From 0a8212f3e87b75e8ab47dc853e612a9a3e1d2544 Mon Sep 17 00:00:00 2001 From: Wilmer Ricciotti Date: Fri, 11 May 2012 17:03:27 +0000 Subject: [PATCH] Definition of the structure of the transition table of a simulated Turing Machine. --- matita/matita/lib/turing/universal/tuples.ma | 158 +++++++++++++++---- 1 file changed, 130 insertions(+), 28 deletions(-) diff --git a/matita/matita/lib/turing/universal/tuples.ma b/matita/matita/lib/turing/universal/tuples.ma index 3492f2e2c..2baf597e7 100644 --- a/matita/matita/lib/turing/universal/tuples.ma +++ b/matita/matita/lib/turing/universal/tuples.ma @@ -20,6 +20,25 @@ definition STape ≝ FinProd … FSUnialpha FinBool. definition only_bits ≝ λl. ∀c.memb STape c l = true → is_bit (\fst c) = true. + +definition no_grids ≝ λl. + ∀c.memb STape c l = true → is_grid (\fst c) = false. + +definition no_bars ≝ λl. + ∀c.memb STape c l = true → is_bar (\fst c) = false. + +definition no_marks ≝ λl. + ∀c.memb STape c l = true → is_marked ? c = false. + +definition tuple_TM : nat → list STape → Prop ≝ + λn,t.∃qin,qout,mv,b1,b2. + only_bits qin ∧ only_bits qout ∧ only_bits mv ∧ + |qin| = n ∧ |qout| = n (* ∧ |mv| = ? *) ∧ + t = qin@〈comma,b1〉::qout@〈comma,b2〉::mv. + +inductive table_TM : nat → list STape → Prop ≝ +| ttm_nil : ∀n.table_TM n [] +| ttm_cons : ∀n,t1,T,b.tuple_TM n t1 → table_TM n T → table_TM n (t1@〈bar,b〉::T). (* l0 x* a l1 x0* a0 l2 ------> l0 x a* l1 x0 a0* l2 @@ -58,19 +77,19 @@ definition R_mark_next_tuple ≝ λt1,t2. ∀ls,c,rs1,rs2. (* c non può essere un separatore ... speriamo *) - t1 = midtape ? ls c (rs1@〈grid,false〉::rs2) → - only_bits rs1 → bar_or_grid c = false → + 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 ? (bar::reverse ? rs3@c::ls) 〈d,true〉 (tail ? (rs4@〈grid,false〉::rs2))) + t2 = midtape STape (〈bar,false〉::reverse ? rs3@c::ls) 〈d,true〉 (tail ? (rs4@〈grid,false〉::rs2))) ∨ - (memb ? bar rs1 = false ∧ - t2 = midtape ? (reverse ? rs1@c::ls) 〈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 c = 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 *) @@ -78,39 +97,41 @@ 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 ? is_bar) (mark ?) (nop ?) 1) ????) -[@sem_if // + (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 #Hc + #ls #c #rs1 #rs2 #Hrs #Hrs1 #Hrs1' #Hc cases (Hleft … Hrs) [ * #Hfalse >Hfalse in Hc; #Htf destruct (Htf) - | * #_ #Hta cases (tech_split ? is_bar rs1) - [ #H1 lapply (Hta rs1 grid rs2 (refl ??) ? ?) - [ (* Hrs1, H1 *) @daemon - | (* bar_or_grid grid = true *) @daemon + | * #_ #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 ?) - [ #Hfalse (* grid is not a bar *) @daemon + @False_ind cases (Hcurrent 〈grid,false〉 ?) + [ normalize #Hfalse destruct (Hfalse) | >Hta % ] | * #tb * whd in ⊢ (%→?); #Hcurrent - cases (Hcurrent grid ?) + cases (Hcurrent 〈grid,false〉 ?) [ #_ #Htb whd in ⊢ (%→?); #Houtc %2 % - [ (* H1 *) @daemon + [ @H1 | >Houtc >Htb >Hta % ] | >Hta % ] ] ] | * #rs3 * #c0 * #rs4 * * #Hc0 #Hsplit #Hrs3 % @(ex_intro ?? rs3) @(ex_intro ?? rs4) - lapply (Hta rs3 c0 (rs4@grid::rs2) ???) - [ #x #Hrs3' (* Hrs1, Hrs3, Hsplit *) @daemon - | (* bar → bar_or_grid *) @daemon + 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' @@ -119,15 +140,15 @@ lapply (sem_seq ? (adv_to_mark_r ? bar_or_grid) [ #_ #Htb' >Htb' in Htb; #Htb generalize in match Hsplit; -Hsplit cases rs4 in Hta; - [ >(eq_pair_fst_snd … grid) - #Hta #Hsplit >(Htb … Hta) - >(?:c0 = bar) - [ @(ex_intro ?? (\fst grid)) @(ex_intro ?? (\snd grid)) - % [ % [ % [ (* Hsplit *) @daemon |(*Hrs3*) @daemon ] | % ] | % ] - | (* Hc0 *) @daemon ] + [ #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) + >(?:c0 = 〈bar,false〉) [ @(ex_intro ?? (\fst r5)) @(ex_intro ?? (\snd r5)) % [ % [ % [ (* Hc0, Hsplit *) @daemon | (*Hrs3*) @daemon ] | % ] | % ] | (* Hc0 *) @daemon ] ] | >Hta % ] @@ -138,4 +159,85 @@ lapply (sem_seq ? (adv_to_mark_r ? bar_or_grid) #Hc0 destruct (Hc0) | >Hta % ] ]]]] -qed. \ No newline at end of file +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 = |l2| → |l2| = |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,true〉::l2 = 〈c1,true〉::l3 ∧ + t2 = midtape ? (reverse ? l2@〈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,true〉::l2 ≠ 〈c1,true〉::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,true〉::l2 ≠ 〈c1,true〉::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. \ No newline at end of file -- 2.39.2