V_____________________________________________________________*)
-include "turing/universal/tuples.ma".
-
-definition write_states ≝ initN 2.
-
-definition wr0 : write_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
-definition wr1 : write_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
-
-definition write ≝ λalpha,c.
- mk_TM alpha write_states
- (λp.let 〈q,a〉 ≝ p in
- match pi1 … q with
- [ O ⇒ 〈wr1,Some ? 〈c,N〉〉
- | S _ ⇒ 〈wr1,None ?〉 ])
- wr0 (λx.x == wr1).
-
-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〉))
(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 %
+ [| <Hta @Helse ]
+ % #Hxc0 >Hxc0 in Hxc; #Hxc lapply (Hxc 〈c,true〉 (refl …)) #Hfalse
+ cases (\Pf Hfalse) #Hfalse0 @Hfalse0 %
+]
+qed.
(*
if current = 0,tt
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' %
- | <Hta @Houtc ] ]
-@daemon
-qed. *)
+#a #l1 #x0 #a0 #l2 #x #l3 #Hintape #Hl1marks cases HR -HR
+[ * #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
+ [ * 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 % // <Hta @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 ?)
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 :
+
+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;
+ * * #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
+ | * #Hnotfalse whd in ⊢ (%→?); #Ht2 cases (Ht2 … Ht3 ?) -Ht2
+ [ * #Hc' #Ht2 % %{true} % // <Hc' @Ht2
+ | * #Hnottrue whd in ⊢ (%→?); #Ht2 cases (Ht2 … Ht3 ?) -Ht2
+ [ * #Hc' #Ht2 %2 <Hc' % // @Ht2
+ | * #Hnotnull @False_ind
+ generalize in match Hnotnull;generalize in match Hnottrue;generalize in match Hnotfalse;
+ cases c in Hc; normalize
+ [ * [ #_ #_ * #Hfalse #_ | #_ * #Hfalse #_ #_ ]
+ | #_ #_ #_ * #Hfalse
+ |*: #Hfalse destruct (Hfalse) ] @Hfalse %
+ | @Hl1marks ]
+ | @Hl1marks ]
+ | @Hl1marks ]
+| #t1 #t2 #t3 whd in ⊢ (%→%→?); #H1 #H2 #ls #c #rs #Ht1
+ >Ht1 in H1; * #Hcur #Ht3 % // @Hcur % ]
+qed.
(*
1) il primo carattere è marcato
>reverse_append >reverse_append normalize #H1 destruct %
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|) [//] #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)
(consequently several lists = []) or not *)
*
[ * #Ha #Houtc1
-(* cut (l1 = [〈a,false〉])
- [ cases l1'' in Hl1cons; // #y #ly #Hly
- >Hly in Hl1; cases l1' in Hl1bits;
- [ #_ normalize #Hfalse destruct (Hfalse)
- | #p #lp #Hl1bits normalize #Heq destruct (Heq)
- @False_ind lapply (Hl1bits 〈a,false〉 ?)
- [ cases lp in e0;
- [ normalize #Hfalse destruct (Hfalse)
- | #p0 #lp0 normalize in ⊢ (%→?); #Heq destruct (Heq)
- @memb_cons @memb_hd ]
- | >Ha normalize #Hfalse destruct (Hfalse) ]
- ]
- ] #Hl1a
- cut (l4 = [〈a0,false〉])
- [ generalize in match Hl4bits; cases l4' in Hl4;
- [ >Hl4cons #Hfalse #_
- lapply (inj_append_singleton_l1 ?? [] ?? Hfalse)
- cases (reverse ? l4'') normalize
- [ #Hfalse1 | #p0 #lp0 #Hfalse1 ] destruct (Hfalse1)
- | #p #lp
-
- cases l4'' in Hl4cons; // #y #ly #Hly
- >Hly in Hl4; cases l4' in Hl4bits;
- [ #_ >reverse_cons #Hfalse
- lapply (inj_append_singleton_l1 ?? [] ?? Hfalse)
- -Hfalse cases ly normalize
- [ #Hfalse | #p #Hp #Hfalse ] destruct (Hfalse)
-
- | #p #lp #Hl1bits normalize #Heq destruct (Heq)
- @False_ind lapply (Hl1bits 〈a,false〉 ?)
- [ cases lp in e0;
- [ normalize #Hfalse destruct (Hfalse)
- | #p0 #lp0 normalize in ⊢ (%→?); #Heq destruct (Heq)
- @memb_cons @memb_hd ]
- | >Ha normalize #Hfalse destruct (Hfalse) ]
- ]
- ] #Hl1a
-
- >Hla normalize #Hl1 destruct (Hl1) lapply (inj_append_ @False_ind
-
- cut (l1'' = [] ∧ l4'' = [])
- [ % [ >Hla in Hl1; normalize #Hl1 destruct (Hl1)
-
- cases l1'' in Hl1bits;
-
- [ #_ normalize #H *)
cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = [])
[ @daemon ] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil
>Hl1cons in Hl1; >Hla
]]]
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
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.
-axiom reverse_merge_config :
+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
∀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,true〉]) → only_bits (〈s1,true〉::l1) →
+ 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.
-axiom sem_copy : Realize ? copy R_copy.
\ No newline at end of file
+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 <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
+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
projects / helm.git / blobdiff