From: Andrea Asperti Date: Tue, 12 Jun 2012 15:34:13 +0000 (+0000) Subject: adding match_machines and removing trans_to_tuples X-Git-Tag: make_still_working~1639 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=c31d09808ffd3866e984c009eb8fc6930fa5e7dc;p=helm.git adding match_machines and removing trans_to_tuples --- diff --git a/matita/matita/lib/turing/universal/match_machines.ma b/matita/matita/lib/turing/universal/match_machines.ma new file mode 100644 index 000000000..ea7ab99db --- /dev/null +++ b/matita/matita/lib/turing/universal/match_machines.ma @@ -0,0 +1,728 @@ +(* + ||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/universal/tuples.ma". +include "turing/universal/marks.ma". + +(* +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 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 ?) tc_true). + +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 daemon :∀P:Prop.P. + +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 ?) tc_true) ????) +[@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 in ⊢ (%→?); #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_on_match ≝ + (seq ? (move_l ?) + (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c))) + (seq ? (move_r ?) (mark ?)))). + +definition R_init_current_on_match ≝ λt1,t2. + ∀l1,l2,c,rs. no_grids l1 → is_grid c = false → + t1 = midtape STape (l1@〈c,false〉::〈grid,false〉::l2) 〈grid,false〉 rs → + t2 = midtape STape (〈grid,false〉::l2) 〈c,true〉 ((reverse ? l1)@〈grid,false〉::rs). + +lemma sem_init_current_on_match : + Realize ? init_current_on_match R_init_current_on_match. +#intape +cases (sem_seq ????? (sem_move_l ?) + (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] -Hloop +#l1 #l2 #c #rs #Hl1 #Hc #Hintape +cases HR -HR #ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hintape) -Hta -Hintape +generalize in match Hl1; cases l1 + [#Hl1 whd in ⊢ ((???(??%%%))→?); #Hta + * #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Hta + [* >Hc #Htemp destruct (Htemp) ] + * #_ #Htc lapply (Htc [ ] 〈grid,false〉 ? (refl ??) (refl …) Hl1) + whd in ⊢ ((???(??%%%))→?); -Htc #Htc + * #td * whd in ⊢ (%→?); #Htd lapply (Htd … Htc) -Htc -Htd + whd in ⊢ ((???(??%%%))→?); #Htd + whd in ⊢ (%→?); #Houtc lapply (Houtc … Htd) -Houtc #Houtc + >Houtc % + |#d #tl #Htl whd in ⊢ ((???(??%%%))→?); #Hta + * #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Htb + [* >(Htl … (memb_hd …)) #Htemp destruct (Htemp)] + * #Hd >append_cons #Htb lapply (Htb … (refl ??) (refl …) ?) + [#x #membx cases (memb_append … membx) -membx #membx + [@Htl @memb_cons @membx | >(memb_single … membx) @Hc]]-Htb #Htb + * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htb -Htc + >reverse_append >associative_append whd in ⊢ ((???(??%%%))→?); #Htc + whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc #Houtc + >Houtc >reverse_cons >associative_append % + ] +qed. + +(* +definition init_current_gen ≝ + seq ? (adv_to_mark_l ? (is_marked ?)) + (seq ? (clear_mark ?) + (seq ? (move_l ?) + (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c))) + (seq ? (move_r ?) (mark ?))))). + +definition R_init_current_gen ≝ λt1,t2. + ∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 → + 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_gen : Realize ? init_current_gen R_init_current_gen. +#intape +cases (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?)) + (sem_seq ????? (sem_clear_mark ?) + (sem_seq ????? (sem_move_l ?) + (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] -Hloop +#l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hintape +cases HR -HR #ta * whd in ⊢ (%→?); #Hta cases (Hta … Hintape) -Hta -Hintape + [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ] +* #_ #Hta lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%] -Hta #Hta +* #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta) -Htb -Hta #Htb +* #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb +generalize in match Hc; generalize in match Hl2; cases l2 + [#_ whd in ⊢ ((???%)→?); #Htemp destruct (Htemp) + whd in ⊢ ((???(??%%%))→?); #Htc + * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd + [2: * whd in ⊢ (??%?→?); #Htemp destruct (Htemp) ] + * #_ #Htd >Htd in Htc; -Htd #Htd + * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Htd + >reverse_append >reverse_cons + whd in ⊢ ((???(??%%%))→?); #Hte + whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc + >Houtc % + |#d #tl #Htl #Hc0 whd in ⊢ ((???(??%%%))→?); #Htc + * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd + [* >(Htl … (memb_hd …)) whd in ⊢ (??%?→?); #Htemp destruct (Htemp)] + * #Hd #Htd lapply (Htd … (refl ??) (refl ??) ?) + [#x #membx @Htl @memb_cons @membx] -Htd #Htd + * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Htd + >reverse_append >reverse_cons >reverse_cons + >reverse_cons in Hc0; >reverse_cons cases (reverse ? tl) + [normalize in ⊢ (%→?); #Hc0 destruct (Hc0) #Hte + whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc + >Houtc % + |* #c2 #b2 #tl2 normalize in ⊢ (%→?); #Hc0 destruct (Hc0) + whd in ⊢ ((???(??%%%))→?); #Hte + whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc + >Houtc >associative_append >associative_append >associative_append % + ] + ] +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,cur,rs.t1 = midtape STape ls cur rs → + \fst cur ≠ grid ∧ + (∀ls0,c,l1,l2,c1,l3,l4,rs0,n. + only_bits_or_nulls l1 → no_marks l1 (* → no_grids l2 *) → + bit_or_null c = true → bit_or_null c1 = true → + only_bits_or_nulls l3 → S n = |l1| → |l1| = |l3| → + table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) → + ls = 〈grid,false〉::ls0 → cur = 〈c,true〉 → + rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0 → + (* facciamo match *) + (〈c,false〉::l1 = 〈c1,false〉::l3 ∧ + t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉 + (l2@〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs0)) + ∨ + (* non facciamo match e marchiamo la prossima tupla *) + (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ + ∃c2,l5,l6.l4 = l5@〈bar,false〉::〈c2,false〉::l6 ∧ + (* condizioni su l5 l6 l7 *) + t2 = midtape STape (〈grid,false〉::ls0) 〈c,true〉 + (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉:: + l5@〈bar,false〉::〈c2,true〉::l6@〈grid,false〉::rs0)) + ∨ + (* 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. + +axiom tech_split2 : ∀A,l1,l2,l3,l4,x. + memb A x l1 = false → memb ? x l3 = false → + l1@x::l2 = l3@x::l4 → l1 = l3 ∧ l2 = l4. + +axiom injective_append : ∀A,l.injective … (λx.append A x l). + +lemma sem_match_tuple_step: + accRealize ? match_tuple_step (inr … (inl … (inr … start_nop))) + 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 | Htapea in Hcur; #Hcur cases (Hcur ? (refl ??)) + -Hcur #Hcur #Htapeb % + [ % #Hfalse >Hfalse in Hcur; normalize #Hfalse1 destruct (Hfalse1)] + #ls0 #c #l1 #l2 #c1 #l3 #l4 #rs0 #n #Hl1bitnull #Hl1marks #Hc #Hc1 #Hl3 #eqn + #eqlen #Htable #Hls #Hcur #Hrs -Htapea >Hls in Htapeb; >Hcur >Hrs #Htapeb + cases (Hcompare … Htapeb) -Hcompare -Htapeb * #_ #_ #Hcompare + cases (Hcompare c c1 l1 l3 l2 (l4@〈grid,false〉::rs0) eqlen Hl1bitnull Hl3 Hl1marks … (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 + % + ] + |* #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 + cut (bit_or_null 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 …)) >(bit_or_null_not_grid ? 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 + cases (Htapee … Htaped ???) -Htaped -Htapee + [* #rs3 * * (* we proceed by cases on rs4 *) + [(* rs4 is empty : the case is absurd since the tape + cannot end with a bar *) + * #d * #b * * * #Heq1 @False_ind + cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut + >Hcut in Htable; >H3 >associative_append + normalize >Heq1 Hcut + Hcut >H3 >associative_append @memb_append_l2 + @memb_cons >Heq1 @memb_append_l2 @memb_cons @memb_hd] #d2intable + cut (is_grid d2 = false) + [@(no_grids_in_table … Htable … 〈d2,b2〉 d2intable)] #Hd2 + cut (b2 = false) + [@(no_marks_in_table … Htable … 〈d2,b2〉 d2intable)] #Hb2 + >Hb2 in Heq1; #Heq1 -Hb2 -b2 + whd in ⊢ ((???%)→?); #Htemp destruct (Htemp) #Htapee >Htapee -Htapee * + [(* we know current is not grid *) + * #tapef * whd in ⊢ (%→?); #Htapef + cases (Htapef … (refl …)) >Hd2 #Htemp destruct (Htemp) + |* #tapef * whd in ⊢ (%→?); #Htapef + cases (Htapef … (refl …)) #_ -Htapef #Htapef + * #tapeg >Htapef -Htapef * + (* move_l *) + whd in ⊢ (%→?); + #H lapply (H … (refl …)) whd in ⊢ (???%→?); -H #Htapeg + >Htapeg -Htapeg + (* init_current *) + whd in ⊢ (%→?); #Htapeout + cases (some_option_hd ? (reverse ? (reverse ? la)) 〈c',true〉) + * #c00 #b00 #Hoption + lapply + (Htapeout (reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? l2@(〈grid,false〉::reverse ? lb)) + c' (reverse ? la) false ls0 bar (〈d2,true〉::rs3'@〈grid,false〉::rs0) c00 b00 ?????) -Htapeout + [whd in ⊢ (??(??%??)?); @eq_f3 [2:%|3: %] + >associative_append + generalize in match (〈c',true〉::reverse ? la@〈grid,false〉::ls0); #l + whd in ⊢ (???(???%)); >associative_append >associative_append % + |>reverse_cons @Hoption + |cases la in H2; + [normalize in ⊢ (%→?); #Htemp destruct (Htemp) + @bit_or_null_not_grid @Hc + |#x #tl normalize in ⊢ (%→?); #Htemp destruct (Htemp) + @bit_or_null_not_grid @(Hl1bitnull 〈c',false〉) @memb_append_l2 @memb_hd + ] + |cut (only_bits_or_nulls (la@(〈c',false〉::lb))) + [

(\P eqc0) @Hc |@Hl1bitnull] + |#Hl1' #x #Hx @bit_or_null_not_grid @Hl1' + @memb_append_l1 @daemon + ] + |@daemon] #Htapeout % %2 % // + @(ex_intro … d2) + cut (∃rs32.rs3 = lc@〈comma,false〉::rs32) + [ (*cases (tech_split STape (λc.c == 〈bar,false〉) l4) + [ + | * #l41 * * #cbar #bfalse * #l42 * * #Hbar #Hl4 #Hl41 + @(ex_intro ?? l41) >Hl4 in Heq1; #Heq1 + + cut (sublist … lc l3) + [ #x #Hx cases la in H3; + [ normalize #H3 destruct (H3) @Hx + | #p #la' normalize #Hla' destruct (Hla') + @memb_append_l2 @memb_cons @Hx ] ] #Hsublist*) + @daemon] + * #rs32 #Hrs3 + (* cut + (〈c1,false〉::l3@〈comma,false〉::l4= la@〈d',false〉::rs3@〈bar,false〉::〈d2,b2〉::rs3') + [@daemon] #Hcut *) + cut (l4=rs32@〈bar,false〉::〈d2,false〉::rs3') + [ >Hrs3 in Heq1; @daemon ] #Hl4 + @(ex_intro … rs32) @(ex_intro … rs3') % [@Hl4] + >Htapeout @eq_f2 + [(* by Hoption, H2 *) @daemon + |(*>Hrs3 *)>append_cons + > (?:l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs + = (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@[〈bar,false〉])@〈d2,true〉::rs3'@〈grid,false〉::rs) + [|>associative_append normalize + >associative_append normalize + >associative_append normalize + >associative_append normalize + % ] + >reverse_append >reverse_append >reverse_cons + >reverse_reverse >reverse_cons >reverse_reverse + >reverse_append >reverse_append >reverse_cons + >reverse_reverse >reverse_reverse >reverse_reverse + >(?:(la@[〈c',false〉])@((((lb@[〈grid,false〉])@l2)@la)@[〈d',false〉])@rs3 + =((la@〈c',false〉::lb)@([〈grid,false〉]@l2@la@[〈d',false〉]@rs3))) + [|>associative_append >associative_append + >associative_append >associative_append >associative_append + >associative_append % ] +

Hrs3 + >associative_append >associative_append normalize + >associative_append >associative_append + @eq_f @eq_f @eq_f + >(?:la@(〈d',false〉::lc@〈comma,false〉::rs32)@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs0 = + (la@〈d',false〉::lc)@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs0 ) + [| >associative_append normalize >associative_append % ] +

Htapee -Htapee * + [whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef + cases (Htapef … (refl …)) -Htapef #_ #Htapef >Htapef -Htapef + whd in ⊢ (%→?); #Htapeout %2 % + [% [//] whd #x #Hx @Hnobars @memb_append_l2 @memb_cons // + | >(Htapeout … (refl …)) % ] + |whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef + cases (Htapef … (refl …)) -Htapef + whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp) + ] + |(* no marks in table *) + #x #membx @(no_marks_in_table … Htable) + @memb_append_l2 + cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut + >H3 >associative_append @memb_append_l2 @memb_cons @membx + |(* no grids in table *) + #x #membx @(no_grids_in_table … Htable) + @memb_append_l2 + cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut + >H3 >associative_append @memb_append_l2 @memb_cons @membx + |whd in ⊢ (??%?); >(bit_or_null_not_grid … Hd') >(bit_or_null_not_bar … Hd') % + ] + ] + |#x #membx @(no_marks_in_table … Htable) + @memb_append_l2 @memb_cons @memb_append_l1 @membx + |#x #membx @(no_marks_in_table … Htable) + @memb_append_l1 @membx + |% + ] + ] +qed. + +(* + 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 … start_nop))). + +lemma is_grid_true : ∀c.is_grid c = true → c = grid. +* normalize [ #b ] #H // destruct (H) +qed. + +(* possible variante ? +definition weakR_match_tuple ≝ λt1,t2. + (∀ls,cur,rs,b. t1 = midtape STape ls 〈grid,b〉 rs → t2 = t1) ∧ + (∀c,l1,c1,l2,l3,ls0,rs0,n. + t1 = midtape STape (〈grid,false〉::ls0) 〈bit c,true〉 rs + (l1@〈grid,false〉::l2@〈bit c1,true〉::l3@〈grid,false〉::rs0) → + only_bits_or_nulls l1 → no_marks l1 → S n = |l1| → + table_TM (S n) (l2@〈c1,false〉::l3) → + (* facciamo match *) + (∃l4,newc,mv,l5. + 〈c1,false〉::l3 = l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5 ∧ + t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉 + (l2@l4@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l5@ + 〈grid,false〉::rs0)) + ∨ + (* 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〉 ∧ + ∀l4,newc,mv,l5. + 〈c1,false〉::l3 ≠ l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)). +*) + +definition R_match_tuple0 ≝ λt1,t2. + ∀ls,cur,rs. + t1 = midtape STape ls cur rs → + (is_grid (\fst cur) = true → t2 = t1) ∧ + (∀c,l1,c1,l2,l3,ls0,rs0,n. + ls = 〈grid,false〉::ls0 → + cur = 〈c,true〉 → + rs = l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈grid,false〉::rs0 → + is_bit c = true → is_bit c1 = true → + only_bits_or_nulls l1 → no_marks l1 → S n = |l1| → + table_TM (S n) (l2@〈bar,false〉::〈c1,false〉::l3) → + (* facciamo match *) + (∃l4,newc,mv,l5. + 〈bar,false〉::〈c1,false〉::l3 = l4@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5 ∧ + t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉 + (l2@l4@〈bar,false〉::〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l5@ + 〈grid,false〉::rs0)) + ∨ + (* 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〉 ∧ + ∀l4,newc,mv,l5. + 〈bar,false〉::〈c1,false〉::l3 ≠ l4@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)). + +axiom table_bit_after_bar : + ∀n,l1,c,l2.table_TM n (l1@〈bar,false〉::〈c,false〉::l2) → is_bit c = true. + +lemma wsem_match_tuple : WRealize ? match_tuple R_match_tuple0. +#intape #k #outc #Hloop +lapply (sem_while … sem_match_tuple_step intape k outc Hloop) [%] -Hloop +* #ta * #Hstar @(star_ind_l ??????? Hstar) +[ #tb whd in ⊢ (%→?); #Hleft + #ls #cur #rs #Htb cases (Hleft … Htb) #Hgrid #Houtc % + [ #_ @Houtc + | #c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs + >Hcur in Hgrid; #Hgrid >(is_grid_true … Hgrid) normalize in ⊢ (%→?); + #Hc destruct (Hc) + ] +| (* in the interesting case, we execute a true iteration, then we restart the + while cycle, finally we end with a false iteration *) + #tb #tc #td whd in ⊢ (%→?); #Htc + #Hstar1 #IH whd in ⊢ (%→?); #Hright lapply (IH Hright) -IH whd in ⊢ (%→?); #IH + #ls #cur #rs #Htb % + [ (* cur can't be true because we assume at least one iteration *) + #Hcur cases (Htc … Htb) * #Hfalse @False_ind @Hfalse @(is_grid_true … Hcur) + | (* current and a tuple are marked *) + #c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs #Hc #Hc1 #Hl1bitnull #Hl1marks + #Hl1len #Htable cases (Htc … Htb) -Htc -Htb * #_ #Htc + (* expose the marked tuple in table *) + cut (∃la,lb,mv,lc.l3 = la@〈comma,false〉::lb@〈comma,false〉::mv::lc ∧ + S n = |la| ∧ only_bits_or_nulls la) + [@daemon] * #la * #lb * #mv * #lc * * #Hl3 #Hlalen #Hlabitnull + >Hl3 in Htable; >append_cons #Htable + >(?: l2@〈bar,false〉::〈c1,true〉::l3@〈grid,false〉::rs0 + = (l2@[〈bar,false〉])@〈c1,true〉::la@〈comma,false〉::(lb@〈comma,false〉::mv:: + lc)@〈grid,false〉::rs0) in Hrs; + [| >associative_append normalize >Hl3 + >associative_append normalize % ] #Hrs + cases (Htc ????????? Hl1bitnull Hl1marks ?? Hlabitnull Hl1len ? Htable Hls Hcur Hrs) + [5: Hc1 % + |3: whd in ⊢ (??%?); >Hc % + |-Htc * + [ (* case 1: match successful *) + * #Heq #Htc % %{[]} %{lb} %{mv} %{lc} destruct (Heq) % + [% + | cases (IH … Htc) -IH #Houtc #_ >(Houtc (refl ??)) + >Htc @eq_f normalize >associative_append normalize + >associative_append normalize % + ] + | (* case 2: tuples don't match, we still have other tuples to try *) + * #Hdiff * #c2 * #l5 * #l6 * #Heqlblc #Htc + cases (IH ??? … Htc) -IH #_ #IH + (* by induction hypothesis *) + lapply (IH ? l1 c2 (l2@〈bar,false〉::〈c1,false〉::la@〈comma,false〉::l5) l6 ? rs0 n (refl ??) (refl ??) ???????) + [ generalize in match Htable; + >associative_append normalize + >associative_append normalize >Heqlblc + >associative_append normalize // + | @Hl1len + | @Hl1marks + | @Hl1bitnull + | (*???*) @daemon + | @Hc + | >associative_append normalize + >associative_append normalize + >associative_append % + |-IH * + [ (* the while finally matches a tuple *) + * #l7 * #newc * #mv0 * #l8 * #Hl7l8 #Houtc % + >Heqlblc @(ex_intro ?? (〈bar,false〉::〈c1,false〉::la@〈comma,false〉::l5@l7)) + %{newc} %{mv0} %{l8} % + [ normalize >Hl7l8 >associative_append normalize + >associative_append % + | >Houtc @eq_f >associative_append normalize + >associative_append normalize >associative_append + normalize >associative_append % + ] + | (* the while fails finding a tuple: there are no matches in the whole table *) + * #Houtc #Hdiff1 %2 % + [ @Houtc + | #l50 #newc #mv0 #l51 >Heqlblc + @daemon + ] + ] + ] + ] + | (* match failed and there is no next tuple: the next while cycle will just exit *) + * * #Hdiff #Hnobars generalize in match (refl ? tc); + cases tc in ⊢ (???% → %); + [ #_ normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse) + |2,3: #x #xs #_ normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ] + #ls1 #cur1 #rs1 #Htc normalize in ⊢ (??%?→?); #Hcur1 + cases (IH … Htc) -IH #IH #_ %2 % + [ destruct (Hcur1) >IH [ >Htc % | % ] + | #l4 #newc #mv0 #l5 + (* no_bars except the first one, where the tuple does not match ⇒ + no match *) + @daemon + ] + ] + ] +qed. + +definition R_match_tuple ≝ λt1,t2. + ∀ls,c,l1,c1,l2,rs,n. + is_bit c = true → is_bit c1 = true → + only_bits_or_nulls l1 → no_marks l1 → S n = |l1| → + table_TM (S n) (〈bar,false〉::〈c1,false〉::l2) → + t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉 + (l1@〈grid,false〉::〈bar,false〉::〈c1,true〉::l2@〈grid,false〉::rs) → + (* facciamo match *) + (∃l3,newc,mv,l4. + 〈bar,false〉::〈c1,false〉::l2 = l3@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4 ∧ + t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉 + (l3@〈bar,false〉::〈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. + 〈bar,false〉::〈c1,false〉::l2 ≠ l3@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4). + +(* we still haven't proved termination *) +axiom sem_match_tuple0 : Realize ? match_tuple R_match_tuple0. + +lemma sem_match_tuple : Realize ? match_tuple R_match_tuple. +generalize in match sem_match_tuple0; @Realize_to_Realize +#t1 #t2 #HR #ls #c #l1 #c1 #l2 #rs #n #Hc #Hc1 #Hl1bitsnulls #Hl1marks #Hl1len #Htable #Ht1 +cases (HR … Ht1) -HR #_ #HR +@(HR ??? [] … (refl ??) (refl ??) (refl ??) Hc Hc1 Hl1bitsnulls Hl1marks + Hl1len Htable) +qed. \ No newline at end of file diff --git a/matita/matita/lib/turing/universal/trans_to_tuples.ma b/matita/matita/lib/turing/universal/trans_to_tuples.ma deleted file mode 100644 index 09171b120..000000000 --- a/matita/matita/lib/turing/universal/trans_to_tuples.ma +++ /dev/null @@ -1,389 +0,0 @@ -(* - ||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/universal/tuples.ma". - -(* p < n is represented with a list of bits of lenght n with the - p-th bit from left set to 1 *) - -let rec to_bitlist n p: list bool ≝ - match n with - [ O ⇒ [ ] - | S q ⇒ (eqb p q)::to_bitlist q p]. - -let rec from_bitlist l ≝ - match l with - [ nil ⇒ 0 (* assert false *) - | cons b tl ⇒ if b then |tl| else from_bitlist tl]. - -lemma bitlist_length: ∀n,p.|to_bitlist n p| = n. -#n elim n normalize // -qed. - -lemma bitlist_inv1: ∀n,p.p(not_eq_to_eqb_false … Hpn) normalize @Hind @ltpn - |#Heq >(eq_to_eqb_true … Heq) normalize eqtl @le_n - ] -qed. - -definition nat_of: ∀n. Nat_to n → nat. -#n normalize * #p #_ @p -qed. - -definition bits_of_state ≝ λn.λh:Nat_to n → bool.λs:Nat_to n. - h s::(to_bitlist n (nat_of n s)). - -definition m_bits_of_state ≝ λn.λh.λp. - map ? (unialpha×bool) (λx.〈bit x,false〉) (bits_of_state n h p). - -lemma no_marks_bits_of_state : ∀n,h,p. no_marks (m_bits_of_state n h p). -#n #h #p #x whd in match (m_bits_of_state n h p); -#H cases (orb_true_l … H) -H - [#H >(\P H) % - |elim (to_bitlist n (nat_of n p)) - [whd in ⊢ ((??%?)→?); #H destruct - |#b #l #Hind #H cases (orb_true_l … H) -H #H - [>(\P H) % - |@Hind @H - ] - ] - ] -qed. - -lemma only_bits_bits_of_state : ∀n,h,p. only_bits (m_bits_of_state n h p). -#n #h #p #x whd in match (m_bits_of_state n h p); -#H cases (orb_true_l … H) -H - [#H >(\P H) % - |elim (to_bitlist n (nat_of n p)) - [whd in ⊢ ((??%?)→?); #H destruct - |#b #l #Hind #H cases (orb_true_l … H) -H #H - [>(\P H) % - |@Hind @H - ] - ] - ] -qed. - -definition tuple_type ≝ λn. - (Nat_to n × (option FinBool)) × (Nat_to n × (option (FinBool × move))). - -definition low_action ≝ λaction. - match action with - [ None ⇒ 〈null,null〉 - | Some act ⇒ let 〈na,m〉 ≝ act in - match m with - [ R ⇒ 〈bit na,bit true〉 - | L ⇒ 〈bit na,bit false〉 - | N ⇒ 〈bit na,null〉] - ]. - -definition tuple_of_pair ≝ λn.λh:Nat_to n→bool. - λp:tuple_type n. - let 〈inp,outp〉 ≝ p in - let 〈q,a〉 ≝ inp in - let cin ≝ match a with [ None ⇒ null | Some b ⇒ bit b ] in - let 〈qn,action〉 ≝ outp in - let 〈cout,mv〉 ≝ low_action action in - let qin ≝ m_bits_of_state n h q in - let qout ≝ m_bits_of_state n h qn in - mk_tuple qin 〈cin,false〉 qout 〈cout,false〉 〈mv,false〉. - -definition WFTuple_conditions ≝ - λn,qin,cin,qout,cout,mv. - no_marks qin ∧ no_marks qout ∧ (* queste fuori ? *) - only_bits qin ∧ only_bits qout ∧ - bit_or_null cin = true ∧ bit_or_null cout = true ∧ bit_or_null mv = true ∧ - (cout = null → mv = null) ∧ - |qin| = n ∧ |qout| = n. - -lemma is_tuple: ∀n,h,p. tuple_TM (S n) (tuple_of_pair n h p). -#n #h * * #q #a * #qn #action -@(ex_intro … (m_bits_of_state n h q)) -letin cin ≝ match a with [ None ⇒ null | Some b ⇒ bit b ] -@(ex_intro … cin) -@(ex_intro … (m_bits_of_state n h qn)) -letin cout ≝ - match action with - [ None ⇒ null | Some act ⇒ bit (\fst act)] -@(ex_intro … cout) -letin mv ≝ match action with - [ None ⇒ null - | Some act ⇒ - match \snd act with - [ R ⇒ bit true | L ⇒ bit false | N ⇒ null] - ] -@(ex_intro … mv) -%[%[%[%[%[%[%[% /3/ - |whd in match cin ; cases a // - ] - |whd in match cout; cases action // - ] - |whd in match mv; cases action // - * #b #m cases m // - ] - |whd in match cout; whd in match mv; cases action - [// | #act whd in ⊢ ((??%?)→?); #Hfalse destruct ] - ] - |>length_map normalize @eq_f // - ] - |>length_map normalize @eq_f // - ] - |normalize cases a cases action normalize // - [* #c #m cases m % - |* #c #m #c1 cases m % - ] - ] -qed. - -definition tuple_length ≝ λn.2*n+6. - -lemma length_of_tuple: ∀n,t. tuple_TM n t → - |t| = tuple_length n. -#n #t * #qin * #cin * #qout * #cout * #mv *** #_ -#Hqin #Hqout #eqt >eqt whd in match (mk_tuple ?????); -normalize >length_append >Hqin -Hqin normalize ->length_append normalize >Hqout -Hqout // -qed. - -definition move_eq ≝ λm1,m2:move. - match m1 with - [R ⇒ match m2 with [R ⇒ true | _ ⇒ false] - |L ⇒ match m2 with [L ⇒ true | _ ⇒ false] - |N ⇒ match m2 with [N ⇒ true | _ ⇒ false]]. - -definition tuples_of_pairs ≝ λn.λh.map … (λp.tuple_of_pair n h p). - -definition flatten ≝ λA.foldr (list A) (list A) (append A) []. - -lemma wftable: ∀n,h,l.table_TM (S n) (flatten ? (tuples_of_pairs n h l)). -#n #h #l elim l // -l #a #tl #Hind -whd in match (flatten … (tuples_of_pairs …)); -@ttm_cons // -qed. - -lemma flatten_to_mem: ∀A,n,l,l1,l2.∀a:list A. 0 < n → - (∀x. mem ? x l → |x| = n) → |a| = n → flatten ? l = l1@a@l2 → - (∃q.|l1| = n*q) → mem ? a l. -#A #n #l elim l - [normalize #l1 #l2 #a #posn #Hlen #Ha #Hnil @False_ind - cut (|a|=0) [@daemon] /2/ - |#hd #tl #Hind #l1 #l2 #a #posn #Hlen #Ha - whd in match (flatten ??); #Hflat * #q cases q - [Hlenl11 @Hlen %1 % - |>Hflat >Heql1 >associative_append % - ] - |@(ex_intro …q1) @(injective_plus_r n) - Hl1 // - ] - ] - ] -qed. - -lemma tuple_to_match: ∀n,h,l,qin,cin,qout,cout,mv,p. - p = mk_tuple qin cin qout cout mv - → mem ? p (tuples_of_pairs n h l) → - match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_of_pairs n h l)). -#n #h #l #qin #cin #qout #cout #mv #p -#Hp elim l - [whd in ⊢ (%→?); @False_ind - |#p1 #tl #Hind * - [#H whd in match (tuples_of_pairs ???); - Hp @mit_hd // - |#H whd in match (tuples_of_pairs ???); - cases (is_tuple n h p1) #qin1 * #cin1 * #qout1 * #cout1 * #mv1 - * #_ #Htuplep1 >Htuplep1 @mit_tl // @Hind // - ] - ] -qed. - -axiom match_decomp: ∀n,l,qin,cin,qout,cout,mv. - match_in_table (S n) qin cin qout cout mv l → - ∃l1,l2. l = l1@(mk_tuple qin cin qout cout mv)@l2 ∧ - (∃q.|l1| = (tuple_length (S n))*q) ∧ tuple_TM (S n) (mk_tuple qin cin qout cout mv). -(* -lemma match_tech: ∀n,l,qin,cin,qout,cout,mv. - (∀t. mem ? t l → |t| = |mk_tuple qin cin qout cout mv|) → - match_in_table (S n) qin cin qout cout mv (flatten ? l) → - ∃p. p = mk_tuple qin cin qout cout mv ∧ mem ? p l. -#n #l #qin #cin #qout #cout #mv #Hlen #Hmatch -@(ex_intro … (mk_tuple qin cin qout cout mv)) % // -@flatten_to_mem *) - -lemma match_to_tuple: ∀n,h,l,qin,cin,qout,cout,mv. - match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_of_pairs n h l)) → - ∃p. p = mk_tuple qin cin qout cout mv ∧ mem ? p (tuples_of_pairs n h l). -#n #h #l #qin #cin #qout #cout #mv #Hmatch -@(ex_intro … (mk_tuple qin cin qout cout mv)) % // -cases (match_decomp … Hmatch) #l1 * #l2 * * #Hflat #Hlen #Htuple -@(flatten_to_mem … Hflat … Hlen) - [// - |@daemon - |@(length_of_tuple … Htuple) - ] -qed. - -lemma mem_map: ∀A,B.∀f:A→B.∀l,b. - mem ? b (map … f l) → ∃a. mem ? a l ∧ f a = b. -#A #B #f #l elim l - [#b normalize @False_ind - |#a #tl #Hind #b normalize * - [#eqb @(ex_intro … a) /3/ - |#memb cases (Hind … memb) #a * #mema #eqb - @(ex_intro … a) /3/ - ] - ] -qed. - -lemma match_to_pair: ∀n,h,l,qin,cin,qout,cout,mv. - match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_of_pairs n h l)) → - ∃p. tuple_of_pair n h p = mk_tuple qin cin qout cout mv ∧ mem ? p l. -#n #h #l #qin #cin #qout #cout #mv #Hmatch -cases (match_to_tuple … Hmatch) -#p * #eqp #memb -cases(mem_map … (λp.tuple_of_pair n h p) … memb) -#p1 * #Hmem #H @(ex_intro … p1) % /2/ -qed. - -(* turning DeqMove into a DeqSet *) -lemma move_eq_true:∀m1,m2. - move_eq m1 m2 = true ↔ m1 = m2. -* - [* normalize [% #_ % |2,3: % #H destruct ] - |* normalize [1,3: % #H destruct |% #_ % ] - |* normalize [1,2: % #H destruct |% #_ % ] -qed. - -definition DeqMove ≝ mk_DeqSet move move_eq move_eq_true. - -unification hint 0 ≔ ; - X ≟ DeqMove -(* ---------------------------------------- *) ⊢ - move ≡ carr X. - -unification hint 0 ≔ m1,m2; - X ≟ DeqMove -(* ---------------------------------------- *) ⊢ - move_eq m1 m2 ≡ eqb X m1 m2. - -(* turning DeqMove into a FinSet *) -definition move_enum ≝ [L;R;N]. - -lemma move_enum_unique: uniqueb ? [L;R;N] = true. -// qed. - -lemma move_enum_complete: ∀x:move. memb ? x [L;R;N] = true. -* // qed. - -definition FinMove ≝ - mk_FinSet DeqMove [L;R;N] move_enum_unique move_enum_complete. - -unification hint 0 ≔ ; - X ≟ FinMove -(* ---------------------------------------- *) ⊢ - move ≡ FinSetcarr X. - -definition trans_source ≝ λn.FinProd (initN n) (FinOption FinBool). -definition trans_target ≝ λn.FinProd (initN n) (FinOption (FinProd FinBool FinMove)). - -lemma match_to_trans: - ∀n.∀trans: trans_source n → trans_target n. - ∀h,qin,cin,qout,cout,mv. - match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_of_pairs n h (graph_enum ?? trans))) → - ∃s,t. tuple_of_pair n h 〈s,t〉 = mk_tuple qin cin qout cout mv - ∧ trans s = t. -#n #trans #h #qin #cin #qout #cout #mv #Hmatch -cases (match_to_pair … Hmatch) -Hmatch * #s #t * #Heq #Hmem -@(ex_intro … s) @(ex_intro … t) % // @graph_enum_correct -@mem_to_memb @Hmem -qed. - -(* da spistare *) -lemma mem_map_forward: ∀A,B.∀f:A→B.∀a,l. - mem A a l → mem B (f a) (map ?? f l). - #A #B #f #a #l elim l - [normalize @False_ind - |#b #tl #Hind * - [#eqab (\P eqab) %1 % |#memtl %2 @Hind @memtl] - ] -qed. - -lemma trans_to_match: - ∀n.∀h.∀trans: trans_source n → trans_target n. - ∀inp,outp,qin,cin,qout,cout,mv. trans inp = outp → - tuple_of_pair n h 〈inp,outp〉 = mk_tuple qin cin qout cout mv → - match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_of_pairs n h (graph_enum ?? trans))). -#n #h #trans #inp #outp #qin #cin #qout #cout #mv #Htrans #Htuple -@(tuple_to_match … (refl…)) loop_S_false [2://] whd in match (step FinBool ??); + [>loopM_unfold >loop_S_false [2://] whd in match (step FinBool ??); >(eq_pair_fst_snd ?? (trans ???)) @Hloop |@Houtc1 ] @@ -382,10 +382,13 @@ lapply (sem_while … sem_uni_step1 intape i outc Hloop) qed. theorem sem_universal2: ∀M:normalTM. ∀R. - WRealize ? M R → WRealize ? universalTM (low_R M (start ? M) R). + M ⊫ R → universalTM ⊫ (low_R M (start ? M) R). #M #R #HMR lapply (sem_universal … M (start ? M)) @WRealize_to_WRealize #t1 #t2 whd in ⊢ (%→%); #H #tape1 #Htape1 cases (H ? Htape1) #q * #tape2 * * #HRTM #Hhalt #Ht2 @(ex_intro … q) @(ex_intro … tape2) % [% [@(R_TM_to_R … HRTM) @HMR | //] | //] qed. +axiom terminate_UTM: ∀M:normalTM.∀t. + M ↓ t → universalTM ↓ (low_config M (mk_config ?? (start ? M) t)). +