X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=inline;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Funistep_aux.ma;h=aa4dbc7ab6a86387ee0a98c85b495f4441f7a9af;hb=2a438f72ba71acd29acaae155d8ef60536140ea5;hp=944674d3ef2559078c33b90edd4b6574dbecd722;hpb=2f8eacc69333b13fe143cc007681f21464e06529;p=helm.git diff --git a/matita/matita/lib/turing/multi_universal/unistep_aux.ma b/matita/matita/lib/turing/multi_universal/unistep_aux.ma index 944674d3e..aa4dbc7ab 100644 --- a/matita/matita/lib/turing/multi_universal/unistep_aux.ma +++ b/matita/matita/lib/turing/multi_universal/unistep_aux.ma @@ -13,6 +13,7 @@ include "turing/multi_universal/moves_2.ma". include "turing/multi_universal/match.ma". include "turing/multi_universal/copy.ma". include "turing/multi_universal/alphabet.ma". +include "turing/multi_universal/tuples.ma". (* @@ -59,7 +60,8 @@ definition obj_to_cfg ≝ mmove cfg FSUnialpha 2 L · (ifTM ?? (inject_TM ? (test_null ?) 2 obj) (copy_step obj cfg FSUnialpha 2 · - mmove cfg FSUnialpha 2 L) + mmove cfg FSUnialpha 2 L · + mmove obj FSUnialpha 2 L) (inject_TM ? (write FSUnialpha null) 2 cfg) tc_true) · inject_TM ? (move_to_end FSUnialpha L) 2 cfg · @@ -70,7 +72,7 @@ definition R_obj_to_cfg ≝ λt1,t2:Vector (tape FSUnialpha) 3. nth cfg ? t1 (niltape ?) = mk_tape FSUnialpha (c::ls) (None ?) [ ] → (∀lso,x,rso.nth obj ? t1 (niltape ?) = midtape FSUnialpha lso x rso → t2 = change_vec ?? t1 - (mk_tape ? [ ] (option_hd ? (reverse ? (c::ls))) (tail ? (reverse ? (c::ls)))) cfg) ∧ + (mk_tape ? [ ] (option_hd ? (reverse ? (x::ls))) (tail ? (reverse ? (x::ls)))) cfg) ∧ (current ? (nth obj ? t1 (niltape ?)) = None ? → t2 = change_vec ?? t1 (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (null::ls))) @@ -103,6 +105,117 @@ qed. lemma sem_obj_to_cfg : obj_to_cfg ⊨ R_obj_to_cfg. @(sem_seq_app FSUnialpha 2 ????? (sem_move_multi ? 2 cfg L ?) + (sem_seq ?????? + (sem_if ?????????? + (sem_test_null_multi ?? obj ?) + (sem_seq ?????? (accRealize_to_Realize … (sem_copy_step …)) + (sem_seq ?????? (sem_move_multi ? 2 cfg L ?) + (sem_move_multi ? 2 obj L ?))) + (sem_inject ???? cfg ? (sem_write FSUnialpha null))) + (sem_seq ?????? (sem_inject ???? cfg ? (sem_move_to_end_l ?)) + (sem_move_multi ? 2 cfg R ?)))) // +#ta #tb * +#tc * whd in ⊢ (%→?); #Htc * +#td * * +[ * #te * * #Hcurtc #Hte + * destruct (Hte) #te * * + [ whd in ⊢ (%→%→?); * #x * #y * * -Hcurtc #Hcurtc1 #Hcurtc2 #Hte + * #tf * whd in ⊢ (%→%→?); #Htf #Htd + * #tg * * * whd in ⊢ (%→%→%→%→?); #Htg1 #Htg2 #Htg3 #Htb + #c #ls #Hta1 % + [ #lso #x0 #rso #Hta2 >Hta1 in Htc; >eq_mk_tape_rightof + whd in match (tape_move ???); #Htc + cut (tg = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[x])) cfg) + [@daemon] -Htg1 -Htg2 -Htg3 #Htg destruct (Htg Htf Hte Htd Htc Htb) + >change_vec_change_vec >change_vec_change_vec + >change_vec_commute // >change_vec_change_vec + >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec + >change_vec_commute // >change_vec_change_vec + >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //] + >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //] + >change_vec_commute [|@sym_not_eq //] @eq_f3 // + [ >Hta2 cases rso in Hta2; whd in match (tape_move_mono ???); + [ #Hta2 whd in match (tape_move ???); tape_move_mk_tape_R [| #_ % %] >reverse_cons + >nth_change_vec_neq in Hcurtc1; [|@sym_not_eq //] >Hta2 + normalize in ⊢ (%→?); #H destruct (H) % + ] + | #Hta2 >Htc in Hcurtc1; >nth_change_vec_neq [| @sym_not_eq //] + >Hta2 #H destruct (H) + ] + | * #Hcurtc0 #Hte #_ #_ #c #ls #Hta1 >Hta1 in Htc; >eq_mk_tape_rightof + whd in match (tape_move ???); #Htc >Htc in Hcurtc0; * + [ >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //] + #Hcurtc #Hcurtc0 >Hcurtc0 in Hcurtc; * #H @False_ind @H % + | >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H) ] + ] +| * #te * * #Hcurtc #Hte + * whd in ⊢ (%→%→?); #Htd1 #Htd2 + * #tf * * * #Htf1 #Htf2 #Htf3 whd in ⊢ (%→?); #Htb + #c #ls #Hta1 % + [ #lso #x #rso #Hta2 >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //] + >Hta2 normalize in ⊢ (%→?); #H destruct (H) + | #_ >Hta1 in Htc; >eq_mk_tape_rightof whd in match (tape_move ???); #Htc + destruct (Hte) cut (td = change_vec ?? tc (midtape ? ls null []) cfg) + [@daemon] -Htd1 -Htd2 #Htd + -Htf1 cut (tf = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[null])) cfg) + [@daemon] -Htf2 -Htf3 #Htf destruct (Htf Htd Htc Htb) + >change_vec_change_vec >change_vec_change_vec >change_vec_change_vec + >change_vec_change_vec >change_vec_change_vec >nth_change_vec // + >reverse_cons >tape_move_mk_tape_R /2/ ] +] +qed. + +definition test_null_char ≝ test_char FSUnialpha (λc.c == null). + +definition R_test_null_char_true ≝ λt1,t2. + current FSUnialpha t1 = Some ? null ∧ t1 = t2. + +definition R_test_null_char_false ≝ λt1,t2. + current FSUnialpha t1 ≠ Some ? null ∧ t1 = t2. + +lemma sem_test_null_char : + test_null_char ⊨ [ tc_true : R_test_null_char_true, R_test_null_char_false]. +#t1 cases (sem_test_char FSUnialpha (λc.c == null) t1) #k * #outc * * #Hloop #Htrue +#Hfalse %{k} %{outc} % [ % +[ @Hloop +| #Houtc cases (Htrue ?) [| @Houtc] * #c * #Hcurt1 #Hcnull lapply (\P Hcnull) + -Hcnull #H destruct (H) #Houtc1 % + [ @Hcurt1 | Hcurt1 in Hc; #Hc lapply (Hc ? (refl ??)) + >(?:((null:FSUnialpha) == null) = true) [|@(\b (refl ??)) ] + #H destruct (H) + | reverse_cons >tape_move_mk_tape_R /2/ ] ] qed. - -lemma wsem_copy : ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n → - copy src dst sig n ⊫ R_copy src dst sig n. -#src #dst #sig #n #Hneq #Hsrc #Hdst #ta #k #outc #Hloop -lapply (sem_while … (sem_copy_step src dst sig n Hneq Hsrc Hdst) … Hloop) // --Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar -[ whd in ⊢ (%→?); * #Hnone #Hout % - [#_ @Hout - |#ls #x #x0 #rs #ls0 #rs0 #Hsrc1 #Hdst1 @False_ind cases Hnone - [>Hsrc1 normalize #H destruct (H) | >Hdst1 normalize #H destruct (H)] - ] -|#tc #td * #x * #y * * #Hcx #Hcy #Htd #Hstar #IH #He lapply (IH He) -IH * - #IH1 #IH2 % - [* [>Hcx #H destruct (H) | >Hcy #H destruct (H)] - |#ls #x' #y' #rs #ls0 #rs0 #Hnth_src #Hnth_dst - >Hnth_src in Hcx; whd in ⊢ (??%?→?); #H destruct (H) - >Hnth_dst in Hcy; whd in ⊢ (??%?→?); #H destruct (H) - >Hnth_src in Htd; >Hnth_dst -Hnth_src -Hnth_dst - cases rs - [(* the source tape is empty after the move *) - #Htd lapply (IH1 ?) - [%1 >Htd >nth_change_vec_neq [2:@(not_to_not … Hneq) //] >nth_change_vec //] - #Hout (* whd in match (tape_move ???); *) %1 %{([])} %{rs0} % - [% [// | // ] - |whd in match (reverse ??); whd in match (reverse ??); - >Hout >Htd @eq_f2 // cases rs0 // - ] - |#c1 #tl1 cases rs0 - [(* the dst tape is empty after the move *) - #Htd lapply (IH1 ?) [%2 >Htd >nth_change_vec //] - #Hout (* whd in match (tape_move ???); *) %2 %{[ ]} %{(c1::tl1)} % - [% [// | // ] - |whd in match (reverse ??); whd in match (reverse ??); - >Hout >Htd @eq_f2 // - ] - |#c2 #tl2 whd in match (tape_move_mono ???); whd in match (tape_move_mono ???); - #Htd - cut (nth src (tape sig) td (niltape sig)=midtape sig (x::ls) c1 tl1) - [>Htd >nth_change_vec_neq [2:@(not_to_not … Hneq) //] @nth_change_vec //] - #Hsrc_td - cut (nth dst (tape sig) td (niltape sig)=midtape sig (x::ls0) c2 tl2) - [>Htd @nth_change_vec //] - #Hdst_td cases (IH2 … Hsrc_td Hdst_td) -Hsrc_td -Hdst_td - [* #rs01 * #rs02 * * #H1 #H2 #H3 %1 - %{(c2::rs01)} %{rs02} % [% [@eq_f //|normalize @eq_f @H2]] - >Htd in H3; >change_vec_commute // >change_vec_change_vec - >change_vec_commute [2:@(not_to_not … Hneq) //] >change_vec_change_vec - #H >reverse_cons >associative_append >associative_append @H - |* #rs11 * #rs12 * * #H1 #H2 #H3 %2 - %{(c1::rs11)} %{rs12} % [% [@eq_f //|normalize @eq_f @H2]] - >Htd in H3; >change_vec_commute // >change_vec_change_vec - >change_vec_commute [2:@(not_to_not … Hneq) //] >change_vec_change_vec - #H >reverse_cons >associative_append >associative_append @H - ] - ] - ] - ] -qed. - - -lemma terminate_copy : ∀src,dst,sig,n,t. - src ≠ dst → src < S n → dst < S n → copy src dst sig n ↓ t. -#src #dst #sig #n #t #Hneq #Hsrc #Hdts -@(terminate_while … (sem_copy_step …)) // -<(change_vec_same … t src (niltape ?)) -cases (nth src (tape sig) t (niltape ?)) -[ % #t1 * #x * #y * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct -|2,3: #a0 #al0 % #t1 * #x * #y * * >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 * #y * * >nth_change_vec // normalize in ⊢ (%→?); - #H1 destruct (H1) #_ >change_vec_change_vec #Ht1 % - #t2 * #x0 * #y0 * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //] - >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H) - |#r0 #rs0 #IH #t #ls #c % #t1 * #x * #y * * >nth_change_vec // - normalize in ⊢ (%→?); #H destruct (H) #Hcur - >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH - ] -] -qed. +*) -lemma sem_copy : ∀src,dst,sig,n. - src ≠ dst → src < S n → dst < S n → - copy src dst sig n ⊨ R_copy src dst sig n. -#i #j #sig #n #Hneq #Hi #Hj @WRealize_to_Realize [/2/| @wsem_copy // ] -qed. +(* macchina che muove il nastro obj a destra o sinistra a seconda del valore + del current di prg, che codifica la direzione in cui ci muoviamo *) + +definition char_to_move ≝ λc.match c with + [ bit b ⇒ if b then R else L + | _ ⇒ N]. + +definition tape_move_obj : mTM FSUnialpha 2 ≝ + ifTM ?? + (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit false)) 2 prg) + (mmove obj FSUnialpha 2 L) + (ifTM ?? + (inject_TM ? (test_char ? (λc:FSUnialpha.c == bit true)) 2 prg) + (mmove obj FSUnialpha 2 R) + (nop ??) + tc_true) + tc_true. + +definition restart_tape ≝ λi. + inject_TM ? (move_to_end FSUnialpha L) 2 i · + mmove i FSUnialpha 2 R. + +definition unistep ≝ + obj_to_cfg · match_m cfg prg FSUnialpha 2 · + restart_tape cfg · copy prg cfg FSUnialpha 2 · + cfg_to_obj · tape_move_obj · restart_tape prg. + +(* +definition legal_tape ≝ λn,l,h,t. + ∃state,char,table. + nth cfg ? t1 (niltape ?) = midtape ? [ ] bar (state@[char]) → + is_config n (bar::state@[char]) → + nth prg ? t1 (niltape ?) = midtape ? [ ] bar table → + bar::table = table_TM n l h → *) + +definition list_of_tape ≝ λsig,t. + left sig t@option_cons ? (current ? t) (right ? t). + +definition low_char' ≝ λc. + match c with + [ None ⇒ null + | Some b ⇒ if (is_bit b) then b else null + ]. + +definition R_unistep ≝ λn,l,h.λt1,t2: Vector ? 3. + ∀state,oldc,table. + (* cfg *) + nth cfg ? t1 (niltape ?) = midtape ? [ ] bar (state@[oldc]) → + is_config n (bar::state@[oldc]) → + (* prg *) + nth prg ? t1 (niltape ?) = midtape ? [ ] bar table → + bar::table = table_TM n l h → + (* obj *) + only_bits (list_of_tape ? (nth obj ? t1 (niltape ?))) → + let char ≝ low_char' (current ? (nth obj ? t1 (niltape ?))) in + let conf ≝ (bar::state@[char]) in + (∃ll,lr.bar::table = ll@conf@lr) → + ∃nstate,nchar,m,t. tuple_encoding n h t = (conf@nstate@[nchar;m]) ∧ + mem ? t l ∧ + t2 = + change_vec ?? + (change_vec ?? t1 (midtape ? [ ] bar (nstate@[nchar])) cfg) + (tape_move_mono ? (nth obj ? t1 (niltape ?)) 〈Some ? nchar,char_to_move m〉) obj. + +definition tape_map ≝ λA,B:FinSet.λf:A→B.λt. + mk_tape B (map ?? f (left ? t)) + (option_map ?? f (current ? t)) + (map ?? f (right ? t)). + +definition low ≝ λM:normalTM.λc:nconfig (no_states M).Vector_of_list ? + [tape_map ?? bit (ctape ?? c); + midtape ? [ ] bar (bits_of_state ? (nhalt M) (cstate ?? c)); + midtape ? [ ] bar (table_TM ? (graph_enum ?? (ntrans M)) (nhalt M)) + ]. + + + + + \ No newline at end of file