From ed36a3d3f81cf251c41bbe1d208376bb0709c600 Mon Sep 17 00:00:00 2001 From: Wilmer Ricciotti Date: Tue, 15 May 2012 09:56:18 +0000 Subject: [PATCH] Added universal machine (mockup) --- matita/matita/lib/turing/universal/copy.ma | 414 +----------------- .../matita/lib/turing/universal/universal.ma | 34 ++ 2 files changed, 42 insertions(+), 406 deletions(-) create mode 100644 matita/matita/lib/turing/universal/universal.ma diff --git a/matita/matita/lib/turing/universal/copy.ma b/matita/matita/lib/turing/universal/copy.ma index 5775eab05..2978b5cc4 100644 --- a/matita/matita/lib/turing/universal/copy.ma +++ b/matita/matita/lib/turing/universal/copy.ma @@ -105,409 +105,11 @@ axiom sem_comp_step : definition copy ≝ whileTM ? copy_step (inr … (inl … (inr … 0))). definition R_copy ≝ λt1,t2. - ∀ls,c,rs.t1 = midtape ? ls 〈c,true〉 rs → - (∀l1,d,l2,l3,l4. - 〈c,false〉::rs = l1@〈d,false〉::l2 → only_bits l1 → is_bit d = false → - ls = l3@l4@〈c0,true〉::l5 → |l4| = |l1@[〈d,false〉]| - - - -axiom no_grids_in_table: ∀n.∀l.table_TM n l → no_grids l. -(* -l0 x* a l1 x0* a0 l2 ------> l0 x a* l1 x0 a0* l2 - ^ ^ - -if current (* x *) = # - then - else if x = 0 - then move_right; ---- - adv_to_mark_r; - if current (* x0 *) = 0 - then advance_mark ---- - adv_to_mark_l; - advance_mark - else STOP - else x = 1 (* analogo *) - -*) - - -(* - MARK NEXT TUPLE machine - (partially axiomatized) - - marks the first character after the first bar (rightwards) - *) - -definition bar_or_grid ≝ λc:STape.is_bar (\fst c) ∨ is_grid (\fst c). - -definition mark_next_tuple ≝ - seq ? (adv_to_mark_r ? bar_or_grid) - (ifTM ? (test_char ? (λc:STape.is_bar (\fst c))) - (move_right_and_mark ?) (nop ?) 1). - -definition R_mark_next_tuple ≝ - λt1,t2. - ∀ls,c,rs1,rs2. - (* c non può essere un separatore ... speriamo *) - t1 = midtape STape ls c (rs1@〈grid,false〉::rs2) → - no_marks rs1 → no_grids rs1 → bar_or_grid c = false → - (∃rs3,rs4,d,b.rs1 = rs3 @ 〈bar,false〉 :: rs4 ∧ - no_bars rs3 ∧ - Some ? 〈d,b〉 = option_hd ? (rs4@〈grid,false〉::rs2) ∧ - t2 = midtape STape (〈bar,false〉::reverse ? rs3@c::ls) 〈d,true〉 (tail ? (rs4@〈grid,false〉::rs2))) - ∨ - (no_bars rs1 ∧ t2 = midtape ? (reverse ? rs1@c::ls) 〈grid,false〉 rs2). - -axiom tech_split : - ∀A:DeqSet.∀f,l. - (∀x.memb A x l = true → f x = false) ∨ - (∃l1,c,l2.f c = true ∧ l = l1@c::l2 ∧ ∀x.memb ? x l1 = true → f x = false). -(*#A #f #l elim l -[ % #x normalize #Hfalse *) - -theorem sem_mark_next_tuple : - Realize ? mark_next_tuple R_mark_next_tuple. -#intape -lapply (sem_seq ? (adv_to_mark_r ? bar_or_grid) - (ifTM ? (test_char ? (λc:STape.is_bar (\fst c))) (move_right_and_mark ?) (nop ?) 1) ????) -[@sem_if [5: // |6: @sem_move_right_and_mark |7: // |*:skip] -| // -|||#Hif cases (Hif intape) -Hif - #j * #outc * #Hloop * #ta * #Hleft #Hright - @(ex_intro ?? j) @ex_intro [|% [@Hloop] ] - -Hloop - #ls #c #rs1 #rs2 #Hrs #Hrs1 #Hrs1' #Hc - cases (Hleft … Hrs) - [ * #Hfalse >Hfalse in Hc; #Htf destruct (Htf) - | * #_ #Hta cases (tech_split STape (λc.is_bar (\fst c)) rs1) - [ #H1 lapply (Hta rs1 〈grid,false〉 rs2 (refl ??) ? ?) - [ * #x #b #Hx whd in ⊢ (??%?); >(Hrs1' … Hx) >(H1 … Hx) % - | % - | -Hta #Hta cases Hright - [ * #tb * whd in ⊢ (%→?); #Hcurrent - @False_ind cases (Hcurrent 〈grid,false〉 ?) - [ normalize #Hfalse destruct (Hfalse) - | >Hta % ] - | * #tb * whd in ⊢ (%→?); #Hcurrent - cases (Hcurrent 〈grid,false〉 ?) - [ #_ #Htb whd in ⊢ (%→?); #Houtc - %2 % - [ @H1 - | >Houtc >Htb >Hta % ] - | >Hta % ] - ] - ] - | * #rs3 * #c0 * #rs4 * * #Hc0 #Hsplit #Hrs3 - % @(ex_intro ?? rs3) @(ex_intro ?? rs4) - lapply (Hta rs3 c0 (rs4@〈grid,false〉::rs2) ???) - [ #x #Hrs3' whd in ⊢ (??%?); >Hsplit in Hrs1;>Hsplit in Hrs3; - #Hrs3 #Hrs1 >(Hrs1 …) [| @memb_append_l1 @Hrs3'|] - >(Hrs3 … Hrs3') @Hrs1' >Hsplit @memb_append_l1 // - | whd in ⊢ (??%?); >Hc0 % - | >Hsplit >associative_append % ] -Hta #Hta - cases Hright - [ * #tb * whd in ⊢ (%→?); #Hta' - whd in ⊢ (%→?); #Htb - cases (Hta' c0 ?) - [ #_ #Htb' >Htb' in Htb; #Htb - generalize in match Hsplit; -Hsplit - cases rs4 in Hta; - [ #Hta #Hsplit >(Htb … Hta) - >(?:c0 = 〈bar,false〉) - [ @(ex_intro ?? grid) @(ex_intro ?? false) - % [ % [ % - [(* Hsplit *) @daemon |(*Hrs3*) @daemon ] | % ] | % ] - | (* Hc0 *) @daemon ] - | #r5 #rs5 >(eq_pair_fst_snd … r5) - #Hta #Hsplit >(Htb … Hta) - >(?:c0 = 〈bar,false〉) - [ @(ex_intro ?? (\fst r5)) @(ex_intro ?? (\snd r5)) - % [ % [ % [ (* Hc0, Hsplit *) @daemon | (*Hrs3*) @daemon ] | % ] - | % ] | (* Hc0 *) @daemon ] ] | >Hta % ] - | * #tb * whd in ⊢ (%→?); #Hta' - whd in ⊢ (%→?); #Htb - cases (Hta' c0 ?) - [ #Hfalse @False_ind >Hfalse in Hc0; - #Hc0 destruct (Hc0) - | >Hta % ] -]]]] -qed. - -definition init_current ≝ - seq ? (adv_to_mark_l ? (is_marked ?)) - (seq ? (clear_mark ?) - (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c))) - (seq ? (move_r ?) (mark ?)))). - -definition R_init_current ≝ λt1,t2. - ∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 → is_grid c = false → - Some ? 〈c0,b0〉 = option_hd ? (reverse ? (〈c,true〉::l2)) → - t1 = midtape STape (l1@〈c,true〉::l2@〈grid,b〉::l3) 〈c1,false〉 rs → - t2 = midtape STape (〈grid,b〉::l3) 〈c0,true〉 - ((tail ? (reverse ? (l1@〈c,false〉::l2))@〈c1,false〉::rs)). - -lemma sem_init_current : Realize ? init_current R_init_current. -#intape -cases (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?)) - (sem_seq ????? (sem_clear_mark ?) - (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c))) - (sem_seq ????? (sem_move_r ?) (sem_mark ?)))) intape) -#k * #outc * #Hloop #HR -@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -cases HR -HR #ta * whd in ⊢ (%→?); #Hta -* #tb * whd in ⊢ (%→?); #Htb -* #tc * whd in ⊢ (%→?); #Htc -* #td * whd in ⊢ (%→%→?); #Htd #Houtc -#l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hc0 #Hintape -cases (Hta … Hintape) [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ] --Hta * #_ #Hta lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%] --Hta #Hta lapply (Htb … Hta) -Htb #Htb cases (Htc … Htb) [ >Hc -Hc * #Hc destruct (Hc) ] --Htc * #_ #Htc lapply (Htc … (refl ??) (refl ??) ?) [@Hl2] --Htc #Htc lapply (Htd … Htc) -Htd ->reverse_append >reverse_cons ->reverse_cons in Hc0; cases (reverse … l2) -[ normalize in ⊢ (%→?); #Hc0 destruct (Hc0) - #Htd >(Houtc … Htd) % -| * #c2 #b2 #tl2 normalize in ⊢ (%→?); - #Hc0 #Htd >(Houtc … Htd) - whd in ⊢ (???%); destruct (Hc0) - >associative_append >associative_append % -] -qed. - -definition match_tuple_step ≝ - ifTM ? (test_char ? (λc:STape.¬ is_grid (\fst c))) - (single_finalTM ? - (seq ? compare - (ifTM ? (test_char ? (λc:STape.is_grid (\fst c))) - (nop ?) - (seq ? mark_next_tuple - (ifTM ? (test_char ? (λc:STape.is_grid (\fst c))) - (mark ?) (seq ? (move_l ?) init_current) tc_true)) tc_true))) - (nop ?) tc_true. - -definition R_match_tuple_step_true ≝ λt1,t2. - ∀ls,c,l1,l2,c1,l3,l4,rs,n. - is_bit c = true → only_bits l1 → no_grids l2 → is_bit c1 = true → - only_bits l3 → n = |l1| → |l1| = |l3| → - table_TM (S n) (〈c1,true〉::l3@〈comma,false〉::l4) → - t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉 - (l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs) → - (* facciamo match *) - (〈c,false〉::l1 = 〈c1,false〉::l3 ∧ - t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉 - (l2@〈bar,false〉::〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs)) - ∨ - (* non facciamo match e marchiamo la prossima tupla *) - ((〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ - ∃c2,l5,l6,l7.l4 = l5@〈bar,false〉::〈c2,false〉::l6@〈comma,false〉::l7 ∧ - (* condizioni su l5 l6 l7 *) - t2 = midtape STape (〈grid,false〉::ls) 〈c,true〉 - (l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈comma,false〉:: - l5@〈bar,false〉::〈c2,true〉::l6@〈comma,false〉::l7)) - ∨ - (* non facciamo match e non c'è una prossima tupla: - non specifichiamo condizioni sul nastro di output, perché - non eseguiremo altre operazioni, quindi il suo formato non ci interessa *) - (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)). - -definition R_match_tuple_step_false ≝ λt1,t2. - ∀ls,c,rs.t1 = midtape STape ls c rs → is_grid (\fst c) = true ∧ t2 = t1. - -include alias "basics/logic.ma". - -(* -lemma eq_f4: ∀A1,A2,A3,A4,B.∀f:A1 → A2 →A3 →A4 →B. - ∀x1,x2,x3,x4,y1,y2,y3,y4. x1 = y1 → x2 = y2 →x3=y3 →x4 = y4 → - f x1 x2 x3 x4 = f y1 y2 y3 y4. -// -qed-. *) - -lemma some_option_hd: ∀A.∀l:list A.∀a.∃b. - Some ? b = option_hd ? (l@[a]) . -#A #l #a cases l normalize /2/ -qed. - -lemma bit_not_grid: ∀d. is_bit d = true → is_grid d = false. -* // normalize #H destruct -qed. - -lemma bit_not_bar: ∀d. is_bit d = true → is_bar d = false. -* // normalize #H destruct -qed. - -axiom sem_match_tuple_step: - accRealize ? match_tuple_step (inr … (inl … (inr … 0))) - R_match_tuple_step_true R_match_tuple_step_false. -(* @(acc_sem_if_app … (sem_test_char ? (λc:STape.¬ is_grid (\fst c))) … - (sem_seq … sem_compare - (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c))) - (sem_nop …) - (sem_seq … sem_mark_next_tuple - (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c))) - (sem_mark ?) (sem_seq … (sem_move_l …) (sem_init_current …)))))) - (sem_nop ?) …) -[(* is_grid: termination case *) - 2:#t1 #t2 #t3 whd in ⊢ (%→?); #H #H1 whd #ls #c #rs #Ht1 - cases (H c ?) [2: >Ht1 %] #Hgrid #Heq % - [@injective_notb @Hgrid | Htapea1 [2:%] - #notgridc -Htapea -Htapea1 -tapea #Htapeb - cases (Hcompare … Htapeb) -Hcompare -Htapeb * #_ #_ #Hcompare - cases (Hcompare c c1 l1 l3 (l2@[〈bar,false〉]) (l4@〈grid,false〉::rs) eqlen … (refl …) Hc ?) - -Hcompare - [* #Htemp destruct (Htemp) #Htapec %1 % [%] - >Htapec in Hor; -Htapec * - [2: * #t3 * whd in ⊢ (%→?); #H @False_ind - cases (H … (refl …)) whd in ⊢ ((??%?)→?); #H destruct (H) - |* #taped * whd in ⊢ (%→?); #Htaped cases (Htaped ? (refl …)) -Htaped * - #Htaped whd in ⊢ (%→?); #Htapeout >Htapeout >Htaped >associative_append - % - ] - |* #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #Htapec - cut (〈c,false〉::l1 ≠ 〈c1,false〉::l3) - [>H2 >H3 elim la - [@(not_to_not …H1) normalize #H destruct % - |#x #tl @not_to_not normalize #H destruct // - ] - ] #Hnoteq %2 - cut (is_bit d' = true) - [cases la in H3; - [normalize in ⊢ (%→?); #H destruct // - |#x #tl #H @(Hl3 〈d',false〉) - normalize in H; destruct @memb_append_l2 @memb_hd - ] - ] #Hd' - >Htapec in Hor; -Htapec * - [* #taped * whd in ⊢ (%→?); #H @False_ind - cases (H … (refl …)) >Hd' #Htemp destruct (Htemp) - |* #taped * whd in ⊢ (%→?); #H cases (H … (refl …)) -H #_ - #Htaped * #tapee * whd in ⊢ (%→?); #Htapee - <(associative_append ? lc (〈comma,false〉::l4)) in Htaped; #Htaped - lapply (Htapee … Htaped ???) -Htaped -Htapee - [whd in ⊢ (??%?); >(bit_not_grid … Hd') >(bit_not_bar … Hd') % - |#x #Hx cases (memb_append … Hx) - [-Hx #Hx @bit_not_grid @Hl3 cases la in H3; normalize - [#H3 destruct (H3) @Hx | #y #tl #H3 destruct (H3) - @memb_append_l2 @memb_cons @Hx ] - |-Hx #Hx @(no_grids_in_table … Htable) - @memb_cons @memb_append_l2 @Hx - ] - |@daemon (* TODO *) - |* - [* #rs3 * * (* we proceed by cases on rs4 *) - [* #d * #b * * * #Heq1 #Hnobars - whd in ⊢ ((???%)→?); #Htemp destruct (Htemp) - #Htapee * - [* #tapef * whd in ⊢ (%→?); >Htapee -Htapee #Htapef - cases (Htapef … (refl …)) -Htapef #_ #Htapef >Htapef -Htapef - whd in ⊢ (%→?); #H lapply (H … ???? (refl …)) #Htapeout - %1 % - [ //| @daemon] - | >Htapeout % - ] - |* #tapef * whd in ⊢ (%→?); >Htapee -Htapee #Htapef - cases (Htapef … (refl …)) whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp) - ] - |* #d2 #b2 #rs3' * #d * #b * * * #Heq1 #Hnobars - cut (is_grid d2 = false) [@daemon (* ??? *)] #Hd2 - whd in ⊢ ((???%)→?); #Htemp destruct (Htemp) #Htapee >Htapee -Htapee * - [* #tapef * whd in ⊢ (%→?); #Htapef - cases (Htapef … (refl …)) >Hd2 #Htemp destruct (Htemp) - |* #tapef * whd in ⊢ (%→?); #Htapef - cases (Htapef … (refl …)) #_ -Htapef #Htapef - * #tapeg >Htapef -Htapef * whd in ⊢ (%→?); - #H lapply (H … (refl …)) whd in ⊢ (???%→?); -H #Htapeg - >Htapeg -Htapeg whd in ⊢ (%→?); #Htapeout - %1 cases (some_option_hd ? (reverse ? (reverse ? la)) 〈c',true〉) - * #c00 #b00 #Hoption - lapply - (Htapeout (reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? (l2@[〈bar,false〉])@(〈grid,false〉::reverse ? lb)) - c' (reverse ? la) false ls bar (〈d2,true〉::rs3'@〈grid,false〉::rs) c00 b00 ?????) -Htapeout - [whd in ⊢ (??(??%??)?); @eq_f3 [2:%|3: %] - >associative_append - generalize in match (〈c',true〉::reverse ? la@〈grid,false〉::ls); #l - whd in ⊢ (???(???%)); >associative_append >associative_append - % - |@daemon - |@daemon - |@daemon - |@daemon - |@daemon - ] - ] - ] - |* #Hnobars #Htapee >Htapee -Htapee * - [whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef - cases (Htapef … (refl …)) -Htapef #_ #Htapef >Htapef -Htapef - whd in ⊢ (%→?); #Htapeout %2 - >(Htapeout … (refl …)) % - [ % - [ @daemon - | @daemon - ] - | % - ] - |whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef - cases (Htapef … (refl …)) -Htapef - whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp) - ] - | - - - - - - - ????? (refl …) Hc ?) -Hcompare - #Hcompare - is_bit c = true → only_bits l1 → no_grids l2 → is_bit c1 = true → - only_bits l3 → n = |l2| → |l2| = |l3| → - table_TM (S n) (〈c1,true〉::l3@〈comma,false〉::l4) →#x - - #intape -cases - (acc_sem_if … (sem_test_char ? (λc:STape.¬ is_grid (\fst c))) - (sem_seq … sem_compare - (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c))) - (sem_nop …) - (sem_seq … sem_mark_next_tuple - (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c))) - (sem_mark ?) (sem_seq … (sem_move_l …) (sem_init_current …)))))) - (sem_nop ?) intape) -#k * #outc * * #Hloop #H1 #H2 -@(ex_intro ?? k) @(ex_intro ?? outc) % -[ % [@Hloop ] ] -Hloop - *) - -(* - MATCH TUPLE - - scrolls through the tuples in the transition table until one matching the - current configuration is found -*) - -definition match_tuple ≝ whileTM ? match_tuple_step (inr … (inl … (inr … 0))). - -definition R_match_tuple ≝ λt1,t2. - ∀ls,c,l1,c1,l2,rs,n. - is_bit c = true → only_bits l1 → is_bit c1 = true → n = |l1| → - table_TM (S n) (〈c1,true〉::l2) → - t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉 - (l1@〈grid,false〉::〈c1,true〉::l2@〈grid,false〉::rs) → - (* facciamo match *) - (∃l3,newc,mv,l4. - 〈c1,false〉::l2 = l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv@l4 ∧ - t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉 - (l3@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv@l4@〈grid,false〉::rs)) - ∨ - (* non facciamo match su nessuna tupla; - non specifichiamo condizioni sul nastro di output, perché - non eseguiremo altre operazioni, quindi il suo formato non ci interessa *) - (current ? t2 = Some ? 〈grid,true〉 ∧ - ∀l3,newc,mv,l4. - 〈c1,false〉::l2 ≠ l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv@l4). + ∀ls,c,rs. ∀l1,d,l2,l3,l4,l5,c0. + t1 = midtape ? ls 〈c,true〉 rs → + (* l1 è la stringa sorgente da copiare, l4@〈c0,true〉 è la sua destinazione *) + 〈c,false〉::rs = l1@〈d,false〉::l2 → only_bits l1 → is_bit d = false → + ls = l3@l4@〈c0,true〉::l5 → |l4@[〈c0,true〉]| = |l1| → + no_marks (l3@l4) → + (∀l4',d'.l4@[〈c0,false〉] = 〈d',false〉::l4' → + t2 = midtape STape (reverse ? l1@l3@〈d',true〉::l4'@l5) 〈d,true〉 l2). \ No newline at end of file diff --git a/matita/matita/lib/turing/universal/universal.ma b/matita/matita/lib/turing/universal/universal.ma new file mode 100644 index 000000000..aa75b326a --- /dev/null +++ b/matita/matita/lib/turing/universal/universal.ma @@ -0,0 +1,34 @@ +(* + ||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_____________________________________________________________*) + + +(* COMPARE BIT + +*) + +include "turing/universal/copy.ma". + +(* + +step : + +init_current; +init_table; +match_tuple; +if is_marked(current) = false (* match *) + then init_current; (* preconditions? *) + adv_to_mark_r; + adv_mark_r; + copy; + ...move... + + +*) \ No newline at end of file -- 2.39.2