X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Funiversal%2Fcopy.ma;h=5d8f4caab57f5f1e6229487beb3079dac99835bc;hb=66d22e9bc8ecc624e93e3e142676045d511ed9b0;hp=3403d555598e23d7e32c1bdf419b3dc836af1267;hpb=64a59f1d6a60237ad78234f2bf90c0e5eb8c89ce;p=helm.git diff --git a/matita/matita/lib/turing/universal/copy.ma b/matita/matita/lib/turing/universal/copy.ma index 3403d5555..5d8f4caab 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 +32,51 @@ 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 #Hta cases (Hta … (refl ??)) -Hta #Hx #Hta + * #tb * whd in ⊢ (%→?); #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 + [ >Htc * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ] + * #_ #Htd lapply (Htd (l1@[〈a0,false〉]) 〈x0,true〉 l2 ? (refl ??) ?) -Htd + [ #x1 #Hx1 cases (memb_append … Hx1) -Hx1 #Hx1 [@(Hl1marks ? Hx1)|>(memb_single … Hx1) %] + | normalize >associative_append % ] #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 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 ] + * #_ #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 #Hta cases (Hta ? (refl ??)) -Hta + #Hxc #Hta >Hta #Houtc %2 % // lapply (\Pf Hxc) @not_to_not #Heq >Heq % ] +qed. (* if current = 0,tt @@ -102,27 +125,51 @@ 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 #Hta cases (Hta … (refl ??)) -Hta #Hx #Hta + * #tb * whd in ⊢ (%→?); #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 + [ >Htc * normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ] + * #_ #Htd lapply (Htd (l1@[〈a0,false〉]) 〈x0,true〉 l2 ? (refl ??) ?) -Htd + [ #x1 #Hx1 cases (memb_append … Hx1) -Hx1 #Hx1 [@(Hl1marks ? Hx1)|>(memb_single … Hx1) %] + | normalize >associative_append % ] >reverse_append #Htd + * #te * whd in ⊢ (%→?); #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 ] + * #_ #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 #Hta cases (Hta ? (refl ??)) -Hta + #Hxc #Hta >Hta whd in ⊢ (%→?); #Houtc %2 % + [ lapply (\Pf Hxc) @not_to_not #Heq >Heq % + | @Houtc ] +qed. definition copy_step ≝ - ifTM ? (test_char STape (λc.is_bit (\fst c))) + ifTM ? (test_char STape (λc.bit_or_null (\fst c))) (single_finalTM ? (copy_step_subcase FSUnialpha (bit false) (copy_step_subcase FSUnialpha (bit true) nocopy_subcase))) (nop ?) @@ -145,9 +192,37 @@ definition R_copy_step_false ≝ λt1,t2. ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs → bit_or_null (\fst c) = false ∧ t2 = t1. - -axiom sem_copy_step : - accRealize ? copy_step (inr … (inl … (inr … 0))) R_copy_step_true R_copy_step_false. + +lemma sem_copy_step : + accRealize ? copy_step (inr … (inl … (inr … start_nop))) R_copy_step_true R_copy_step_false. +#intape +@(acc_sem_if_app … (sem_test_char ? (λc:STape.bit_or_null (\fst c))) … + (sem_copy_step_subcase FSUnialpha (bit false) … + (sem_copy_step_subcase FSUnialpha (bit true) … (sem_nocopy_subcase …))) + (sem_nop …)) +[ #t1 #t2 #t3 whd in ⊢ (%→%→?); #H1 #H2 #ls #c #rs #Ht1 >Ht1 in H1; #H1 + cases (H1 … (refl ??)) #Hc #Ht3 % [ @Hc ] + #a #l1 #x0 #a0 #l2 #l3 #Hls #Hrs #Hl1marks >Hls in Ht3; >Hrs #Ht3 + cases (H2 … Ht3 ?) + [ * #Hc' #Ht2 % %{false} % // Ht1 in H1; #H1 cases (H1 … (refl ??)) #_ #Ht3 cases (H1 ? (refl ??)) -H1 + #Hc #Ht3 % // +] +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 @@ -165,8 +240,16 @@ let rec merge_config (l1,l2:list STape) ≝ | cons p2 l2' ⇒ let 〈c1,b1〉 ≝ p1 in let 〈c2,b2〉 ≝ p2 in match c2 with - [ null ⇒ p1 :: merge_config l1' l2' - | _ ⇒ p2 :: merge_config l1' l2' ] ] ]. + [ null ⇒ p1 + | _ ⇒ p2 ] :: merge_config l1' l2' ] ]. + +lemma merge_config_append : + ∀l1,l2,l3,l4.|l1| = |l2| → + merge_config (l1@l3) (l2@l4) = merge_config l1 l2@merge_config l3 l4. +#l1 #l2 #l3 #l4 #Hlen @(list_ind2 … Hlen) +[normalize // +| #t1 #t2 * #c1 #b1 * #c2 #b2 #IH whd in ⊢ (??%%); >IH % ] +qed. definition R_copy0 ≝ λt1,t2. ∀ls,c,c0,rs,l1,l3,l4. @@ -179,6 +262,21 @@ definition R_copy0 ≝ λt1,t2. t2 = midtape ? (reverse ? l1'@l3@〈grid,true〉:: merge_config l4' (reverse ? l1')@ls) 〈comma,true〉 rs). + +lemma inj_append_singleton_l1 : + ∀A.∀l1,l2:list A.∀a1,a2.l1@[a1] = l2@[a2] → l1 = l2. +#A #l1 #l2 #a1 #a2 #H lapply (eq_f … (reverse ?) … H) +>reverse_append >reverse_append normalize #H1 destruct +lapply (eq_f … (reverse ?) … e0) >reverse_reverse >reverse_reverse // +qed. + +lemma inj_append_singleton_l2 : + ∀A.∀l1,l2:list A.∀a1,a2.l1@[a1] = l2@[a2] → a1 = a2. +#A #l1 #l2 #a1 #a2 #H lapply (eq_f … (reverse ?) … H) +>reverse_append >reverse_append normalize #H1 destruct % +qed. + +axiom daemon : ∀P:Prop.P. lemma wsem_copy0 : WRealize ? copy0 R_copy0. #intape #k #outc #Hloop @@ -200,24 +298,59 @@ lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop #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|) [@daemon] #Hlen1 - @(list_cases_2 … Hlen1) - [ #Hl1nil @False_ind >Hl1nil in Hl1; cases l1' normalize + cut (|l1| = |reverse ? l4|) [>length_reverse @Hlen] #Hlen1 + @(list_cases2 … 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) | #p0 #l0 normalize #Hfalse destruct (Hfalse) cases l0 in e0; [ normalize #Hfalse1 destruct (Hfalse1) | #p0' #l0' normalize #Hfalse1 destruct (Hfalse1) ] ] - | * #a #ba * #a0 #ba0 #l1'' #l4'' #Hl1cons #Hl4cons + | (* case c::l1 = c::a::l1'' *) + * #a #ba * #a0 #ba0 #l1'' #l4'' #Hl1cons #Hl4cons lapply (eq_f ?? (reverse ?) ?? Hl4cons) >reverse_reverse >reverse_cons -Hl4cons #Hl4cons - cut (ba = false) [ @daemon ] #Hba - cut (ba0 = false) [ @daemon ] #Hba0 + cut (ba = false) + [ >Hl1cons in Hl1nomarks; #Hl1nomarks lapply (Hl1nomarks 〈a,ba〉 ?) + [ @memb_hd | normalize // ] ] #Hba + cut (ba0 = false) + [ >Hl4cons in Hl3l4nomarks; #Hl3l4nomarks lapply (Hl3l4nomarks 〈a0,ba0〉 ?) + [ @memb_append_l2 @memb_append_l2 @memb_hd | normalize // ] ] #Hba0 >Hba0 in Hl4cons; >Hba in Hl1cons; -Hba0 -Hba #Hl1cons #Hl4cons >Hl4cons in Htc; >Hl1cons #Htc lapply (Htc a (l3@reverse ? l4'') c0 a0 ls (l1''@rs) ? (refl ??) ?) [ #x #Hx @Hl3l4nomarks >Hl4cons associative_append >associative_append % - | -Htc * + | -Htc + cut (∃la.l1' = 〈c,false〉::la) + [ >Hl1cons in Hl1; cases l1' + [normalize #Hfalse destruct (Hfalse) + | #p #la normalize #Hla destruct (Hla) @(ex_intro ?? la) % ] ] + * #la #Hla + cut (∃lb.l4' = lb@[〈c0,false〉]) + [ >Hl4cons in Hl4; + @(list_elim_left … l4') + [ #Heq lapply (eq_f … (length ?) … Heq) + >length_append >length_append + >commutative_plus normalize >commutative_plus normalize + #Hfalse destruct + | #a1 #tl #_ #Heq + >(inj_append_singleton_l2 ? (reverse ? l4''@[〈a0,false〉]) (〈grid,bg〉::tl) 〈c0,false〉 a1 Heq) + @ex_intro // + ] ] * #lb #Hlb + cut (|lb| = |reverse ? la|) + [ >Hla in Hl1; >Hlb in Hl4; #Hl4 #Hl1 + >(?:l1 = la@[〈comma,bv〉]) in Hlen; + [|normalize in Hl1; destruct (Hl1) %] + >(?:l4 = 〈grid,bg〉::lb) + [|@(inj_append_singleton_l1 ?? (〈grid,bg〉::lb) ?? Hl4) ] + >length_append >commutative_plus >length_reverse + normalize #Hlalb destruct (Hlalb) // + ] #Hlen2 + * + (* by hyp on the first iteration step, + we consider whether c = bit x or c = null *) + (* c = bit x *) [ * #x * #Hx #Htc lapply (Hind (〈bit x,false〉::ls) a a0 rs l1'' (〈bit x,false〉::l3) (reverse ? l4'') ????) @@ -230,36 +363,218 @@ lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop | #x0 #Hx0 @Hl1nomarks >Hl1cons @memb_cons // | >Htc >associative_append % | -Hind - cut (∃la.l1' = 〈c,false〉::la) - [ >Hl1cons in Hl1; cases l1' - [normalize #Hfalse destruct (Hfalse) - | #p #la normalize #Hla destruct (Hla) @(ex_intro ?? la) % ] ] - * #la #Hla - cut (∃lb.l4' = lb@[〈c0,false〉]) - [ >Hl4cons in Hl4; - @(list_elim_left … l4') - (* si usa l'iniettività del "cons destro" - [ normalize - | #p #lb - cases l4' - [normalize - | #p #lb *) - - @(list_elim_left … l4') - Hlb @memb_append_l1 // + | >Hlb in Hl4; normalize in ⊢ (%→?); #Hl4 + @(inj_append_singleton_l1 ? l4 (〈grid,bg〉::lb) … Hl4) + | #x0 #Hx0 @Hl1bits >Hla @memb_cons // + | >Hla in Hl1; normalize in ⊢ (%→?); #Hl1 + destruct (Hl1) // ] -Hind + (* by IH, we proceed by cases, whether a = comma + (consequently several lists = []) or not *) + * + [ * #Ha #Houtc1 + cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = []) + [ @daemon ] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil + >Hl1cons in Hl1; >Hla + >Houtc1 >Htc #Hl1 + >Hl4cons in Hl4; >Hlb #Hl4 + >Hla1 >Hlb1 >Hl1nil >Hl4nil >Hx + cut (a0 = grid) [ @daemon ] #Ha0 associative_append % + | * #Ha #Houtc1 >Houtc1 @eq_f3 // + >Hla >reverse_cons >associative_append @eq_f + >Hx whd in ⊢ (??%?); @eq_f whd in ⊢ (???%); @eq_f @eq_f + >Hlb >append_cons @eq_f2 // >(merge_config_append … Hlen2) % + ] + ] + | (* c = null *) + * #Hc #Htc + lapply (Hind (〈c0,false〉::ls) a a0 rs l1'' (〈null,false〉::l3) (reverse ? l4'') ????) + [ >Hl1cons in Hlen; >Hl4cons >length_append >commutative_plus normalize + #Hlen destruct (Hlen) @e0 + | #x0 #Hx0 cases (memb_append STape ? [〈null,false〉] (l3@reverse ? l4'') … Hx0) -Hx0 #Hx0 + [ >(memb_single … Hx0) % + | @Hl3l4nomarks cases (memb_append … Hx0) -Hx0 #Hx0 + [ @memb_append_l1 // + | @memb_append_l2 >Hl4cons @memb_append_l1 // ] + ] + | >Hl1cons #x' #Hx0 @Hl1nomarks >Hl1cons @memb_cons // + | >Htc @eq_f3 // >associative_append % ] -Hind Hlb @memb_append_l1 // + | >Hlb in Hl4; normalize in ⊢ (%→?); #Hl4 + @(inj_append_singleton_l1 ? l4 (〈grid,bg〉::lb) … Hl4) + | #x0 #Hx0 @Hl1bits >Hla @memb_cons // + | >Hla in Hl1; normalize in ⊢ (%→?); #Hl1 + destruct (Hl1) // ] -Hind * + (* by IH, we proceed by cases, whether a = comma + (consequently several lists = []) or not *) + [ * #Ha #Houtc1 >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 >Hc + cut (a0 = grid) [ @daemon ] #Ha0 associative_append % + | * #Ha #Houtc1 >Houtc1 @eq_f3 // + >Hla >reverse_cons >associative_append @eq_f + >Hc whd in ⊢ (??%?); @eq_f whd in ⊢ (???%); @eq_f @eq_f + >Hlb >append_cons @eq_f2 // >(merge_config_append … Hlen2) % + ] + ] +]]] +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. + +axiom sem_copy0 : Realize ? copy0 R_copy0. + +definition option_cons ≝ λA.λa:option A.λl. + match a with + [ None ⇒ l + | Some a' ⇒ a'::l ]. + +lemma sem_copy : Realize ? copy R_copy. +#intape +cases (sem_seq … (sem_copy0 …) + (sem_seq … (sem_move_l …) + (sem_seq … (sem_adv_to_mark_l … (is_marked ?)) + (sem_seq … (sem_clear_mark …) + (sem_seq … (sem_adv_to_mark_r … (is_marked ?)) (sem_clear_mark …))))) intape) +#k * #outc * #Hloop #HR %{k} %{outc} % [@Hloop] -Hloop +#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 * whd in ⊢ (%→?); #Htb +lapply (Htb … Hta) -Htb #Htb change with (midtape ????) in Htb:(???%); +* #tc * whd in ⊢ (%→?); #Htc +cases (Htc … Htb) +[ * #Hfalse normalize in Hfalse; destruct (Hfalse) ] +* #_ #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 whd in ⊢ (%→?); #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