X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Funiversal%2Fcopy.ma;h=b2fcf29f11e72c744719e499d0363f2b79754637;hb=df4cfc76ab059f6b3d5daf324712ad27ec281088;hp=65bf961b996826ca491dd301e165cdc4b57c74ec;hpb=70a6a8146e5815a97330ea291bea09cf798c0008;p=helm.git diff --git a/matita/matita/lib/turing/universal/copy.ma b/matita/matita/lib/turing/universal/copy.ma index 65bf961b9..b2fcf29f1 100644 --- a/matita/matita/lib/turing/universal/copy.ma +++ b/matita/matita/lib/turing/universal/copy.ma @@ -66,17 +66,20 @@ cases (sem_if ? (test_char ? (λx. x == 〈c,true〉)) ?????? tc_true (sem_test_ * * #_ #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 + #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 + * * #_ #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 % ] + #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. (* @@ -138,35 +141,33 @@ cases (sem_if ? (test_char ? (λx:STape.x == 〈null,true〉)) ?????? tc_true (sem_adv_to_mark_r … (is_marked ?))))))) (sem_nop ?) intape) #k * #outc * #Hloop #HR @(ex_intro ?? k) @(ex_intro ?? 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 % ] >reverse_append #Htd - * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Hte -Htd -Htc #Hte - * #tf * whd in ⊢ (%→?); #Htf lapply (Htf … Hte) -Hte -Htf +[ * #ta * whd in ⊢ (%→?); >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 + [ #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 + * * #_ #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 + #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 + * * #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 #Hta cases (Hta ? (refl ??)) -Hta - #Hxc #Hta >Hta whd in ⊢ (%→?); #Houtc %2 % - [ lapply (\Pf Hxc) @not_to_not #Heq >Heq % - | @Houtc ] +| * #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; #H1 - cases (H1 … (refl ??)) #Hc #Ht3 % [ @Hc ] +[ #t1 #t2 #t3 whd in ⊢ (%→%→?); #H1 #H2 #ls #c #rs #Ht1 >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; #H1 cases (H1 … (refl ??)) #_ #Ht3 cases (H1 ? (refl ??)) -H1 - #Hc #Ht3 % // -] + >Ht1 in H1; * #Hcur #Ht3 % // @Hcur % ] qed. (* @@ -279,11 +278,21 @@ qed. 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 % % @@ -293,14 +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|) [>length_reverse @Hlen] #Hlen1 - @(list_cases2 … 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) @@ -428,6 +437,178 @@ lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop ]]] 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 @@ -495,21 +676,81 @@ definition R_copy ≝ λt1,t2. 〈merge_char c0 c1,false〉::reverse ? l1@〈s1,false〉::ls) 〈comma,false〉 rs. -axiom sem_copy0 : Realize ? copy0 R_copy0. +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 : 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 + +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 @@ -542,13 +783,13 @@ cut (ta = midtape STape (〈c1,false〉::reverse ? l1@〈s1,false〉::l2@〈grid |3: >length_append >length_append >length_reverse >Hlen % ] normalize >associative_append normalize >associative_append % ] -] -Hta #Hta * #tb * whd in ⊢ (%→?); #Htb +] -Hta #Hta * #tb * * #_ #Htb lapply (Htb … Hta) -Htb #Htb change with (midtape ????) in Htb:(???%); -* #tc * whd in ⊢ (%→?); #Htc +* #tc * * #_ #Htc cases (Htc … Htb) -[ * #Hfalse normalize in Hfalse; destruct (Hfalse) ] -* #_ #Htc -lapply (Htc (reverse ? l1@〈s1,false〉::l2) 〈grid,true〉 +(* [ * #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 @@ -556,10 +797,10 @@ lapply (Htc (reverse ? l1@〈s1,false〉::l2) 〈grid,true〉 [ >(\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 +* #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 +* * #_ #Hte #_ lapply (Hte (reverse ? (reverse ? l1@〈s1,false〉::l2)@[〈c1,false〉]) 〈comma,true〉 rs ? (refl ??) ?) -Hte [ >reverse_append >reverse_cons >reverse_reverse #x #Hx @@ -573,7 +814,7 @@ lapply (Hte (reverse ? (reverse ? l1@〈s1,false〉::l2)@[〈c1,false〉]) | >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 +#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