include "basics/lists/list.ma".
include "basics/sets.ma".
+include "basics/deqsets.ma".
+
+(********* isnilb *********)
+let rec isnilb A (l: list A) on l ≝
+match l with
+[ nil ⇒ true
+| cons hd tl ⇒ false
+].
(********* search *********)
let rec memb (S:DeqSet) (x:S) (l: list S) on l ≝
match l with
[ nil ⇒ false
- | cons a tl ⇒ (a == x) ∨ memb S x tl
+ | cons a tl ⇒ (x == a) ∨ memb S x tl
].
-
+
+interpretation "boolean membership" 'mem a l = (memb ? a l).
+
lemma memb_hd: ∀S,a,l. memb S a (a::l) = true.
#S #a #l normalize >(proj2 … (eqb_true S …) (refl S a)) //
qed.
lemma memb_cons: ∀S,a,b,l.
memb S a l = true → memb S a (b::l) = true.
-#S #a #b #l normalize cases (b==a) normalize //
+#S #a #b #l normalize cases (a==b) normalize //
+qed.
+
+lemma memb_single: ∀S,a,x. memb S a [x] = true → a = x.
+#S #a #x normalize cases (true_or_false … (a==x)) #H
+ [#_ >(\P H) // |>H normalize #abs @False_ind /2/]
qed.
lemma memb_append: ∀S,a,l1,l2.
memb S a (l1@l2) = true →
memb S a l1= true ∨ memb S a l2 = true.
#S #a #l1 elim l1 normalize [#l2 #H %2 //]
-#b #tl #Hind #l2 cases (b==a) normalize /2/
+#b #tl #Hind #l2 cases (a==b) normalize /2/
qed.
lemma memb_append_l1: ∀S,a,l1,l2.
memb S a l1= true → memb S a (l1@l2) = true.
#S #a #l1 elim l1 normalize
[normalize #le #abs @False_ind /2/
- |#b #tl #Hind #l2 cases (b==a) normalize /2/
+ |#b #tl #Hind #l2 cases (a==b) normalize /2/
]
qed.
lemma memb_append_l2: ∀S,a,l1,l2.
memb S a l2= true → memb S a (l1@l2) = true.
#S #a #l1 elim l1 normalize //
-#b #tl #Hind #l2 cases (b==a) normalize /2/
+#b #tl #Hind #l2 cases (a==b) normalize /2/
qed.
lemma memb_exists: ∀S,a,l.memb S a l = true →
∃l1,l2.l=l1@(a::l2).
#S #a #l elim l [normalize #abs @False_ind /2/]
#b #tl #Hind #H cases (orb_true_l … H)
- [#eqba @(ex_intro … (nil S)) @(ex_intro … tl)
- >(proj1 … (eqb_true …) eqba) //
+ [#eqba @(ex_intro … (nil S)) @(ex_intro … tl) >(\P eqba) //
|#mem_tl cases (Hind mem_tl) #l1 * #l2 #eqtl
@(ex_intro … (b::l1)) @(ex_intro … l2) >eqtl //
]
lemma not_memb_to_not_eq: ∀S,a,b,l.
memb S a l = false → memb S b l = true → a==b = false.
#S #a #b #l cases (true_or_false (a==b)) //
-#eqab >(proj1 … (eqb_true …) eqab) #H >H #abs @False_ind /2/
+#eqab >(\P eqab) #H >H #abs @False_ind /2/
qed.
lemma memb_map: ∀S1,S2,f,a,l. memb S1 a l= true →
memb S2 (f a) (map … f l) = true.
#S1 #S2 #f #a #l elim l normalize [//]
-#x #tl #memba cases (true_or_false (x==a))
- [#eqx >eqx >(proj1 … (eqb_true …) eqx)
- >(proj2 … (eqb_true …) (refl … (f a))) normalize //
- |#eqx >eqx cases (f x==f a) normalize /2/
+#x #tl #memba cases (true_or_false (a==x))
+ [#eqx >eqx >(\P eqx) >(\b (refl … (f x))) normalize //
+ |#eqx >eqx cases (f a==f x) normalize /2/
]
qed.
memb S3 (op a1 a2) (compose S1 S2 S3 op l1 l2) = true.
#S1 #S2 #S3 #op #a1 #a2 #l1 elim l1 [normalize //]
#x #tl #Hind #l2 #memba1 #memba2 cases (orb_true_l … memba1)
- [#eqa1 >(proj1 … (eqb_true …) eqa1) @memb_append_l1 @memb_map //
+ [#eqa1 >(\P eqa1) @memb_append_l1 @memb_map //
|#membtl @memb_append_l2 @Hind //
]
qed.
+lemma memb_reverse: ∀S:DeqSet.∀a:S.∀l.
+ memb ? a l = true → memb ? a (reverse ? l) = true.
+#S #a #l elim l [normalize //]
+#b #tl #Hind #memba change with ([b]@tl) in match (b::tl);
+>reverse_append cases (orb_true_l … memba) #Hcase
+ [@memb_append_l2 >(\P Hcase) whd in match (reverse ??); @memb_hd
+ |@memb_append_l1 /2/
+ ]
+qed.
+
+lemma mem_to_memb: ∀S:DeqSet.∀a,l. mem S a l → memb S a l = true.
+#S #a #l elim l normalize
+ [@False_ind
+ |#hd #tl #Hind *
+ [#eqa >(\b eqa) %
+ |#Hmem cases (a==hd) // normalize /2/
+ ]
+ ]
+qed.
+
+lemma memb_to_mem: ∀S:DeqSet.∀l,a. memb S a l =true → mem S a l.
+#S #l #a elim l
+ [normalize #H destruct
+ |#b #tl #Hind #mema cases (orb_true_l … mema)
+ [#eqab >(\P eqab) %1 % |#memtl %2 @Hind @memtl]
+ ]
+qed.
+
(**************** unicity test *****************)
let rec uniqueb (S:DeqSet) l on l : bool ≝
if memb S a r then r else a::r
].
-axiom unique_append_elim: ∀S:DeqSet.∀P: S → Prop.∀l1,l2.
+lemma memb_unique_append: ∀S,a,l1,l2.
+memb S a (unique_append S l1 l2) = true →
+ memb S a l1= true ∨ memb S a l2 = true.
+#S #a #l1 elim l1 normalize [#l2 #H %2 //]
+#b #tl #Hind #l2 cases (true_or_false … (a==b)) #Hab >Hab normalize /2/
+cases (memb S b (unique_append S tl l2)) normalize
+ [@Hind | >Hab normalize @Hind]
+qed.
+
+lemma unique_append_elim: ∀S:DeqSet.∀P: S → Prop.∀l1,l2.
(∀x. memb S x l1 = true → P x) → (∀x. memb S x l2 = true → P x) →
∀x. memb S x (unique_append S l1 l2) = true → P x.
+#S #P #l1 #l2 #Hl1 #Hl2 #x #membx cases (memb_unique_append … membx)
+/2/
+qed.
lemma unique_append_unique: ∀S,l1,l2. uniqueb S l2 = true →
uniqueb S (unique_append S l1 l2) = true.
#H >H normalize [@Hind //] >H normalize @Hind //
qed.
+lemma uniqueb_append: ∀A,l1,l2. uniqueb A l1 = true → uniqueb A l2 =true →
+ (∀a. memb A a l1 =true → ¬ memb A a l2 =true) → uniqueb A (l1@l2) = true.
+#A #l1 elim l1 [normalize //] #a #tl #Hind #l2 #Hatl #Hl2
+#Hmem normalize cut (memb A a (tl@l2)=false)
+ [2:#Hcut >Hcut normalize @Hind //
+ [@(andb_true_r … Hatl) |#x #Hmemx @Hmem @orb_true_r2 //]
+ |@(noteq_to_eqnot ? true) % #Happend cases (memb_append … Happend)
+ [#H1 @(absurd … H1) @sym_not_eq @eqnot_to_noteq
+ @sym_eq @(andb_true_l … Hatl)
+ |#H @(absurd … H) @Hmem normalize >(\b (refl ? a)) //
+ ]
+ ]
+qed.
+
+lemma memb_map_to_exists: ∀A,B:DeqSet.∀f:A→B.∀l,b.
+ memb ? b (map ?? f l) = true → ∃a. memb ? a l = true ∧ f a = b.
+#A #B #f #l elim l
+ [#b normalize #H destruct (H)
+ |#a #tl #Hind #b #H cases (orb_true_l … H)
+ [#eqb @(ex_intro … a) <(\P eqb) % //
+ |#memb cases (Hind … memb) #a * #mema #eqb
+ @(ex_intro … a) /3/
+ ]
+ ]
+qed.
+
+lemma memb_map_inj: ∀A,B:DeqSet.∀f:A→B.∀l,a. injective A B f →
+ memb ? (f a) (map ?? f l) = true → memb ? a l = true.
+#A #B #f #l #a #injf elim l
+ [normalize //
+ |#a1 #tl #Hind #Ha cases (orb_true_l … Ha)
+ [#eqf @orb_true_r1 @(\b ?) >(injf … (\P eqf)) //
+ |#membtl @orb_true_r2 /2/
+ ]
+ ]
+qed.
+
+lemma unique_map_inj: ∀A,B:DeqSet.∀f:A→B.∀l. injective A B f →
+ uniqueb A l = true → uniqueb B (map ?? f l) = true .
+#A #B #f #l #injf elim l
+ [normalize //
+ |#a #tl #Hind #Htl @true_to_andb_true
+ [@sym_eq @noteq_to_eqnot @sym_not_eq
+ @(not_to_not ?? (memb_map_inj … injf …) )
+ <(andb_true_l ?? Htl) /2/
+ |@Hind @(andb_true_r ?? Htl)
+ ]
+ ]
+qed.
+
(******************* sublist *******************)
definition sublist ≝
λS,l1,l2.∀a. memb S a l1 = true → memb S a l2 = true.
cases (memb_append … (sub x (orb_true_r2 … membx)))
[#membxl3 @memb_append_l1 //
|#membxal4 cases (orb_true_l … membxal4)
- [#eqax @False_ind lapply (andb_true_l … unique)
- >(proj1 … (eqb_true …) eqax) >membx normalize /2/
- |#membxl4 @memb_append_l2 //
+ [#eqxa @False_ind lapply (andb_true_l … unique)
+ <(\P eqxa) >membx normalize /2/ |#membxl4 @memb_append_l2 //
]
]
qed.
∀S,l1,l2. sublist S l1 (unique_append S l1 l2).
#S #l1 elim l1 normalize [#l2 #S #abs @False_ind /2/]
#x #tl #Hind #l2 #a
-normalize cases (true_or_false … (x==a)) #eqxa >eqxa
-[>(proj1 … (eqb_true …) eqxa) cases (true_or_false (memb S a (unique_append S tl l2)))
- [#H >H normalize // | #H >H normalize >(proj2 … (eqb_true …) (refl … a)) //]
+normalize cases (true_or_false … (a==x)) #eqax >eqax
+[<(\P eqax) cases (true_or_false (memb S a (unique_append S tl l2)))
+ [#H >H normalize // | #H >H normalize >(\b (refl … a)) //]
|cases (memb S x (unique_append S tl l2)) normalize
- [/2/ |>eqxa normalize /2/]
+ [/2/ |>eqax normalize /2/]
]
qed.
∀S,l1,l2. sublist S l2 (unique_append S l1 l2).
#S #l1 elim l1 [normalize //] #x #tl #Hind normalize
#l2 #a cases (memb S x (unique_append S tl l2)) normalize
-[@Hind | cases (x==a) normalize // @Hind]
+[@Hind | cases (a==x) normalize // @Hind]
+qed.
+
+lemma decidable_sublist:∀S,l1,l2.
+ (sublist S l1 l2) ∨ ¬(sublist S l1 l2).
+#S #l1 #l2 elim l1
+ [%1 #a normalize in ⊢ (%→?); #abs @False_ind /2/
+ |#a #tl * #subtl
+ [cases (true_or_false (memb S a l2)) #memba
+ [%1 whd #x #membx cases (orb_true_l … membx)
+ [#eqax >(\P eqax) // |@subtl]
+ |%2 @(not_to_not … (eqnot_to_noteq … true memba)) #H1 @H1 @memb_hd
+ ]
+ |%2 @(not_to_not … subtl) #H1 #x #H2 @H1 @memb_cons //
+ ]
+ ]
qed.
(********************* filtering *****************)
+lemma memb_filter_memb: ∀S,f,a,l.
+ memb S a (filter S f l) = true → memb S a l = true.
+#S #f #a #l elim l [normalize //]
+#b #tl #Hind normalize (cases (f b)) normalize
+cases (a==b) normalize // @Hind
+qed.
+
+lemma uniqueb_filter : ∀S,l,f.
+ uniqueb S l = true → uniqueb S (filter S f l) = true.
+#S #l #f elim l //
+#a #tl #Hind #Hunique cases (andb_true … Hunique)
+#memba #uniquetl cases (true_or_false … (f a)) #Hfa
+ [>filter_true // @true_to_andb_true
+ [@sym_eq @noteq_to_eqnot @(not_to_not … (eqnot_to_noteq … (sym_eq … memba)))
+ #H @sym_eq @(memb_filter_memb … f) <H //
+ |/2/
+ ]
+ |>filter_false /2/
+ ]
+qed.
+
lemma filter_true: ∀S,f,a,l.
memb S a (filter S f l) = true → f a = true.
#S #f #a #l elim l [normalize #H @False_ind /2/]
#b #tl #Hind cases (true_or_false (f b)) #H
normalize >H normalize [2:@Hind]
-cases (true_or_false (b==a)) #eqab
- [#_ <(proj1 … (eqb_true …) eqab) // | >eqab normalize @Hind]
+cases (true_or_false (a==b)) #eqab
+ [#_ >(\P eqab) // | >eqab normalize @Hind]
qed.
-lemma memb_filter_memb: ∀S,f,a,l.
- memb S a (filter S f l) = true → memb S a l = true.
-#S #f #a #l elim l [normalize //]
-#b #tl #Hind normalize (cases (f b)) normalize
-cases (b==a) normalize // @Hind
-qed.
-
lemma memb_filter: ∀S,f,l,x. memb S x (filter ? f l) = true →
memb S x l = true ∧ (f x = true).
/3/ qed.
lemma memb_filter_l: ∀S,f,x,l. (f x = true) → memb S x l = true →
memb S x (filter ? f l) = true.
#S #f #x #l #fx elim l normalize //
-#b #tl #Hind cases (true_or_false (b==x)) #eqbx
- [>(proj1 … (eqb_true … ) eqbx) >(proj2 … (eqb_true …) (refl … x))
- >fx normalize >(proj2 … (eqb_true …) (refl … x)) normalize //
- |>eqbx cases (f b) normalize [>eqbx normalize @Hind| @Hind]
+#b #tl #Hind cases (true_or_false (x==b)) #eqxb
+ [<(\P eqxb) >(\b (refl … x)) >fx normalize >(\b (refl … x)) normalize //
+ |>eqxb cases (f b) normalize [>eqxb normalize @Hind| @Hind]
]
qed.
projects / helm.git / blobdiff