X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Funiversal%2Fcopy.ma;h=b2fcf29f11e72c744719e499d0363f2b79754637;hb=637ff9311e16f1d58e03d873f84c354e1cf1e716;hp=6b77ef9b7ee8704d0429d65f4faf440af9390d43;hpb=a1607a9e8140658889cb964b004d22cdd8d9f61d;p=helm.git diff --git a/matita/matita/lib/turing/universal/copy.ma b/matita/matita/lib/turing/universal/copy.ma index 6b77ef9b7..b2fcf29f1 100644 --- a/matita/matita/lib/turing/universal/copy.ma +++ b/matita/matita/lib/turing/universal/copy.ma @@ -10,26 +10,7 @@ V_____________________________________________________________*) -(* COMPARE BIT - -*) - -include "turing/universal/tuples.ma". - -definition write_states ≝ initN 2. - -definition write ≝ λalpha,c. - mk_TM alpha write_states - (λp.let 〈q,a〉 ≝ p in - match q with - [ O ⇒ 〈1,Some ? 〈c,N〉〉 - | S _ ⇒ 〈1,None ?〉 ]) - O (λx.x == 1). - -definition R_write ≝ λalpha,c,t1,t2. - ∀ls,x,rs.t1 = midtape alpha ls x rs → t2 = midtape alpha ls c rs. - -axiom sem_write : ∀alpha,c.Realize ? (write alpha c) (R_write alpha c). +include "turing/universal/marks.ma". definition copy_step_subcase ≝ λalpha,c,elseM.ifTM ? (test_char ? (λx.x == 〈c,true〉)) @@ -51,9 +32,55 @@ definition R_copy_step_subcase ≝ (x = c ∧ t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x,false〉::l2) 〈a,true〉 l3) ∨ (x ≠ c ∧ RelseM t1 t2). -axiom sem_copy_step_subcase : - ∀alpha,c,elseM,RelseM. +lemma sem_copy_step_subcase : + ∀alpha,c,elseM,RelseM. Realize ? elseM RelseM → Realize ? (copy_step_subcase alpha c elseM) (R_copy_step_subcase alpha c RelseM). +#alpha #c #elseM #RelseM #HelseM #intape +cases (sem_if ? (test_char ? (λx. x == 〈c,true〉)) ?????? tc_true (sem_test_char ? (λx.x == 〈c,true〉)) + (sem_seq ????? (sem_adv_mark_r alpha) + (sem_seq ????? (sem_move_l …) + (sem_seq ????? (sem_adv_to_mark_l … (is_marked alpha)) + (sem_seq ????? (sem_write ? 〈c,false〉) + (sem_seq ????? (sem_move_r …) + (sem_seq ????? (sem_mark …) + (sem_seq ????? (sem_move_r …) (sem_adv_to_mark_r … (is_marked alpha))))))))) + HelseM intape) +#k * #outc * #Hloop #HR %{k} %{outc} % [@Hloop] -Hloop +#a #l1 #x0 #a0 #l2 #x #l3 #Hintape #Hl1marks cases HR -HR +[ * #ta * whd in ⊢ (%→?); >Hintape * * #c0 * whd in ⊢ (??%?→?); #Hx #Hc #Hta + * #tb * whd in ⊢ (%→?); * #Htb cases (Htb (l1@〈a0,false〉::〈x0,true〉::l2) x) -Htb + #Htb lapply (Htb … Hta) -Htb #Htb #_ #_ + * #tc * whd in ⊢ (%→?); * #_ #Htc lapply (Htc … Htb) -Htb -Htc #Htc + * #td * whd in ⊢ (%→?); * #_ #Htd cases (Htd … Htc) -Htd #_ #Htd cases (Htd (refl ??)) + -Htd #Htd lapply (Htd (l1@[〈a0,false〉]) 〈x0,true〉 l2 ???) // + [#x1 #Hx1 cases (memb_append … Hx1) [ @Hl1marks | #Hsingle >(memb_single … Hsingle) % ] + |whd in ⊢ (??%?); // ] + -Htd #Htd #_ + * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Hte -Htd -Htc #Hte + * #tf * whd in ⊢ (%→?); * #_ #Htf lapply (Htf … Hte) -Hte -Htf >reverse_append #Htf + * #tg * whd in ⊢ (%→?); * #Htg #_ lapply (Htg … Htf) -Htf -Htg >reverse_single #Htg + * #th * whd in ⊢ (%→?); * #_ #Hth lapply (Hth … Htg) -Htg -Hth + generalize in match Hl1marks; -Hl1marks @(list_elim_left … l1) + [ #Hl1marks #Hth whd in ⊢ (%→?); * #_ #Houtc cases (Houtc … Hth) -Houtc + [ * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ] + * * #_ #Houtc #_ lapply (Houtc [] ?? (refl ??) (refl ??) Hl1marks) -Houtc + #Houtc % >(\P Hc) in Hx; #Hx destruct (Hx) % // @Houtc + | -l1 #c1 #l1 #_ #Hl1marks >reverse_append >reverse_single + #Hth whd in ⊢ (%→?); * #_ #Houtc cases (Houtc … Hth) -Houtc + [ * >Hl1marks [ #Hfalse destruct (Hfalse) ] @memb_append_l2 @memb_hd ] + * * #_ #Houtc lapply (Houtc (reverse ? l1@[〈x,false〉]) 〈a,true〉 l3 ? (refl ??) ?) -Houtc + [ #x1 #Hx1 cases (memb_append … Hx1) -Hx1 #Hx1 + [@Hl1marks @memb_append_l1 <(reverse_reverse … l1) @memb_reverse @Hx1 + |>(memb_single … Hx1) % ] + | normalize >associative_append % ] + #Houtc #_ % destruct (Hx) lapply (\P Hc) -Hc #Hc destruct (Hc) % // + >Houtc >reverse_append >reverse_reverse >associative_append >associative_append % ] +| * #ta * whd in ⊢ (%→?); >Hintape * #Hxc #Hta #Helse %2 % + [| Hxc0 in Hxc; #Hxc lapply (Hxc 〈c,true〉 (refl …)) #Hfalse + cases (\Pf Hfalse) #Hfalse0 @Hfalse0 % +] +qed. (* if current = 0,tt @@ -102,27 +129,49 @@ definition R_nocopy_subcase ≝ t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x0,false〉::l2) 〈a,true〉 l3) ∨ (x ≠ null ∧ t2 = t1). -axiom sem_nocopy_subcase : Realize ? nocopy_subcase R_nocopy_subcase. -(* #intape +lemma sem_nocopy_subcase : Realize ? nocopy_subcase R_nocopy_subcase. +#intape cases (sem_if ? (test_char ? (λx:STape.x == 〈null,true〉)) ?????? tc_true - (sem_test_char ? (λx:STape.x == 〈null,true〉)) - (sem_seq … (sem_adv_mark_r …) - (sem_seq … (sem_move_l …) - (sem_seq … (sem_adv_to_mark_l … (is_marked ?)) - (sem_seq … (sem_adv_mark_r …) - (sem_seq … (sem_move_r …) (sem_adv_to_mark_r … (is_marked ?)) - ))))) (sem_nop ?) intape) + (sem_test_char ? (λx:STape.x == 〈null,true〉)) + (sem_seq … (sem_adv_mark_r …) + (sem_seq … (sem_move_l …) + (sem_seq … (sem_adv_to_mark_l … (is_marked ?)) + (sem_seq … (sem_adv_mark_r …) + (sem_seq … (sem_move_r …) + (sem_adv_to_mark_r … (is_marked ?))))))) (sem_nop ?) intape) #k * #outc * #Hloop #HR @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop -cases HR -HR -[| * #ta * whd in ⊢ (%→%→?); #Hta #Houtc - #ls #x #rs #Hintape %2 >Hintape in Hta; #Hta cases (Hta ? (refl ??)) -Hta #Hx #Hta % - [ lapply (\Pf Hx) @not_to_not #Hx' >Hx' % - | Hintape * * #c * whd in ⊢ (??%?→?); #Hc destruct (Hc) #Hx #Hta + * #tb * whd in ⊢ (%→?); * #Htb #_ cases (Htb (l1@〈a0,false〉::〈x0,true〉::l2) x) -Htb #Htb #_ lapply (Htb … Hta) -Hta -Htb #Htb + * #tc * whd in ⊢ (%→?); * #_ #Htc lapply (Htc … Htb) -Htb -Htc #Htc + * #td * whd in ⊢ (%→?); * #_ #Htd cases (Htd … Htc) -Htd #_ #Htd cases (Htd (refl …)) -Htd #Htd #_ + lapply (Htd (l1@[〈a0,false〉]) 〈x0,true〉 l2 ? (refl …) ?) + [#x1 #Hx1 cases (memb_append … Hx1) [@Hl1marks| -Hx1 #Hx1 >(memb_single … Hx1) % ] + |>associative_append % ] -Htd >reverse_append in ⊢ (???%→?); >associative_append in ⊢ (???%→?); #Htd + * #te * whd in ⊢ (%→?); * #Hte cases (Hte l2 x0) -Hte #Hte #_ #_ lapply (Hte … Htd) -Hte -Htd -Htc #Hte + * #tf * whd in ⊢ (%→?); * #_ #Htf lapply (Htf … Hte) -Hte -Htf + generalize in match Hl1marks; -Hl1marks @(list_elim_left … l1) + [ #Hl1marks #Hth whd in ⊢ (%→?); * #_ #Houtc cases (Houtc … Hth) -Houtc + [ * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ] + * * #_ #Houtc lapply (Houtc [] ?? (refl ??) (refl ??) Hl1marks) -Houtc + #Houtc lapply (\P Hx) -Hx #Hx destruct (Hx) #_ % % [%] @Houtc + | -l1 #c1 #l1 #_ #Hl1marks >reverse_append >reverse_single + #Hth whd in ⊢ (%→?); * #_ #Houtc cases (Houtc … Hth) -Houtc + [ * >Hl1marks [ #Hfalse destruct (Hfalse) ] @memb_append_l2 @memb_hd ] + * * #Hc1 #Houtc #_ lapply (Houtc (reverse ? l1@[〈x,false〉]) 〈a,true〉 l3 ? (refl ??) ?) -Houtc + [ #x1 #Hx1 cases (memb_append … Hx1) -Hx1 #Hx1 + [@Hl1marks @memb_append_l1 <(reverse_reverse … l1) @memb_reverse @Hx1 + |>(memb_single … Hx1) % ] + | normalize >associative_append % ] + #Houtc lapply (\P Hx) -Hx #Hx destruct (Hx) % % [%] >Houtc + >reverse_append >reverse_reverse >associative_append >associative_append % ] +| * #ta * whd in ⊢ (%→?); >Hintape * #Hxc + cut (x ≠ null) [ % #Hx cases (\Pf (Hxc ? (refl …))) #Hfalse @Hfalse >Hx % ] -Hxc #Hxnull + #Hta whd in ⊢ (%→?); #Houtc %2 % // Ht1 in H1; + * * #c0 * whd in ⊢ (??%?→?); #Hc0 destruct (Hc0) #Hc #Ht3 % // + #a #l1 #x0 #a0 #l2 #l3 #Hls #Hrs #Hl1marks >Hls in Ht3; >Hrs #Ht3 + cases (H2 … Ht3 ?) + [ * #Hc' #Ht2 % %{false} % // Ht1 in H1; * #Hcur #Ht3 % // @Hcur % ] +qed. (* 1) il primo carattere è marcato @@ -155,7 +230,7 @@ axiom sem_copy_step : 3) il terminatore non è né bit, né null *) -definition copy0 ≝ whileTM ? copy_step (inr … (inl … (inr … 0))). +definition copy0 ≝ whileTM ? copy_step (inr … (inl … (inr … start_nop))). let rec merge_config (l1,l2:list STape) ≝ match l1 with @@ -201,16 +276,23 @@ lemma inj_append_singleton_l2 : >reverse_append >reverse_append normalize #H1 destruct % qed. -lemma length_reverse : ∀A,l.|reverse A l| = |l|. -#A #l elim l // -#a0 #l0 #IH normalize >rev_append_def >length_append >IH normalize // +axiom daemon : ∀P:Prop.P. + +lemma list_cases2_full : + ∀T1,T2:Type[0].∀l1:list T1.∀l2:list T2.∀P:Prop. + length ? l1 = length ? l2 → + (l1 = [] → l2 = [] → P) → + (∀hd1,hd2,tl1,tl2.l1 = hd1::tl1 → l2 = hd2::tl2 → P) → P. +#T1 #T2 #l1 #l2 #P #Hl @(list_ind2 … Hl) +[ #Pnil #Pcons @Pnil // +| #tl1 #tl2 #hd1 #hd2 #IH1 #IH2 #Hp @Hp // ] qed. lemma wsem_copy0 : WRealize ? copy0 R_copy0. #intape #k #outc #Hloop lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop * #ta * #Hstar @(star_ind_l ??????? Hstar) -[ #tb whd in ⊢ (%→?); #Hleft +[ whd in ⊢ (%→?); #Hleft #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits cases (Hleft … Htb) -Hleft #Hc #Houtc % % @@ -220,14 +302,14 @@ lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop #Hl1bits lapply (Hl1bits 〈c',false〉 ?) [ @memb_hd ] >Hc #Hfalse destruct ] | @Houtc ] -| #tb #tc #td whd in ⊢ (%→?→(?→%)→%→?); #Htc #Hstar1 #Hind #Htd +| #tc #td whd in ⊢ (%→?→(?→%)→%→?); #Htc #Hstar1 #Hind #Htd lapply (Hind Htd) -Hind #Hind #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits %2 cases (Htc … Htb) -Htc #Hcbitnull #Htc % [ % #Hc' >Hc' in Hcbitnull; normalize #Hfalse destruct (Hfalse) ] - cut (|l1| = |reverse ? l4|) [//] #Hlen1 - @(list_cases_2 … Hlen1) + cut (|l1| = |reverse ? l4|) [>length_reverse @Hlen] #Hlen1 + @(list_cases2_full … Hlen1) [ (* case l1 = [] is discriminated because l1 contains at least comma *) #Hl1nil @False_ind >Hl1nil in Hl1; cases l1' normalize [ #Hl1 destruct normalize in Hcbitnull; destruct (Hcbitnull) @@ -272,7 +354,7 @@ lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop [|normalize in Hl1; destruct (Hl1) %] >(?:l4 = 〈grid,bg〉::lb) [|@(inj_append_singleton_l1 ?? (〈grid,bg〉::lb) ?? Hl4) ] - >length_append >commutative_plus >length_reverse + >length_append >commutative_plus >length_reverse normalize #Hlalb destruct (Hlalb) // ] #Hlen2 * @@ -301,11 +383,12 @@ lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop (* by IH, we proceed by cases, whether a = comma (consequently several lists = []) or not *) * - [ * #Ha #Houtc1 >Hl1cons in Hl1; >Hla + [ * #Ha #Houtc1 + cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = []) + [ @daemon ] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil + >Hl1cons in Hl1; >Hla >Houtc1 >Htc #Hl1 >Hl4cons in Hl4; >Hlb #Hl4 - cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = []) - [@daemon] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil >Hla1 >Hlb1 >Hl1nil >Hl4nil >Hx cut (a0 = grid) [ @daemon ] #Ha0 associative_append % @@ -352,20 +435,388 @@ lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop ] ] ]]] -qed. +qed. + +definition Pre_copy0 ≝ λt1. + ∃ls,c,c0,rs,l1,l3,l4. + t1 = midtape STape (l3@l4@〈c0,true〉::ls) 〈c,true〉 (l1@rs) ∧ + no_marks l1 ∧ no_marks (l3@l4) ∧ |l1| = |l4| ∧ + (∃l1',bv.〈c,false〉::l1 = l1'@[〈comma,bv〉] ∧ only_bits_or_nulls l1') ∧ + (∃l4',bg.l4@[〈c0,false〉] = 〈grid,bg〉::l4' ∧ only_bits_or_nulls l4'). + +definition Pre_copy ≝ λt1. + ∃ls,s0,s1,c0,c1,rs,l1,l3,l4. + t1 = midtape STape (l3@〈grid,false〉::〈c0,false〉::l4@〈s0,true〉::ls) 〈s1,true〉 (l1@〈c1,false〉::〈comma,false〉::rs) ∧ + no_marks l1 ∧ no_marks l3 ∧ no_marks l4 ∧ |l1| = |l4| ∧ + only_bits (l4@[〈s0,false〉]) ∧ only_bits (〈s1,false〉::l1) ∧ + bit_or_null c0 = true ∧ bit_or_null c1 = true. + +lemma list_last: ∀A.∀l:list A. + l = [ ] ∨ ∃a,l1. l = l1@[a]. +#A #l <(reverse_reverse ? l) cases (reverse A l) + [%1 // + |#a #l1 %2 @(ex_intro ?? a) @(ex_intro ?? (reverse ? l1)) // + ] +qed. + +lemma terminate_copy0 : ∀t.Pre_copy0 t → Terminate ? copy0 t. +#t #HPre +@(terminate_while_guarded ??? + Pre_copy0 … + (acc_Realize_to_acc_GRealize ??? Pre_copy0 … sem_copy_step) + … HPre) [%] + [ -HPre -t #t1 #t2 * #ls * #c * #c0 * #rs * #l1 * #l3 * #l4 + * * * * * #Ht1 #Hl1nomarks #Hl3l4nomarks #Hlen + * #l1' * #bv * #Hl1 #Hbitsnullsl1' * #l4' * #bg * #Hl4 #Hbitsnullsl4' + #HR cases (HR … Ht1) -HR #Hbitnullc + cut (∃d1,l1''.l1 = 〈d1,false〉::l1'') + [ lapply Hl1nomarks cases l1 in Hl1; + [ whd in ⊢ (???%→?); #Hl1 + lapply (append_l2_injective_r ? [] l1' [〈c,false〉] [〈comma,bv〉] (refl ??) Hl1) + #Hc destruct (Hc) normalize in Hbitnullc; destruct (Hbitnullc) + | * #d #bd #l1'' #_ #Hl1nomarks >(?:bd = false) [| @(Hl1nomarks 〈d,bd〉) @memb_hd ] /3/ ] ] + * #d1 * #l1'' #Hl1'' + cut (∃d2,l4''.l4 = l4''@[〈d2,false〉]) + [ lapply Hl4 cases (list_last ? l4) + [ #Hl4' >Hl4' in Hlen; >Hl1'' normalize in ⊢ (%→?); #HFalse destruct (HFalse) + | * * #d2 #bd2 * #l4'' #Hl4'' >Hl4'' in Hl3l4nomarks; #Hl3l4nomarks #_ + <(?:bd2 = false) [| @(Hl3l4nomarks 〈d2,bd2〉) @memb_append_l2 @memb_append_l2 @memb_hd ] + /3/ ] ] + * #d2 * #l4'' #Hl4'' >Hl1'' >Hl4'' + cut (∃l1''',bv0.〈d1,false〉::l1''=l1'''@[〈comma,bv0〉]∧only_bits_or_nulls l1''') + [ Hl1''#HFalse destruct(HFalse) + | * #a * #l1''' #Hl1''' >Hl1''' in Hl1; #Hl1 #_ + lapply (append_l2_injective_r ? (〈c,false〉::l1''') l1' [a] [〈comma,bv〉] (refl …) Hl1) + #Ha destruct (Ha) %{l1'''} %{bv} % // + #x #Hx @Hbitsnullsl1' + lapply (append_l1_injective_r ? (〈c,false〉::l1''') l1' [〈comma,bv〉] [〈comma,bv〉] (refl …) Hl1) + #H Hl4' >Hl4'' cases l4'' + [ normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) + | #y #yl normalize in ⊢ (%→?); #H1 destruct (H1) cases yl in e0; + [ normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) + | #z #zl normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ] ] + | * #a * #l4''' #Hl4''' >Hl4''' #Hl4 + lapply (append_l1_injective_r ? l4 (〈grid,bg〉::l4''') [〈c0,false〉] [a] (refl …) Hl4) + -Hl4 #Hl4 >Hl4 %{l4'''} %{bg} % // + #x #Hx @Hbitsnullsl4' >Hl4''' @memb_append_l1 @Hx ] ] + #Hl4''' #Hl1''' + #HR cases (HR d1 (l3@l4'') c0 d2 ls (l1''@rs) ? ? ?) + [3: >associative_append >associative_append % + |4: % + |5: #x #Hx @Hl3l4nomarks cases (memb_append … Hx) + [ @memb_append_l1 + | >Hl4'' -Hx #Hx @memb_append_l2 @memb_append_l1 @Hx ] + | * #x1 * #Hc #Ht2 whd >Ht2 + %{(〈bit x1,false〉::ls)} %{d1} %{d2} %{rs} %{l1''} %{(〈bit x1,false〉::l3)} %{l4''} + % [ % [ % [ % [ % + [ >associative_append % + | #x #Hx @Hl1nomarks >Hl1'' @memb_cons @Hx ] + | #x #Hx cases (orb_true_l … Hx) -Hx #Hx + [ >(\P Hx) % + | @Hl3l4nomarks cases (memb_append … Hx) + [ @memb_append_l1 + | -Hx #Hx >Hl4'' @memb_append_l2 @memb_append_l1 @Hx ] ] ] + | >Hl1'' in Hlen; >Hl4'' >length_append >commutative_plus + normalize in ⊢ (%→?); #Hlen destruct (Hlen) @e0 ] + | @Hl1''' ] + | @Hl4''' ] + | * #Hc #Ht2 whd >Ht2 + %{(〈c0,false〉::ls)} %{d1} %{d2} %{rs} %{l1''} %{(〈null,false〉::l3)} %{l4''} + % [ % [ % [ % [ % + [ >associative_append % + | #x #Hx @Hl1nomarks >Hl1'' @memb_cons @Hx ] + | #x #Hx cases (orb_true_l … Hx) -Hx #Hx + [ >(\P Hx) % + | @Hl3l4nomarks cases (memb_append … Hx) + [ @memb_append_l1 + | -Hx #Hx >Hl4'' @memb_append_l2 @memb_append_l1 @Hx ] ] ] + | >Hl1'' in Hlen; >Hl4'' >length_append >commutative_plus + normalize in ⊢ (%→?); #Hlen destruct (Hlen) @e0 ] + | @Hl1''' ] + | @Hl4''' ] + ] + |cases HPre -HPre #ls * #c * #c0 * #rs * #l1 * #l3 * #l4 + * * * * * #Ht #Hl1nomarks #Hl3l4nomarks #Hlen + * #l1' * #bv * #Hl1 #Hbitsnullsl1' + * #l4' * #bg * #Hl4 #Hbitsnullsl4' + >Ht lapply Hbitsnullsl1' lapply Hl1 lapply l1' lapply Hl3l4nomarks + lapply Hl1nomarks lapply l3 lapply c0 lapply c lapply ls + -Hbitsnullsl1' -Hl1 -l1' -Hl3l4nomarks -Hl1nomarks -l3 -Hl4 -c0 -c -ls + <(reverse_reverse ? l4) <(length_reverse ? l4) in Hlen; #Hlen + @(list_ind2 … Hlen) + [ #ls #c #c0 #l3 #_ #_ #l1' #Hl1 #Hbitsnullsl1' cases l1' in Hl1; + [| #x * [| #x0 #xs ] normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ] + normalize in ⊢ (%→?); #Hl1' destruct (Hl1') % #t1 whd in ⊢ (%→?); + #HR cases (HR … (refl …)) whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse) + | #xs #ys * #x #bx * #y #by #IH + #ls #c #c0 #l3 #Hl1nomarks #Hl3l4nomarks #l1' #Hl1 + #Hbitsnullsl1' % + #t1 whd in ⊢ (%→?); #HR cases (HR … (refl …)) -HR + #Hbitnullc #HR + lapply (Hl1nomarks 〈x,bx〉 (memb_hd …)) normalize in ⊢ (%→?); #Hbx destruct (Hbx) + (*cut (∃d2,l4''.〈y,by〉::ys = l4''@[〈d2,false〉]) + [ cases (list_last ? ys) in Hl3l4nomarks; + [ #Hl4' #Hl3l4nomarks >Hl4' >(?:by = false) [%{y} %{([])} %] + @(Hl3l4nomarks 〈y,by〉) @memb_append_l2 @memb_hd + | * * #d2 #bd2 * #l4'' #Hl4'' >Hl4'' #Hl3l4nomarks + <(?:bd2 = false) [| @(Hl3l4nomarks 〈d2,bd2〉) @memb_append_l2 @memb_cons @memb_append_l2 @memb_hd ] + %{d2} %{(〈y,by〉::l4'')} % ] ] + * #a0 * #l4'' #Hl4'' >Hl4'' in HR; #HR *) + cases (HR x (l3@reverse ? ys) c0 y ls (xs@rs) ? (refl …) ?) + [3: >reverse_cons >associative_append >associative_append + >(?:by = false) [|@(Hl3l4nomarks 〈y,by〉) @memb_append_l2 >reverse_cons @memb_append_l2 @memb_hd ] % + |4: #x #Hx @Hl3l4nomarks >reverse_cons Ht1 + <(?: (〈bit x1,false〉::l3)@reverse (FSUnialpha×bool) ys@〈y,true〉::〈bit x1,false〉::ls = + 〈bit x1,false〉::(l3@reverse (FSUnialpha×bool) ys)@〈y,true〉::〈bit x1,false〉::ls) + [cut (∃l1''.〈x,false〉::xs = l1''@[〈comma,bv〉] ∧ only_bits_or_nulls l1'') + [lapply Hbitsnullsl1' lapply Hl1 -Hl1 -Hbitsnullsl1' cases l1' + [normalize in ⊢(%→?); #Hfalse destruct (Hfalse) + |#x2 #l1'' normalize in ⊢ (%→?); #Heq #Hbitsnullsl1' destruct (Heq) + %{l1''} % [@e0] + #x #Hx @Hbitsnullsl1' @memb_cons @Hx ] ] + * #l1'' * #Hl1'' #Hbitsnullsl1'' + @(IH (〈bit x1,false〉::ls) x y (〈bit x1,false〉::l3) ?? l1'' Hl1'' Hbitsnullsl1'') + [#x #Hx @Hl1nomarks @memb_cons @Hx + |#x #Hx cases (orb_true_l … Hx) -Hx #Hx + [>(\P Hx) % + |cases (memb_append … Hx) -Hx #Hx @Hl3l4nomarks + [@memb_append_l1 @Hx + |@memb_append_l2 >reverse_cons @memb_append_l1 @Hx ] ] ] + |>associative_append % ] + | * #Hc #Ht1 >Ht1 + <(?: (〈null,false〉::l3)@reverse (FSUnialpha×bool) ys@〈y,true〉::〈c0,false〉::ls = + 〈null,false〉::(l3@reverse (FSUnialpha×bool) ys)@〈y,true〉::〈c0,false〉::ls) + [cut (∃l1''.〈x,false〉::xs = l1''@[〈comma,bv〉] ∧ only_bits_or_nulls l1'') + [lapply Hbitsnullsl1' lapply Hl1 -Hl1 -Hbitsnullsl1' cases l1' + [normalize in ⊢(%→?); #Hfalse destruct (Hfalse) + |#x2 #l1'' normalize in ⊢ (%→?); #Heq #Hbitsnullsl1' destruct (Heq) + %{l1''} % [@e0] + #x #Hx @Hbitsnullsl1' @memb_cons @Hx ] ] + * #l1'' * #Hl1'' #Hbitsnullsl1'' + @(IH (〈c0,false〉::ls) x y (〈null,false〉::l3) ?? l1'' Hl1'' Hbitsnullsl1'') + [#x #Hx @Hl1nomarks @memb_cons @Hx + |#x #Hx cases (orb_true_l … Hx) -Hx #Hx + [>(\P Hx) % + |cases (memb_append … Hx) -Hx #Hx @Hl3l4nomarks + [@memb_append_l1 @Hx + |@memb_append_l2 >reverse_cons @memb_append_l1 @Hx ] ] ] + |>associative_append % ] +]]] +qed. + +definition merge_char ≝ λc1,c2. + match c2 with + [ null ⇒ c1 + | _ ⇒ c2 ]. + +lemma merge_cons : + ∀c1,c2,conf1,conf2. + merge_config (〈c1,false〉::conf1) (〈c2,false〉::conf2) = + 〈merge_char c1 c2,false〉::merge_config conf1 conf2. +#c1 #c2 #conf1 #conf2 normalize @eq_f2 // +cases c2 /2/ +qed. + +lemma merge_bits : ∀l1,l2.|l1| = |l2| → only_bits l2 → merge_config l1 l2 = l2. +#l1 #l2 #Hlen @(list_ind2 … Hlen) // +#tl1 #tl2 #hd1 #hd2 #IH +>(eq_pair_fst_snd … hd1) >(eq_pair_fst_snd … hd2) #Hbits +change with (cons ???) in ⊢ (??%?); @eq_f2 +[ cases (\fst hd2) in Hbits; + [ #b #_ % + |*: #Hfalse lapply (Hfalse … (memb_hd …)) normalize #Hfalse1 destruct (Hfalse1) ] +| @IH #x #Hx @Hbits @memb_cons // ] +qed. + +lemma merge_config_c_nil : + ∀c.merge_config c [] = []. +#c cases c normalize // +qed. + +lemma reverse_merge_config : + ∀c1,c2.|c1| = |c2| → reverse ? (merge_config c1 c2) = + merge_config (reverse ? c1) (reverse ? c2). +#c1 #c2 <(length_reverse ? c1) <(length_reverse ? c2) #Hlen +<(reverse_reverse ? c1) in ⊢ (??%?); <(reverse_reverse ? c2) in ⊢ (??%?); +generalize in match Hlen; @(list_ind2 … Hlen) -Hlen // +#tl1 #tl2 #hd1 #hd2 #IH whd in ⊢ (??%%→?); #Hlen destruct (Hlen) -Hlen +<(length_reverse ? tl1) in e0; <(length_reverse ? tl2) #Hlen +>reverse_cons >reverse_cons >(merge_config_append ???? Hlen) +>reverse_append >(eq_pair_fst_snd ?? hd1) >(eq_pair_fst_snd ?? hd2) +whd in ⊢ (??%%); @eq_f2 // @IH // +qed. definition copy ≝ - seq STape (move_l …) (seq ? (adv_to_mark_l … (is_marked ?)) - (seq ? (clear_mark …) (seq ? (adv_to_mark_r … (is_marked ?)) (clear_mark …)))). + seq STape copy0 (seq ? (move_l …) (seq ? (adv_to_mark_l … (is_marked ?)) + (seq ? (clear_mark …) (seq ? (adv_to_mark_r … (is_marked ?)) (clear_mark …))))). + +(* + s0, s1 = caratteri di testa dello stato + c0 = carattere corrente del nastro oggetto + c1 = carattere in scrittura sul nastro oggetto + + questa dimostrazione sfrutta il fatto che + merge_config (l0@[c0]) (l1@[c1]) = l1@[merge_char c0 c1] + se l0 e l1 non contengono null +*) definition R_copy ≝ λt1,t2. - ∀ls,c,c0,rs,l1,l3,l4. - t1 = midtape STape (l3@〈grid,false〉::l4@〈c0,true〉::ls) 〈c,true〉 (l1@〈comma,false〉::rs) → + ∀ls,s0,s1,c0,c1,rs,l1,l3,l4. + t1 = midtape STape (l3@〈grid,false〉::〈c0,false〉::l4@〈s0,true〉::ls) 〈s1,true〉 (l1@〈c1,false〉::〈comma,false〉::rs) → no_marks l1 → no_marks l3 → no_marks l4 → |l1| = |l4| → - only_bits_or_nulls (〈c0,true〉::l4) → only_bits_or_nulls (〈c,true〉::l1) → - t2 = midtape STape (reverse ? l1@l3@〈grid,false〉:: - merge_config (l4@[〈c0,false〉]) (reverse ? (〈c,false〉::l1))@ls) - 〈comma,false〉 rs. - -axiom sem_copy : Realize ? copy R_copy. + only_bits (l4@[〈s0,false〉]) → only_bits (〈s1,false〉::l1) → + bit_or_null c0 = true → bit_or_null c1 = true → + t2 = midtape STape (〈c1,false〉::reverse ? l1@〈s1,false〉::l3@〈grid,false〉:: + 〈merge_char c0 c1,false〉::reverse ? l1@〈s1,false〉::ls) + 〈comma,false〉 rs. + +lemma sem_copy0 : GRealize ? copy0 Pre_copy0 R_copy0. +@WRealize_to_GRealize +[ @terminate_copy0 +| @wsem_copy0 ] +qed. + +definition option_cons ≝ λA.λa:option A.λl. + match a with + [ None ⇒ l + | Some a' ⇒ a'::l ]. + +lemma sem_copy : GRealize ? copy Pre_copy R_copy. +@(GRealize_to_GRealize_2 ?? Pre_copy0 ? R_copy) // +(* preamble: Pre_copy0 implies Pre_copy *) +[ #t1 * #ls * #s0 * #s1 * #c0 * #c1 * #rs * #l1 * #l3 * #l4 * * * * * * * * + #Ht1 #Hl1nomarks #Hl3nomarks #Hl4nomarks #Hlen #Hbitsl4 #Hbitsl1 + #Hbitnullc0 #Hbitnullc1 whd + %{ls} %{s1} %{s0} %{rs} %{(l1@[〈c1,false〉;〈comma,false〉])} + %{l3} %{(〈grid,false〉::〈c0,false〉::l4)} + % [ % [ % [ % [ % + [ >Ht1 -Ht1 @eq_f >associative_append % + | #x #Hx cases (memb_append … Hx) -Hx #Hx + [ @(Hl1nomarks ? Hx) + | cases (orb_true_l … Hx) -Hx #Hx + [ >(\P Hx) % + | >(memb_single … Hx) % ] ] ] + | #x #Hx cases (memb_append … Hx) -Hx #Hx + [ @(Hl3nomarks ? Hx) + | cases (orb_true_l … Hx) -Hx #Hx + [ >(\P Hx) % + | cases (orb_true_l … Hx) -Hx #Hx + [ >(\P Hx) % + | @(Hl4nomarks ? Hx) ]]]] + |>length_append >Hlen >commutative_plus % ] + | %{(〈s1,false〉::l1@[〈c1,false〉])} %{false} % + [ @eq_f >associative_append % + | #x #Hx cases (orb_true_l … Hx) -Hx #Hx + [ >(\P Hx) @is_bit_to_bit_or_null @(Hbitsl1 〈s1,false〉) @memb_hd + | cases (memb_append … Hx) -Hx #Hx + [ @is_bit_to_bit_or_null @(Hbitsl1 〈\fst x,\snd x〉) @memb_cons (memb_single… Hx) // ] + ] + ] ] + | %{(〈c0,false〉::l4@[〈s0,false〉])} %{false} % + [ % + | #x #Hx cases (orb_true_l … Hx) -Hx #Hx + [ >(\P Hx) // + | cases (memb_append … Hx) -Hx #Hx + [ @is_bit_to_bit_or_null @(Hbitsl4 〈\fst x,\snd x〉) @memb_append_l1 (memb_single… Hx) @is_bit_to_bit_or_null @(Hbitsl4 〈s0,false〉) @memb_append_l2 @memb_hd ] + ] +] ] ] +(*whd in ⊢ (%→%) +change with (?·?) in match copy;*) +@(sem_seq_app_guarded ???? (λx.True) ??? sem_copy0) +[2:@(sem_seq_app_guarded ???????? (Realize_to_GRealize … (sem_move_l (FinProd FSUnialpha FinBool)))) + [@(λx.True) + |4:// + |5:@sub_reflexive + |3:@(sem_seq_app_guarded ???????? (Realize_to_GRealize … (sem_adv_to_mark_l … (is_marked FSUnialpha)))) + [@(λx.True) + |4:// + |5:@sub_reflexive + |3:@(sem_seq_app_guarded ???????? (Realize_to_GRealize … (sem_clear_mark ?))) + [@(λx.True) + |4:// + |5:@sub_reflexive + |3:@(sem_seq_app_guarded ???????? (Realize_to_GRealize … (sem_adv_to_mark_r ? (is_marked ?)))) + [@(λx.True) + |4:// + |5:@sub_reflexive + |3:@(Realize_to_GRealize … (sem_clear_mark ?)) ] ] ] ] +|3: // +|1:skip] +#intape #outtape #HR +#ls #s0 #s1 #c0 #c1 #rs #l1 #l2 #l3 #Hintape #Hl1marks #Hl2marks #Hl3marks #Hlen +#Hbits1 #Hbits2 #Hc0bits #Hc1bits +cases HR -HR #ta * whd in ⊢ (%→?); #Hta +cut (ta = midtape STape (〈c1,false〉::reverse ? l1@〈s1,false〉::l2@〈grid,true〉:: + 〈merge_char c0 c1,false〉::reverse ? l1@〈s1,false〉::ls) + 〈comma,true〉 rs) +[lapply (Hta ls s1 s0 rs (l1@[〈c1,false〉;〈comma,false〉]) l2 (〈grid,false〉::〈c0,false〉::l3) ?) + [>associative_append in ⊢ (???(????%)); normalize in ⊢ (???(??%%%)); @Hintape ] + -Hta #Hta cases (Hta ??? (〈s1,false〉::l1@[〈c1,false〉]) false ? ? ?? (refl ??) ?) + [3: #x #Hx cases (memb_append … Hx) -Hx #Hx + [ @Hl1marks // + | cases (orb_true_l … Hx) -Hx #Hx [ >(\P Hx) % | >(memb_single … Hx) % ]] + |4: #x #Hx cases (memb_append … Hx) -Hx #Hx + [ @Hl2marks // + | cases (orb_true_l … Hx) -Hx #Hx [ >(\P Hx) % | cases (orb_true_l … Hx) [-Hx #Hx >(\P Hx) % | @Hl3marks ] ] ] + |5: >length_append normalize >Hlen >commutative_plus % + |6: normalize >associative_append % + |7: #x #Hx cases (memb_append ?? (〈s1,false〉::l1) … Hx) -Hx #Hx + [ whd in ⊢ (??%?); >(Hbits2 … Hx) % + | >(memb_single … Hx) // ] + |8: #x #Hx cases (memb_append … Hx) -Hx #Hx + [ cases (orb_true_l … Hx) -Hx #Hx [ >(\P Hx) // | whd in ⊢ (??%?); >Hbits1 // @memb_append_l1 // ] + | >(memb_single … Hx) whd in ⊢ (??%?); >(Hbits1 〈s0,false〉) // @memb_append_l2 @memb_hd ] + | * #Hs1 @False_ind >Hs1 in Hbits2; #Hbits2 lapply (Hbits2 〈comma,false〉 ?) // + normalize #Hfalse destruct (Hfalse) + | * #Hs1 #Ht2 >Ht2 >reverse_cons >reverse_append >reverse_single @eq_f3 // + >merge_cons >merge_bits + [2: #x #Hx @Hbits2 cases (memb_append STape ? (reverse ? l1) ? Hx) -Hx #Hx + [@daemon | >(memb_single … Hx) @memb_hd ] + |3: >length_append >length_append >length_reverse >Hlen % ] + normalize >associative_append normalize >associative_append % + ] +] -Hta #Hta * #tb * * #_ #Htb +lapply (Htb … Hta) -Htb #Htb change with (midtape ????) in Htb:(???%); +* #tc * * #_ #Htc +cases (Htc … Htb) +(* [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ] *) +#_ #Htc cases (Htc (refl ??)) -Htc #Htc #_ +lapply (Htc (reverse ? l1@〈s1,false〉::l2) 〈grid,true〉 + (〈merge_char c0 c1,false〉::reverse ? l1@〈s1,false〉::ls)???) +[ #x #Hx cases (memb_append … Hx) -Hx #Hx + [ @Hl1marks @daemon + | cases (orb_true_l … Hx) -Hx #Hx + [ >(\P Hx) % | @(Hl2marks … Hx) ] ] +| % +| whd in ⊢ (??%?); >associative_append % ] -Htc #Htc +* #td * whd in ⊢ (%→?); * #_ #Htd lapply (Htd … Htc) -Htd #Htd +* #te * whd in ⊢ (%→?); * #_ #Hte cases (Hte … Htd) -Hte -Htd +[ * #Hfalse normalize in Hfalse; destruct (Hfalse) ] +* * #_ #Hte #_ +lapply (Hte (reverse ? (reverse ? l1@〈s1,false〉::l2)@[〈c1,false〉]) + 〈comma,true〉 rs ? (refl ??) ?) -Hte +[ >reverse_append >reverse_cons >reverse_reverse #x #Hx + cases (memb_append … Hx) -Hx #Hx + [ cases (memb_append … Hx) -Hx #Hx + [ cases (memb_append … Hx) -Hx #Hx + [ @daemon + | lapply (memb_single … Hx) -Hx #Hx >Hx % ] + | @(Hl1marks … Hx) ] + | lapply (memb_single … Hx) -Hx #Hx >Hx % ] +| >reverse_append >reverse_reverse >reverse_cons + >associative_append >associative_append >associative_append + >associative_append >associative_append % ] +#Hte * #_ #Houtc lapply (Houtc … Hte) -Houtc #Houtc >Houtc +@eq_f3 // +>reverse_append >reverse_append >reverse_single >reverse_cons +>reverse_append >reverse_append >reverse_reverse >reverse_reverse +>reverse_single >associative_append >associative_append % +qed. \ No newline at end of file