]
qed.
-(*
-lemma sem_match_and_adv :
- ∀alpha,f.Realize ? (match_and_adv alpha f) (R_match_and_adv alpha f).
-#alpha #f #intape
-cases (sem_if ? (test_char ? f) … tc_true (sem_test_char ? f) (sem_adv_both_marks alpha) (sem_clear_mark ?) intape)
-#k * #outc * #Hloop #Hif @(ex_intro ?? k) @(ex_intro ?? outc)
-% [ @Hloop ] -Hloop
-cases Hif
-[ * #ta * whd in ⊢ (%→%→?); #Hta #Houtc
- #l0 #x #a #l1 #x0 #a0 #l2 #Hl1 #Hintape >Hintape in Hta;
- * * #x1 * whd in ⊢ ((??%?)→?); #H destruct (H) #Hf #Hta % %
- [ @Hf | >append_cons >append_cons in Hta; #Hta @(proj1 ?? (Houtc …) …Hta)
- [ #x #memx cases (memb_append …memx)
- [@Hl1 | -memx #memx >(memb_single … memx) %]
- |>reverse_cons >reverse_append % ] ]
-| * #ta * whd in ⊢ (%→%→?); #Hta #Houtc
- #l0 #x #a #l1 #x0 #a0 #l2 #Hl1 #Hintape >Hintape in Hta;
- * #Hf #Hta %2 % [ @Hf % | >(proj2 ?? Houtc … Hta) % ]
-]
-qed.
-*)
-
definition R_match_and_adv_of ≝
λalpha,t1,t2.current (FinProd … alpha FinBool) t1 = None ? → t2 = t1.
>(loop_eq … Hloop Hloop0) //
qed.
-(*
- if x = c
- then move_right; ----
- adv_to_mark_r;
- if current (* x0 *) = 0
- then advance_mark ----
- adv_to_mark_l;
- advance_mark
- else STOP
- else M
-*)
-
definition comp_step_subcase ≝ λalpha,c,elseM.
ifTM ? (test_char ? (λx.x == c))
(move_r … · adv_to_mark_r ? (is_marked alpha) · match_and_adv ? (λx.x == c))
if f a then 〈acc,a::tl〉 else split_on A tl f (a::acc)
].
-lemma split_on_spec: ∀A,l,f,acc,res1,res2.
+lemma split_on_spec: ∀A:DeqSet.∀l,f,acc,res1,res2.
split_on A l f acc = 〈res1,res2〉 →
(∃l1. res1 = l1@acc ∧
reverse ? l1@res2 = l ∧
- ∀x. mem ? x l1 → f x = false) ∧
+ ∀x. memb ? x l1 =true → f x = false) ∧
∀a,tl. res2 = a::tl → f a = true.
#A #l #f elim l
[#acc #res1 #res2 normalize in ⊢ (%→?); #H destruct %
- [@(ex_intro … []) % normalize [% % | #x @False_ind]
+ [@(ex_intro … []) % normalize [% % | #x #H destruct]
|#a #tl #H destruct
]
|#a #tl #Hind #acc #res1 #res2 normalize in ⊢ (%→?);
cases (true_or_false (f a)) #Hfa >Hfa normalize in ⊢ (%→?);
#H destruct
- [% [@(ex_intro … []) % normalize [% % | #x @False_ind]
+ [% [@(ex_intro … []) % normalize [% % | #x #H destruct]
|#a1 #tl1 #H destruct (H) //]
|cases (Hind (a::acc) res1 res2 H) * #l1 * *
#Hres1 #Htl #Hfalse #Htrue % [2:@Htrue] @(ex_intro … (l1@[a])) %
[% [>associative_append @Hres1 | >reverse_append <Htl % ]
- |#x #Hmemx cases (mem_append ???? Hmemx)
- [@Hfalse | normalize * [#H >H //| @False_ind]
+ |#x #Hmemx cases (memb_append ???? Hmemx)
+ [@Hfalse | #H >(memb_single … H) //]
]
]
]
axiom mem_reverse: ∀A,l,x. mem A x (reverse ? l) → mem A x l.
-lemma split_on_spec_ex: ∀A,l,f.∃l1,l2.
- l1@l2 = l ∧ (∀x:A. mem ? x l1 → f x = false) ∧
+lemma split_on_spec_ex: ∀A:DeqSet.∀l,f.∃l1,l2.
+ l1@l2 = l ∧ (∀x:A. memb ? x l1 = true → f x = false) ∧
∀a,tl. l2 = a::tl → f a = true.
#A #l #f @(ex_intro … (reverse … (\fst (split_on A l f []))))
@(ex_intro … (\snd (split_on A l f [])))
cases (split_on_spec A l f [ ] ?? (eq_pair_fst_snd …)) * #l1 * *
>append_nil #Hl1 >Hl1 #Hl #Hfalse #Htrue %
- [% [@Hl|#x #memx @Hfalse @mem_reverse //] | @Htrue]
+ [% [@Hl|#x #memx @Hfalse <(reverse_reverse … l1) @memb_reverse //] | @Htrue]
qed.
(* versione esistenziale *)
]
qed.
-(* old universal version
-
-definition R_comp_step_true ≝ λt1,t2.
- ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls 〈c,true〉 rs →
- (* bit_or_null c = false *)
- (bit_or_null c = false → t2 = midtape ? ls 〈c,false〉 rs) ∧
- (* no marks in rs *)
- (bit_or_null c = true →
- (∀c.memb ? c rs = true → is_marked ? c = false) →
- ∀a,l. (a::l) = reverse ? (〈c,true〉::rs) →
- t2 = rightof (FinProd FSUnialpha FinBool) a (l@ls)) ∧
- (∀l1,c0,l2.
- bit_or_null c = true →
- (∀c.memb ? c l1 = true → is_marked ? c = false) →
- rs = l1@〈c0,true〉::l2 →
- (c = c0 →
- l2 = [ ] → (* test true but l2 is empty *)
- t2 = rightof ? 〈c0,false〉 ((reverse ? l1)@〈c,true〉::ls)) ∧
- (c = c0 →
- ∀a,a0,b,l1',l2'. (* test true and l2 is not empty *)
- 〈a,false〉::l1' = l1@[〈c0,false〉] →
- l2 = 〈a0,b〉::l2' →
- t2 = midtape ? (〈c,false〉::ls) 〈a,true〉 (l1'@〈a0,true〉::l2')) ∧
- (c ≠ c0 →(* test false *)
- t2 = midtape (FinProd … FSUnialpha FinBool)
- ((reverse ? l1)@〈c,true〉::ls) 〈c0,false〉 l2)).
-
-definition R_comp_step_false ≝
- λt1,t2.
- ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
- is_marked ? c = false ∧ t2 = t1.
-
-(*
-lemma is_marked_to_exists: ∀alpha,c. is_marked alpha c = true →
- ∃c'. c = 〈c',true〉.
-#alpha * c *)
-
-lemma sem_comp_step :
- accRealize ? comp_step (inr … (inl … (inr … start_nop)))
- R_comp_step_true R_comp_step_false.
-@(acc_sem_if_app … (sem_test_char ? (is_marked ?))
- (sem_comp_step_subcase FSUnialpha 〈bit false,true〉 ??
- (sem_comp_step_subcase FSUnialpha 〈bit true,true〉 ??
- (sem_comp_step_subcase FSUnialpha 〈null,true〉 ??
- (sem_clear_mark …))))
- (sem_nop …) …)
-[#intape #outape #ta #Hta #Htb #ls #c #rs #Hintape whd in Hta;
- >Hintape in Hta; * #_ -Hintape (* forse non serve *)
- cases (true_or_false (c==bit false)) #Hc
- [>(\P Hc) #Hta %
- [%[whd in ⊢ ((??%?)→?); #Hdes destruct
- |#Hc @(proj1 ?? (proj1 ?? (Htb … Hta) (refl …)))
- ]
- |#l1 #c0 #l2 #Hc @(proj2 ?? (proj1 ?? (Htb … Hta) (refl …)))
- ]
- |cases (true_or_false (c==bit true)) #Hc1
- [>(\P Hc1) #Hta
- cut (〈bit true, true〉 ≠ 〈bit false, true〉) [% #Hdes destruct] #Hneq %
- [%[whd in ⊢ ((??%?)→?); #Hdes destruct
- |#Hc @(proj1 … (proj1 ?? (proj2 ?? (Htb … Hta) Hneq … Hta) (refl …)))
- ]
- |#l1 #c0 #l2 #Hc @(proj2 ?? (proj1 ?? (proj2 ?? (Htb … Hta) Hneq … Hta)(refl …)))
- ]
- |cases (true_or_false (c==null)) #Hc2
- [>(\P Hc2) #Hta
- cut (〈null, true〉 ≠ 〈bit false, true〉) [% #Hdes destruct] #Hneq
- cut (〈null, true〉 ≠ 〈bit true, true〉) [% #Hdes destruct] #Hneq1 %
- [%[whd in ⊢ ((??%?)→?); #Hdes destruct
- |#Hc @(proj1 … (proj1 ?? (proj2 ?? (proj2 ?? (Htb … Hta) Hneq … Hta) Hneq1 … Hta) (refl …)))
- ]
- |#l1 #c0 #l2 #Hc @(proj2 ?? (proj1 ?? (proj2 ?? (proj2 ?? (Htb … Hta) Hneq … Hta) Hneq1 … Hta) (refl …)))
- ]
- |#Hta cut (bit_or_null c = false)
- [lapply Hc; lapply Hc1; lapply Hc2 -Hc -Hc1 -Hc2
- cases c normalize [* normalize /2/] /2/] #Hcut %
- [%[cases (Htb … Hta) #_ -Htb #Htb
- cases (Htb … Hta) [2: % #H destruct (H) normalize in Hc; destruct] #_ -Htb #Htb
- cases (Htb … Hta) [2: % #H destruct (H) normalize in Hc1; destruct] #_ -Htb #Htb
- lapply (Htb ?) [% #H destruct (H) normalize in Hc2; destruct]
- * #_ #Houttape #_ @(Houttape … Hta)
- |>Hcut #H destruct
- ]
- |#l1 #c0 #l2 >Hcut #H destruct
- ]
- ]
- ]
- ]
-|#intape #outape #ta #Hta #Htb #ls #c #rs #Hintape
- >Hintape in Hta; whd in ⊢ (%→?); * #Hmark #Hta % [@Hmark //]
- whd in Htb; >Htb //
-]
-qed. *)
-
-(*
-definition R_comp_step_true ≝
- λt1,t2.
- ∀l0,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) l0 c rs →
- ∃c'. c = 〈c',true〉 ∧
- ((bit_or_null c' = true ∧
- ∀a,l1,c0,a0,l2.
- rs = 〈a,false〉::l1@〈c0,true〉::〈a0,false〉::l2 →
- (∀c.memb ? c l1 = true → is_marked ? c = false) →
- (c0 = c' ∧
- t2 = midtape ? (〈c',false〉::l0) 〈a,true〉 (l1@〈c0,false〉::〈a0,true〉::l2)) ∨
- (c0 ≠ c' ∧
- t2 = midtape (FinProd … FSUnialpha FinBool)
- (reverse ? l1@〈a,false〉::〈c',true〉::l0) 〈c0,false〉 (〈a0,false〉::l2))) ∨
- (bit_or_null c' = false ∧ t2 = midtape ? l0 〈c',false〉 rs)).
-
-definition R_comp_step_false ≝
- λt1,t2.
- ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
- is_marked ? c = false ∧ t2 = t1.
-
-lemma sem_comp_step :
- accRealize ? comp_step (inr … (inl … (inr … start_nop)))
- R_comp_step_true R_comp_step_false.
-#intape
-cases (acc_sem_if … (sem_test_char ? (is_marked ?))
- (sem_comp_step_subcase FSUnialpha 〈bit false,true〉 ??
- (sem_comp_step_subcase FSUnialpha 〈bit true,true〉 ??
- (sem_comp_step_subcase FSUnialpha 〈null,true〉 ??
- (sem_clear_mark …))))
- (sem_nop …) intape)
-#k * #outc * * #Hloop #H1 #H2
-@(ex_intro ?? k) @(ex_intro ?? outc) %
-[ % [@Hloop ] ] -Hloop
-[ #Hstate lapply (H1 Hstate) -H1 -Hstate -H2 *
- #ta * whd in ⊢ (%→?); #Hleft #Hright #ls #c #rs #Hintape
- >Hintape in Hleft; * *
- cases c in Hintape; #c' #b #Hintape #x * whd in ⊢ (??%?→?); #H destruct (H)
- whd in ⊢ (??%?→?); #Hb >Hb #Hta @(ex_intro ?? c') % //
- cases (Hright … Hta)
- [ * #Hc' #H1 % % [destruct (Hc') % ]
- #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
- cases (H1 … Hl1 Hrs)
- [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
- | * #Hneq #Houtc %2 %
- [ @sym_not_eq //
- | @Houtc ]
- ]
- | * #Hc #Helse1 cases (Helse1 … Hta)
- [ * #Hc' #H1 % % [destruct (Hc') % ]
- #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
- cases (H1 … Hl1 Hrs)
- [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
- | * #Hneq #Houtc %2 %
- [ @sym_not_eq //
- | @Houtc ]
- ]
- | * #Hc' #Helse2 cases (Helse2 … Hta)
- [ * #Hc'' #H1 % % [destruct (Hc'') % ]
- #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
- cases (H1 … Hl1 Hrs)
- [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
- | * #Hneq #Houtc %2 %
- [ @sym_not_eq //
- | @Houtc ]
- ]
- | * #Hc'' whd in ⊢ (%→?); #Helse3 %2 %
- [ generalize in match Hc''; generalize in match Hc'; generalize in match Hc;
- cases c'
- [ * [ #_ #Hfalse @False_ind @(absurd ?? Hfalse) %
- | #Hfalse @False_ind @(absurd ?? Hfalse) % ]
- | #_ #_ #Hfalse @False_ind @(absurd ?? Hfalse) %
- |*: #_ #_ #_ % ]
- | @(Helse3 … Hta)
- ]
- ]
- ]
- ]
-| #Hstate lapply (H2 Hstate) -H1 -Hstate -H2 *
- #ta * whd in ⊢ (%→%→?); #Hleft #Hright #ls #c #rs #Hintape
- >Hintape in Hleft; * #Hc #Hta % [@Hc % | >Hright //]
-]
-qed.*)
+(* compare *)
definition compare ≝
whileTM ? comp_step (inr … (inl … (inr … start_nop))).
>reverse_append >reverse_append >associative_append
>associative_append %
]
- |lapply Hbs1 lapply Hbs2 lapply Hrs -Hbs1 -Hbs2 -Hrs
+ |lapply Hbs1 lapply Hb0s1 lapply Hbs2 lapply Hb0s2 lapply Hrs
+ -Hbs1 -Hb0s1 -Hbs2 -Hb0s2 -Hrs
@(list_cases2 … Hlen)
- [#Hrs #_ #_ >associative_append >associative_append #Htapeb #_
+ [#Hrs #_ #_ #_ #_ >associative_append >associative_append #Htapeb #_
lapply (Htapeb … (\P eqbb0) … (refl …) (refl …)) -Htapeb #Htapeb
cases (IH … Htapeb) -IH * #Hout #_ #_ %1 %
[>(\P eqbb0) %
|>(Hout grid (refl …) (refl …)) @eq_f
normalize >associative_append %
]
- |* #a1 #ba1 * #a2 #ba2 #tl1 #tl2 #HlenS #Hrs #Hbs1 #Hbs2
- cut (ba1 = false) [@(Hbs1 〈a1,ba1〉) @memb_hd] #Hba1 >Hba1
+ |* #a1 #ba1 * #a2 #ba2 #tl1 #tl2 #HlenS #Hrs #Hb0s2 #Hbs2 #Hb0s1 #Hbs1
+ cut (ba1 = false) [@(Hbs2 〈a1,ba1〉) @memb_hd] #Hba1 >Hba1
>associative_append >associative_append #Htapeb #_
lapply (Htapeb … (\P eqbb0) … (refl …) (refl …)) -Htapeb #Htapeb
cases (IH … Htapeb) -IH * #_ #_
+ cut (ba2=false) [@(Hb0s2 〈a2,ba2〉) @memb_hd] #Hba2 >Hba2
#IH cases(IH a1 a2 ?? (l1@[〈b0,false〉]) l2 HlenS ????? (refl …) ??)
- [
-
-
-(*
- cut (∃a,l1'.〈a,false〉::l1'=((bs@[〈grid,false〉])@l1)@[〈b0,false〉])
- [generalize in match Hbs2; cases bs
- [#_ @(ex_intro … grid) @(ex_intro … (l1@[〈b0,false〉]))
- >associative_append %
- |* #bsc #bsb #bstl #Hbs2 @(ex_intro … bsc)
- @(ex_intro … (((bstl@[〈grid,false〉])@l1)@[〈b0,false〉]))
- normalize @eq_f2 [2:%] @eq_f @sym_eq @(Hbs2 〈bsc,bsb〉) @memb_hd
- ]
- ]
- * #a * #l1' #H2
- cut (∃a0,b1,l2'.b0s@〈comma,false〉::l2=〈a0,b1〉::l2')
- [cases b0s
- [@(ex_intro … comma) @(ex_intro … false) @(ex_intro … l2) %
- |* #bsc #bsb #bstl @(ex_intro … bsc) @(ex_intro … bsb)
- @(ex_intro … (bstl@〈comma,false〉::l2)) %
- ]
- ] *)
- * #a0 * #b1 * #l2' #H3
- lapply (Htapeb … (\P eqbb0) a a0 b1 l1' l2' H2 H3) -Htapeb #Htapeb
- cases (IH … Htapeb) -IH *
-
-
- [2: * >Hc' #Hfalse @False_ind destruct ] * #_
- @(list_cases2 … Hlen)
- [ #Hbs #Hb0s generalize in match Hrs; >Hbs in ⊢ (%→?); >Hb0s in ⊢ (%→?);
- -Hrs #Hrs normalize in Hrs; #Hleft cases (Hleft ????? Hrs ?) -Hleft
- [ * #Heqb #Htapeb cases (IH … Htapeb) -IH * #IH #_ #_
- % %
- [ >Heqb >Hbs >Hb0s %
- | >Hbs >Hb0s @IH %
- ]
- |* #Hneqb #Htapeb %2
- @(ex_intro … [ ]) @(ex_intro … b)
- @(ex_intro … b0) @(ex_intro … [ ])
- @(ex_intro … [ ]) %
- [ % [ % [@sym_not_eq //| >Hbs %] | >Hb0s %]
- | cases (IH … Htapeb) -IH * #_ #IH #_ >(IH ? (refl ??))
- @Htapeb
- ]
- | @Hl1 ]
- | * #b' #bitb' * #b0' #bitb0' #bs' #b0s' #Hbs #Hb0s
- generalize in match Hrs; >Hbs in ⊢ (%→?); >Hb0s in ⊢ (%→?);
- cut (bit_or_null b' = true ∧ bit_or_null b0' = true ∧
- bitb' = false ∧ bitb0' = false)
- [ % [ % [ % [ >Hbs in Hbs1; #Hbs1 @(Hbs1 〈b',bitb'〉) @memb_hd
- | >Hb0s in Hb0s1; #Hb0s1 @(Hb0s1 〈b0',bitb0'〉) @memb_hd ]
- | >Hbs in Hbs2; #Hbs2 @(Hbs2 〈b',bitb'〉) @memb_hd ]
- | >Hb0s in Hb0s2; #Hb0s2 @(Hb0s2 〈b0',bitb0'〉) @memb_hd ]
- | * * * #Ha #Hb #Hc #Hd >Hc >Hd
- #Hrs #Hleft
- cases (Hleft b' (bs'@〈grid,false〉::l1) b0 b0'
- (b0s'@〈comma,false〉::l2) ??) -Hleft
- [ 3: >Hrs normalize @eq_f >associative_append %
- | * #Hb0 #Htapeb cases (IH …Htapeb) -IH * #_ #_ #IH
- cases (IH b' b0' bs' b0s' (l1@[〈b0,false〉]) l2 ??????? Ha ?) -IH
- [ * #Heq #Houtc % %
- [ >Hb0 @eq_f >Hbs in Heq; >Hb0s in ⊢ (%→?); #Heq
- destruct (Heq) >Hb0s >Hc >Hd %
- | >Houtc >Hbs >Hb0s >Hc >Hd >reverse_cons >associative_append
- >associative_append %
+ [3:#x #memx @Hbs1 @memb_cons @memx
+ |4:#x #memx @Hb0s1 @memb_cons @memx
+ |5:#x #memx @Hbs2 @memb_cons @memx
+ |6:#x #memx @Hb0s2 @memb_cons @memx
+ |7:#x #memx cases (memb_append …memx) -memx #memx
+ [@Hl1 @memx | >(memb_single … memx) %]
+ |8:@(Hbs1 〈a1,ba1〉) @memb_hd
+ |9: >associative_append >associative_append %
+ |-IH -Hbs1 -Hb0s1 -Hbs2 -Hrs *
+ #Ha1a2 #Houtc %1 %
+ [>(\P eqbb0) @eq_f destruct (Ha1a2) %
+ |>Houtc @eq_f3
+ [>reverse_cons >associative_append %
+ |%
+ |>associative_append %
+ ]
]
- | * #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #H4 %2
- @(ex_intro … (〈b,false〉::la)) @(ex_intro … c') @(ex_intro … d')
- @(ex_intro … lb) @(ex_intro … lc)
- % [ % [ % // >Hbs >Hc >H2 % | >Hb0s >Hd >H3 >Hb0 % ]
- | >H4 >Hbs >Hb0s >Hc >Hd >Hb0 >reverse_append
- >reverse_cons >reverse_cons
- >associative_append >associative_append
- >associative_append >associative_append %
+ |-IH -Hbs1 -Hb0s1 -Hbs2 -Hrs *
+ #la * #c' * #d' * #lb * #lc * * *
+ #Hcd #H1 #H2 #Houtc %2
+ @(ex_intro … (〈b,false〉::la)) @(ex_intro … c') @(ex_intro … d')
+ @(ex_intro … lb) @(ex_intro … lc) %
+ [%[%[@Hcd | >H1 %] |>(\P eqbb0) >Hba2 >H2 %]
+ |>Houtc @eq_f3
+ [>(\P eqbb0) >reverse_append >reverse_cons
+ >reverse_cons >associative_append >associative_append
+ >associative_append >associative_append %
+ |%
+ |%
]
- | generalize in match Hlen; >Hbs >Hb0s
- normalize #Hlen destruct (Hlen) @e0
- | #c0 #Hc0 @Hbs1 >Hbs @memb_cons //
- | #c0 #Hc0 @Hb0s1 >Hb0s @memb_cons //
- | #c0 #Hc0 @Hbs2 >Hbs @memb_cons //
- | #c0 #Hc0 @Hb0s2 >Hb0s @memb_cons //
- | #c0 #Hc0 cases (memb_append … Hc0)
- [ @Hl1 | #Hc0' >(memb_single … Hc0') % ]
- | %
- | >associative_append >associative_append % ]
- | * #Hneq #Htapeb %2
- @(ex_intro … []) @(ex_intro … b) @(ex_intro … b0)
- @(ex_intro … bs) @(ex_intro … b0s) %
- [ % // % // @sym_not_eq //
- | >Hbs >Hb0s >Hc >Hd >reverse_cons >associative_append
- >reverse_append in Htapeb; >reverse_cons
- >associative_append >associative_append
- #Htapeb <Htapeb
- cases (IH … Htapeb) -Htapeb -IH * #_ #IH #_ @(IH ? (refl ??))
- ]
- | #c1 #Hc1 cases (memb_append … Hc1) #Hyp
- [ @Hbs2 >Hbs @memb_cons @Hyp
- | cases (orb_true_l … Hyp)
- [ #Hyp2 >(\P Hyp2) %
- | @Hl1
- ]
- ]
- ]
-]]]]]
-qed.
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+]
+qed.
+
+lemma WF_cst_niltape:
+ WF ? (inv ? R_comp_step_true) (niltape (FinProd FSUnialpha FinBool)).
+@wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * #H destruct
+qed.
+
+lemma WF_cst_rightof:
+ ∀a,ls. WF ? (inv ? R_comp_step_true) (rightof (FinProd FSUnialpha FinBool) a ls).
+#a #ls @wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * #H destruct
+qed.
+
+lemma WF_cst_leftof:
+ ∀a,ls. WF ? (inv ? R_comp_step_true) (leftof (FinProd FSUnialpha FinBool) a ls).
+#a #ls @wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * #H destruct
+qed.
+
+lemma WF_cst_midtape_false:
+ ∀ls,c,rs. WF ? (inv ? R_comp_step_true)
+ (midtape (FinProd … FSUnialpha FinBool) ls 〈c,false〉 rs).
+#ls #c #rs @wf #t1 whd in ⊢ (%→?); * #ls' * #c' * #rs' * #H destruct
+qed.
+
+(* da spostare *)
+lemma not_nil_to_exists:∀A.∀l: list A. l ≠ [ ] →
+ ∃a,tl. a::tl = l.
+ #A * [* #H @False_ind @H // | #a #tl #_ @(ex_intro … a) @(ex_intro … tl) //]
+ qed.
+
+lemma terminate_compare:
+ ∀t. Terminate ? compare t.
+#t @(terminate_while … sem_comp_step) [%]
+cases t // #ls * #c * //
+#rs
+(* we cannot proceed by structural induction on the right tape,
+ since compare moves the marks! *)
+cut (∃len. |rs| = len) [/2/]
+* #len lapply rs lapply c lapply ls -ls -c -rs elim len
+ [#ls #c #rs #Hlen >(lenght_to_nil … Hlen) @wf #t1 whd in ⊢ (%→?); * #ls0 * #c0 * #rs0 * #Hmid destruct (Hmid)
+ * * #H1 #H2 #_ cases (true_or_false (bit_or_null c0)) #Hc0
+ [>(H2 Hc0 … (refl …)) // #x whd in ⊢ ((??%?)→?); #Hdes destruct
+ |>(H1 Hc0) //
+ ]
+ |-len #len #Hind #ls #c #rs #Hlen @wf #t1 whd in ⊢ (%→?); * #ls0 * #c0 * #rs0 * #Hmid destruct (Hmid)
+ * * #H1 #H2 #H3 cases (true_or_false (bit_or_null c0)) #Hc0
+ [-H1 cases (split_on_spec_ex ? rs0 (is_marked ?)) #rs1 * #rs2
+ cases rs2
+ [(* no marks in right tape *)
+ * * >append_nil #H >H -H #Hmarks #_
+ cases (not_nil_to_exists ? (reverse (FSUnialpha×bool) (〈c0,true〉::rs0)) ?)
+ [2: % >reverse_cons #H cases (nil_to_nil … H) #_ #H1 destruct]
+ #a0 * #tl #H4 >(H2 Hc0 Hmarks a0 tl H4) //
+ |(* the first marked element is a0 *)
+ * #a0 #a0b #rs3 * * #H4 #H5 #H6 lapply (H3 ? a0 rs3 … Hc0 H5 ?)
+ [<H4 @eq_f @eq_f2 [@eq_f @(H6 〈a0,a0b〉 … (refl …)) | %]
+ |cases (true_or_false (c0==a0)) #eqc0a0 (* comparing a0 with c0 *)
+ [* * (* we check if we have elements at the right of a0 *)
+ lapply H4 -H4 cases rs3
+ [#_ #Ht1 #_ #_ >(Ht1 (\P eqc0a0) (refl …)) //
+ |(* a1 will be marked *)
+ cases (not_nil_to_exists ? (rs1@[〈a0,false〉]) ?)
+ [2: % #H cases (nil_to_nil … H) #_ #H1 destruct]
+ * #a2 #a2b * #tl2 #H7 * #a1 #a1b #rs4 #H4 #_ #Ht1 #_
+ cut (a2b =false)
+ [lapply (memb_hd ? 〈a2,a2b〉 tl2) >H7 #mema2
+ cases (memb_append … mema2)
+ [@H5 |#H lapply(memb_single … H) #H2 destruct %]
+ ]
+ #Ha2b >Ha2b in H7; #H7
+ >(Ht1 (\P eqc0a0) … H7 (refl …)) @Hind -Hind -Ht1 -Ha2b -H2 -H3 -H5 -H6
+ <H4 in Hlen; >length_append normalize <plus_n_Sm #Hlen1
+ >length_append normalize <(injective_S … Hlen1) @eq_f2 //
+ cut (|〈a2,false〉::tl2|=|rs1@[〈a0,false〉]|) [>H7 %]
+ >length_append normalize <plus_n_Sm <plus_n_O //
+ ]
+ |(* c0 =/= a0 *) * * #_ #_ #Ht1 >(Ht1 (\Pf eqc0a0)) //
+ ]
+ ]
+ ]
+ |>(H1 Hc0) //
+ ]
+qed.
-axiom sem_compare : Realize ? compare R_compare.
+lemma sem_compare : Realize ? compare R_compare.
+/2/ qed.
projects / helm.git / blobdiff