X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Funiversal%2Ftuples.ma;h=0faa4eece58844e8925519b42899d97c2b799be3;hb=0460fd3dc2909efe0baa6592281d0cf0527165ff;hp=59c993adaaecccac7477ac8dfd6f3be6620d0c20;hpb=b14ce4adebec4078cf662290a7d611c1d54bf388;p=helm.git diff --git a/matita/matita/lib/turing/universal/tuples.ma b/matita/matita/lib/turing/universal/tuples.ma index 59c993ada..0faa4eece 100644 --- a/matita/matita/lib/turing/universal/tuples.ma +++ b/matita/matita/lib/turing/universal/tuples.ma @@ -19,8 +19,286 @@ include "turing/universal/marks.ma". definition STape ≝ FinProd … FSUnialpha FinBool. definition only_bits ≝ λl. - ∀c.memb STape c l = true → is_bit (\fst c) = true. + ∀c.memb STape c l + = true → is_bit (\fst c) = true. + +definition only_bits_or_nulls ≝ λl. + ∀c.memb STape c l = true → bit_or_null (\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. + +lemma bit_not_grid: ∀d. is_bit d = true → is_grid d = false. +* // normalize #H destruct +qed. + +lemma bit_or_null_not_grid: ∀d. bit_or_null 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. + +lemma bit_or_null_not_bar: ∀d. bit_or_null d = true → is_bar d = false. +* // normalize #H destruct +qed. + +definition mk_tuple ≝ λqin,cin,qout,cout,mv. + 〈bar,false〉 :: qin @ cin :: 〈comma,false〉:: qout @ cout :: 〈comma,false〉 :: [mv]. + +(* by definition, a tuple is not marked *) +definition tuple_TM : nat → list STape → Prop ≝ + λn,t.∃qin,cin,qout,cout,mv. + no_marks qin ∧ no_marks qout ∧ + 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 ∧ + t = mk_tuple qin 〈cin,false〉 qout 〈cout,false〉 〈mv,false〉. + +inductive table_TM (n:nat) : list STape → Prop ≝ +| ttm_nil : table_TM n [] +| ttm_cons : ∀t1,T.tuple_TM n t1 → table_TM n T → table_TM n (t1@T). + +inductive match_in_table (n:nat) (qin:list STape) (cin: STape) + (qout:list STape) (cout:STape) (mv:STape) +: list STape → Prop ≝ +| mit_hd : + ∀tb. + tuple_TM n (mk_tuple qin cin qout cout mv) → + match_in_table n qin cin qout cout mv + (mk_tuple qin cin qout cout mv @tb) +| mit_tl : + ∀qin0,cin0,qout0,cout0,mv0,tb. + tuple_TM n (mk_tuple qin0 cin0 qout0 cout0 mv0) → + match_in_table n qin cin qout cout mv tb → + match_in_table n qin cin qout cout mv + (mk_tuple qin0 cin0 qout0 cout0 mv0@tb). + +axiom tuple_len : ∀n,t.tuple_TM n t → |t| = 2*n+6. +axiom append_eq_tech1 : + ∀A,l1,l2,l3,l4.l1@l2 = l3@l4 → |l1| < |l3| → ∃la:list A.l1@la = l3. +axiom append_eq_tech2 : + ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l4 = false → ∃la:list A.l3 = l1@a::la. +(*axiom list_decompose_cases : + ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → ∃la,lb:list A.l3 = la@a::lb ∨ l4 = la@a::lb. +axiom list_decompose_l : + ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l4 = false → + ∃la,lb.l2 = la@lb ∧ l3 = l1@a::la.*) +axiom list_decompose_r : + ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l3 = false → + ∃la,lb.l1 = la@lb ∧ l4 = lb@a::l2. +(*axiom list_decompose_memb : + ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → |l1| < |l3| → memb A a l3 = true.*) + +lemma table_invert_r : ∀n,t,T. + tuple_TM n t → table_TM n (t@T) → table_TM n T. +#n #t #T #Htuple #Htable inversion Htable +[ cases Htuple #qin * #cin * #qout * #cout * #mv * #_ #Ht >Ht + normalize #Hfalse destruct (Hfalse) +| #t0 #T0 #Htuple0 #Htable0 #_ #Heq + lapply (append_l2_injective ?????? Heq) + [ >(tuple_len … Htuple) >(tuple_len … Htuple0) % ] + -Heq #Heq destruct (Heq) // ] +qed. + +lemma match_in_table_to_tuple : + ∀n,T,qin,cin,qout,cout,mv. + match_in_table n qin cin qout cout mv T → table_TM n T → + tuple_TM n (mk_tuple qin cin qout cout mv). +#n #T #qin #cin #qout #cout #mv #Hmatch elim Hmatch +[ // +| #qin0 #cin0 #qout0 #cout0 #mv0 #tb #Htuple #Hmatch #IH #Htable + @IH @(table_invert_r ???? Htable) @Htuple +] +qed. + +lemma match_in_table_append : + ∀n,T,qin,cin,qout,cout,mv,t. + tuple_TM n t → + match_in_table n qin cin qout cout mv (t@T) → + t = mk_tuple qin cin qout cout mv ∨ match_in_table n qin cin qout cout mv T. +#n #T #qin #cin #qout #cout #mv #t #Ht #Hmatch inversion Hmatch +[ #T0 #H #H1 % >(append_l1_injective … H1) // + >(tuple_len … Ht) >(tuple_len … H) % +| #qin0 #cin0 #qout0 #cout0 #mv0 #T0 #H #H1 #_ #H2 %2 + >(append_l2_injective … H2) // >(tuple_len … Ht) >(tuple_len … H) % +] +qed. + +lemma generic_match_to_match_in_table_tech : + ∀n,t,T0,T1,T2.tuple_TM n t → table_TM n (T1@〈bar,false〉::T2) → + t@T0 = T1@〈bar,false〉::T2 → T1 = [] ∨ ∃T3.T1 = t@T3. +#n #t #T0 #T1 #T2 #Ht cases T1 +[ #_ #_ % % +| normalize #c #T1c #Htable #Heq %2 + cases Ht in Heq; #qin * #cin * #qout * #cout * #mv ********** + #Hqin1 #Hqout1 #Hqin2 #Hqout2 #Hcin #Hcout #Hmv #Hcoutmv #Hqinlen #Hqoutlen + #Heqt >Heqt whd in ⊢ (??%%→?); #Ht lapply (cons_injective_r ????? Ht) + #Ht' cases (list_decompose_r STape … (sym_eq … Ht') ?) + [ #la * #lb * #HT1c #HT0 %{lb} >HT1c @(eq_f2 ??? (append ?) (c::la)) // + >HT0 in Ht'; >HT1c >associative_append in ⊢ (???%→?); #Ht' + <(append_l1_injective_r … Ht') // <(cons_injective_l ????? Ht) % + |@(noteq_to_eqnot ? true) @(not_to_not … not_eq_true_false) #Hbar @sym_eq + cases (memb_append … Hbar) -Hbar #Hbar + [@(Hqin2 … Hbar) + |cases (orb_true_l … Hbar) -Hbar + [#Hbar lapply (\P Hbar) -Hbar #Hbar destruct (Hbar) @Hcin + |whd in ⊢ ((??%?)→?); #Hbar cases (memb_append … Hbar) -Hbar #Hbar + [@(Hqout2 … Hbar) + |cases (orb_true_l … Hbar) -Hbar + [#Hbar lapply (\P Hbar) -Hbar #Hbar destruct (Hbar) @Hcout + |#Hbar cases (orb_true_l … Hbar) -Hbar + [whd in ⊢ ((??%?)→?); #Hbar @Hbar + |#Hbar lapply (memb_single … Hbar) -Hbar #Hbar destruct (Hbar) @Hmv + ] + ] + ] + ] + ] + ] +qed. + +lemma generic_match_to_match_in_table : + ∀n,T.table_TM n T → + ∀qin,cin,qout,cout,mv.|qin| = n → |qout| = n → + only_bits qin → only_bits qout → + bit_or_null (\fst cin) = true → bit_or_null (\fst cout) = true → + bit_or_null (\fst mv) = true → + ∀t1,t2. + T = (t1@〈bar,false〉::qin@cin::〈comma,false〉::qout@cout::〈comma,false〉::[mv])@t2 → + match_in_table n qin cin qout cout mv T. +#n #T #Htable #qin #cin #qout #cout #mv #Hlenqin #Hlenqout +#Hqinbits #Hqoutbits #Hcin #Hcout #Hmv +elim Htable +[ * [ #t2 normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) + | #c0 #t1 #t2 normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ] +| #tuple #T0 #H1 #Htable0#IH #t1 #t2 #HT cases H1 #qin0 * #cin0 * #qout0 * #cout0 * #mv0 + * * * * * * * * * * + #Hqin0marks #Hqout0marks #Hqin0bits #Hqout0bits #Hcin0 #Hcout0 #Hmv0 #Hcout0mv0 + #Hlenqin0 #Hlenqout0 #Htuple + lapply (generic_match_to_match_in_table_tech n ? T0 t1 + (qin@cin::〈comma,false〉::qout@[cout;〈comma,false〉;mv]@t2) H1) #Htmp + >Htuple in H1; #H1 + lapply (ttm_cons … T0 H1 Htable0) HT + >associative_append normalize >associative_append normalize + >associative_append #Htable cases (Htmp Htable ?) + [ #Ht1 >Htuple in HT; >Ht1 normalize in ⊢ (??%%→?); + >associative_append >associative_append #HT + cut (qin0 = qin ∧ (〈cin0,false〉 = cin ∧ (qout0 = qout ∧ + (〈cout0,false〉 = cout ∧ (〈mv0,false〉 = mv ∧ T0 = t2))))) + [ lapply (cons_injective_r ????? HT) -HT #HT + lapply (append_l1_injective … HT) [ >Hlenqin @Hlenqin0 ] + #Hqin % [ @Hqin ] -Hqin + lapply (append_l2_injective … HT) [ >Hlenqin @Hlenqin0 ] -HT #HT + lapply (cons_injective_l ????? HT) #Hcin % [ @Hcin ] -Hcin + lapply (cons_injective_r ????? HT) -HT #HT + lapply (cons_injective_r ????? HT) -HT + >associative_append >associative_append #HT + lapply (append_l1_injective … HT) [ >Hlenqout @Hlenqout0 ] + #Hqout % [ @Hqout ] -Hqout + lapply (append_l2_injective … HT) [ >Hlenqout @Hlenqout0 ] -HT normalize #HT + lapply (cons_injective_l ????? HT) #Hcout % [ @Hcout ] -Hcout + lapply (cons_injective_r ????? HT) -HT #HT + lapply (cons_injective_r ????? HT) -HT #HT + lapply (cons_injective_l ????? HT) #Hmv % [ @Hmv ] -Hmv + @(cons_injective_r ????? HT) ] + -HT * #Hqin * #Hcin * #Hqout * #Hcout * #Hmv #HT0 + >(?:〈bar,false〉::qin0@(〈cin0,false〉::〈comma,false〉::qout0@ + [〈cout0,false〉;〈comma,false〉;〈mv0,false〉])@T0 = tuple@T0) + [ >Htuple >Hqin >Hqout >Hcin >Hcout >Hmv % // + | >Htuple normalize >associative_append normalize >associative_append + normalize >associative_append % ] + | * #T3 #HT3 >HT3 in HT; >associative_append; >associative_append #HT + lapply (append_l2_injective … HT) // -HT #HT %2 // + @(IH T3 t2) >HT >associative_append % + |>HT >associative_append normalize >associative_append normalize + >associative_append % ] +] +qed. +(* +lemma table_invert_l : ∀n,T0,qin,cin,qout,cout,mv. + table_TM n (mk_tuple qin cin qout cout mv@〈bar,false〉::T0) → + tuple_TM n (mk_tuple qin cin qout cout mv). +#n #T #qin #cin #qout #cout #mv #HT inversion HT +[ change with (append ???) in ⊢ (??(??%?)?→?);cases qin [ #Hfalse | #t0 #ts0 #Hfalse] normalize in Hfalse; destruct (Hfalse) +| #t0 #T0 #Ht0 #HT0 #_ + + +lemma table_invert_r : ∀n,T0,qin,cin,qout,cout,mv. + table n (mk_tuple qin cin qout cout mv@〈bar,false〉::T0) → table n T0. +*) + +lemma no_grids_in_tuple : ∀n,l.tuple_TM n l → no_grids l. +#n #l * #qin * #cin * #qout * #cout * #mv * * * * * * * * * * +#_ #_ #Hqin #Hqout #Hcin #Hcout #Hmv #_ #_ #_ #Hl >Hl +#c #Hc cases (orb_true_l … Hc) -Hc #Hc +[ >(\P Hc) % +| cases (memb_append … Hc) -Hc #Hc +[ @bit_not_grid @(Hqin … Hc) +| cases (orb_true_l … Hc) -Hc #Hc +[ change with (c == 〈cin,false〉 = true) in Hc; >(\P Hc) @bit_or_null_not_grid // +| cases (orb_true_l … Hc) -Hc #Hc +[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) % +| cases (memb_append …Hc) -Hc #Hc +[ @bit_not_grid @(Hqout … Hc) +| cases (orb_true_l … Hc) -Hc #Hc +[ change with (c == 〈cout,false〉 = true) in Hc; >(\P Hc) @bit_or_null_not_grid // +| cases (orb_true_l … Hc) -Hc #Hc +[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) % +| >(memb_single … Hc) @bit_or_null_not_grid @Hmv +]]]]]] +qed. + +lemma no_marks_in_tuple : ∀n,l.tuple_TM n l → no_marks l. +#n #l * #qin * #cin * #qout * #cout * #mv * * * * * * * * * * +#Hqin #Hqout #_ #_ #_ #_ #_ #_ #_ #_ #Hl >Hl +#c #Hc cases (orb_true_l … Hc) -Hc #Hc +[ >(\P Hc) % +| cases (memb_append … Hc) -Hc #Hc +[ @(Hqin … Hc) +| cases (orb_true_l … Hc) -Hc #Hc +[ change with (c == 〈cin,false〉 = true) in Hc; >(\P Hc) % +| cases (orb_true_l … Hc) -Hc #Hc +[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) % +| cases (memb_append … Hc) -Hc #Hc +[ @(Hqout … Hc) +| cases (orb_true_l … Hc) -Hc #Hc +[ change with (c == 〈cout,false〉 = true) in Hc; >(\P Hc) % +| cases (orb_true_l … Hc) -Hc #Hc +[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) % +| >(memb_single … Hc) % +]]]]]] +qed. + +lemma no_grids_in_table: ∀n.∀l.table_TM n l → no_grids l. +#n #l #t elim t + [normalize #c #H destruct + |#t1 #t2 #Ht1 #Ht2 #IH lapply (no_grids_in_tuple … Ht1) -Ht1 #Ht1 #x #Hx + cases (memb_append … Hx) -Hx #Hx + [ @(Ht1 … Hx) + | @(IH … Hx) ] ] +qed. + +lemma no_marks_in_table: ∀n.∀l.table_TM n l → no_marks l. +#n #l #t elim t + [normalize #c #H destruct + |#t1 #t2 #Ht1 #Ht2 #IH lapply (no_marks_in_tuple … Ht1) -Ht1 #Ht1 #x #Hx + cases (memb_append … Hx) -Hx #Hx + [ @(Ht1 … Hx) + | @(IH … Hx) ] ] +qed. + +axiom last_of_table: ∀n,l,b.¬ table_TM n (l@[〈bar,b〉]). + (* l0 x* a l1 x0* a0 l2 ------> l0 x a* l1 x0 a0* l2 ^ ^ @@ -47,37 +325,32 @@ if current (* x *) = # marks the first character after the first bar (rightwards) *) -check unialpha - -axiom is_bar : FinProd … myalpha FinBool → bool. -axiom is_grid : FinProd … myalpha FinBool → bool. -definition bar_or_grid ≝ λc.is_bar c ∨ is_grid c. -axiom bar : FinProd … myalpha FinBool. -axiom grid : FinProd … myalpha FinBool. +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 ? is_bar) - (move_r_and_mark ?) (nop ?) 1). + (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 ? ls c (rs1@grid::rs2) → - memb ? grid rs1 = false → bar_or_grid c = false → - (∃rs3,rs4,d,b.rs1 = rs3 @ bar :: rs4 ∧ - memb ? bar rs3 = false ∧ - Some ? 〈d,b〉 = option_hd ? (rs4@grid::rs2) ∧ - t2 = midtape ? (bar::reverse ? rs3@c::ls) 〈d,true〉 (tail ? (rs4@grid::rs2))) + 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))) ∨ - (memb ? bar rs1 = false ∧ - t2 = midtape ? (reverse ? rs1@c::ls) grid 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 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 *) @@ -85,39 +358,41 @@ 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 ?) 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 #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 in ⊢ (%→?); #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' @@ -126,15 +401,15 @@ 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 % ] @@ -145,4 +420,600 @@ lapply (sem_seq ? (adv_to_mark_r ? bar_or_grid) #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. + +axiom Realize_to_Realize : + ∀alpha,M,R1,R2.(∀t1,t2.R1 t1 t2 → R2 t1 t2) → Realize alpha M R1 → Realize alpha M R2. + +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