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
+[ * #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
+ * #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
+ [ #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 % >(\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 %
+ [| <Hta @Helse ]
+ % #Hxc0 >Hxc0 in Hxc; #Hxc lapply (Hxc 〈c,true〉 (refl …)) #Hfalse
+ cases (\Pf Hfalse) #Hfalse0 @Hfalse0 %
+]
qed.
(*
(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 % // <Hta @Houtc ]
qed.
definition copy_step ≝
(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 ]
+[ #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} % // <Hc' @Ht2
| @Hl1marks ]
| @Hl1marks ]
| #t1 #t2 #t3 whd in ⊢ (%→%→?); #H1 #H2 #ls #c #rs #Ht1
- >Ht1 in H1; #H1 cases (H1 … (refl ??)) #_ #Ht3 cases (H1 ? (refl ??)) -H1
- #Hc #Ht3 % //
-]
+ >Ht1 in H1; * #Hcur #Ht3 % // @Hcur % ]
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 % %
#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)
]]]
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'' cases (list_last ? l1) in Hl1'';
+ [ #Hl1'' >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 <H @memb_cons @Hx ] ]
+ cut (∃l4''',bg0.l4''@[〈d2,false〉]=〈grid,bg0〉::l4'''∧only_bits_or_nulls l4''')
+ [ <Hl4'' cases (list_last ? l4') in Hl4;
+ [ #Hl4' >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 <associative_append @memb_append_l1 @Hx
+ | * #x1 * #Hc #Ht1 >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
〈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 <eq_pair_fst_snd @Hx
+ | >(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 <eq_pair_fst_snd @Hx
+ | >(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
|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
[ >(\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
| >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
projects / helm.git / blobdiff