ∀A,n,v,a,b,i,j. i ≠ j → + change_vec A n (change_vec A n v a i) b j + = change_vec A n (change_vec A n v b j) a i. +#A #n #v #a #b #i #j #Hij @(eq_vec … a) +#k #Hk cases (decidable_eq_nat k i) #Hki +[ >Hki >nth_change_vec // >(nth_change_vec_neq ??????? (sym_not_eq … Hij)) + >nth_change_vec // +| cases (decidable_eq_nat k j) #Hkj + [ >Hkj >nth_change_vec // >(nth_change_vec_neq ??????? Hij) >nth_change_vec // + | >(nth_change_vec_neq ??????? (sym_not_eq … Hki)) + >(nth_change_vec_neq ??????? (sym_not_eq … Hkj)) + >(nth_change_vec_neq ??????? (sym_not_eq … Hki)) + >(nth_change_vec_neq ??????? (sym_not_eq … Hkj)) // + ] +] +qed. + lemma copy_q0_q1 : - ∀src,dst,sig,n,is_sep,v,t.src < S n → dst < S n → + ∀src,dst,sig,n,is_sep,v,t.src ≠ dst → src < S n → dst < S n → ∀s.current ? t = Some ? s → is_sep s = false → step sig n (copy_step src dst sig n is_sep) (mk_mconfig ??? copy0 (change_vec ? (S n) v t src)) = mk_mconfig ??? copy1 (change_vec ? (S n) (change_vec ?? v - (tape_move ? (nth src ? v (niltape ?)) (Some ? 〈s,R〉)) src) + (tape_move ? t (Some ? 〈s,R〉)) src) (tape_move ? (nth dst ? v (niltape ?)) (Some ? 〈s,R〉)) dst). -#src #dst #sig #n #is_sep #v #t #Hsrc #Hdst #s #Hcurrent #Hsep +#src #dst #sig #n #is_sep #v #t #Hneq #Hsrc #Hdst #s #Hcurrent #Hsep whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2 [ >current_chars_change_vec // whd in match (trans ????); >nth_change_vec // >Hcurrent whd in ⊢ (??(???%)?); >Hsep % | >current_chars_change_vec // whd in match (trans ????); >nth_change_vec // >Hcurrent whd in ⊢ (??(???????(???%))?); - >Hsep whd in ⊢ (??(???????(???%))?); >pmap_change - (* le due change commutano *) + >Hsep whd in ⊢ (??(???????(???%))?); >change_vec_commute // >pmap_change + >change_vec_commute // @eq_f3 // + <(change_vec_same ?? v dst (niltape ?)) in ⊢(??%?); + >pmap_change @eq_f3 // ] +qed. lemma sem_copy_step : - ∀src,dst,sig,n,is_sep.src < S n → dst < S n → - copy_step src dst sig n is_sep ⊨ R_copy_step src dst sig n is_sep. -#src #dst #sig #n #is_sep #Hsrc #Hdst #int -<(change_vec_same ?? int src (niltape ?)) cases (nth src ? int (niltape ?)) -[ %{2} % [| % - [ whd in ⊢ (??%?); >copy_q0_q2 // - | % // #x1 #x2 #xls #xrs >nth_change_vec // #Hfalse destruct ] ] -| #a #al %{2} % [| % - [ whd in ⊢ (??%?); >copy_q0_q2 // - | % // #x1 #x2 #xls #xrs >nth_change_vec // #Hfalse destruct ] ] -| #a #al %{2} % [| % - [ whd in ⊢ (??%?); >copy_q0_q2 // - | % // #x1 #x2 #xls #xrs >nth_change_vec // #Hfalse destruct ] ] -| #ls #c #rs %{2} % [| % - [ \ No newline at end of file + ∀src,dst,sig,n,is_sep.src ≠ dst → src < S n → dst < S n → + copy_step src dst sig n is_sep ⊨ + [ copy1: R_copy_step_true src dst sig n is_sep, + R_copy_step_false src dst sig n is_sep ]. +#src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst #int +lapply (refl ? (current ? (nth src ? int (niltape ?)))) +cases (current ? (nth src ? int (niltape ?))) in ⊢ (???%→?); +[ #Hcur <(change_vec_same … int src (niltape ?)) %{2} % + [| % [ % + [ whd in ⊢ (??%?); >copy_q0_q2_null /2/ + | normalize in ⊢ (%→?); #H destruct (H) ] + | #_ %2 >nth_change_vec >Hcur // % // ] ] +| #c #Hcur cases (true_or_false (is_sep c)) #Hsep + [ <(change_vec_same … int src (niltape ?)) %{2} % + [| % [ % + [ whd in ⊢ (??%?); >copy_q0_q2_sep /2/ + | normalize in ⊢ (%→?); #H destruct (H) ] + | #_ % >nth_change_vec // %{c} % [ % /2/ | // ] ] ] + | %{2} % [| % [ % + [ whd in ⊢ (??%?); + <(change_vec_same … int src (niltape ?)) in ⊢ (??%?); + >Hcur in ⊢ (??%?); whd in ⊢ (??%?); >(copy_q0_q1 … Hsep) /2/ + | #_ whd %{c} % % /2/ ] + | * #Hfalse @False_ind /2/ ] ] ] ] +qed. + +definition copy ≝ λsrc,dst,sig,n,is_sep. + whileTM … (copy_step src dst sig n is_sep) copy1. + +definition R_copy ≝ + λsrc,dst,sig,n,is_sep.λint,outt: Vector (tape sig) (S n). + (∀ls,x,xs,rs,sep. + nth src ? int (niltape ?) = midtape sig ls x (xs@sep::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 ls0 x0 (target@c::rs0) → + outt = change_vec ?? + (change_vec ?? int (midtape sig (reverse ? xs@x::ls) sep rs) src) + (midtape sig (reverse ? xs@x::ls0) c rs0) dst) ∧ + (∀c.current ? (nth src ? int (niltape ?)) = Some ? c → is_sep c = true → + outt = int) ∧ + (current ? (nth src ? int (niltape ?)) = None ? → outt = int). + +lemma change_vec_change_vec : ∀A,n,v,a,b,i. + change_vec A n (change_vec A n v a i) b i = change_vec A n v b i. +#A #n #v #a #b #i @(eq_vec … a) #i0 #Hi0 +cases (decidable_eq_nat i i0) #Hii0 +[ >Hii0 >nth_change_vec // >nth_change_vec // +| >nth_change_vec_neq // >nth_change_vec_neq // + >nth_change_vec_neq // ] +qed. + +lemma wsem_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 #ta #k #outc #Hloop +lapply (sem_while … (sem_copy_step src dst sig n 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) + | #c #Hc #Hsepc @Houtc ] + | #_ @Houtc ] + | * #Hcur #Houtc % [ % + [ #ls #x0 #xs #rs #sep #Hsrctc >Hsrctc in Hcur; normalize in ⊢ (%→?); + #Hcur destruct (Hcur) + | #c #Hc #Hsepc @Houtc ] + | #_ @Houtc ] + ] +| #tc #td #te * #c0 * * #Hc0 #Hc0nosep #Hd #Hstar #IH #He + lapply (IH He) -IH * * #IH1 #IH2 #IH3 % [ % + [ #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 >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 #Hd>(IH2 … Hsep) + [ >Hd -Hd >Hxsnil >Htargetnil @(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 // + >(nth_change_vec_neq … src dst) [|@(sym_not_eq … Hneq)] + >nth_change_vec // whd in ⊢ (??%?); % + | cases (decidable_eq_nat i dst) #Hidst + [ >Hidst >nth_change_vec // >nth_change_vec // + >nth_change_vec_neq // >Hdst_tc >Htargetnil % + | >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 // >nth_change_vec // >Hxsnil % ] + |#hd1 #hd2 #tl1 #tl2 #Hxs #Htarget >Hxs >Htarget #Hd + >(IH1 (c0::ls) hd1 tl1 rs sep ?? Hsep (c0::ls0) hd2 tl2 c 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 >associative_append % + | >Hd >nth_change_vec // >nth_change_vec_neq // >Hdst_tc >Htarget // + | >Hxs in Hlen; >Htarget normalize #Hlen destruct (Hlen) // + | Hd >nth_change_vec_neq [|@sym_not_eq //] + >nth_change_vec // >nth_change_vec // ] + ] + | #c #Hc #Hsepc >Hc in Hc0; #Hcc0 destruct (Hcc0) >Hc0nosep in Hsepc; + #H destruct (H) + ] +| #HNone >HNone in Hc0; #Hc0 destruct (Hc0) ] ] +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. \ No newline at end of file -- 2.39.2