2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
8 \ / This file is distributed under the terms of the
9 \ / GNU General Public License Version 2
10 V_______________________________________________________________ *)
13 (* boolean functions over lists *)
15 include "basics/lists/list.ma".
16 include "basics/sets.ma".
18 (********* search *********)
20 let rec memb (S:DeqSet) (x:S) (l: list S) on l ≝
23 | cons a tl ⇒ (a == x) ∨ memb S x tl
26 lemma memb_hd: ∀S,a,l. memb S a (a::l) = true.
27 #S #a #l normalize >(proj2 … (eqb_true S …) (refl S a)) //
30 lemma memb_cons: ∀S,a,b,l.
31 memb S a l = true → memb S a (b::l) = true.
32 #S #a #b #l normalize cases (b==a) normalize //
35 lemma memb_append: ∀S,a,l1,l2.
36 memb S a (l1@l2) = true →
37 memb S a l1= true ∨ memb S a l2 = true.
38 #S #a #l1 elim l1 normalize [#l2 #H %2 //]
39 #b #tl #Hind #l2 cases (b==a) normalize /2/
42 lemma memb_append_l1: ∀S,a,l1,l2.
43 memb S a l1= true → memb S a (l1@l2) = true.
44 #S #a #l1 elim l1 normalize
45 [normalize #le #abs @False_ind /2/
46 |#b #tl #Hind #l2 cases (b==a) normalize /2/
50 lemma memb_append_l2: ∀S,a,l1,l2.
51 memb S a l2= true → memb S a (l1@l2) = true.
52 #S #a #l1 elim l1 normalize //
53 #b #tl #Hind #l2 cases (b==a) normalize /2/
56 lemma memb_exists: ∀S,a,l.memb S a l = true →
58 #S #a #l elim l [normalize #abs @False_ind /2/]
59 #b #tl #Hind #H cases (orb_true_l … H)
60 [#eqba @(ex_intro … (nil S)) @(ex_intro … tl)
61 >(proj1 … (eqb_true …) eqba) //
62 |#mem_tl cases (Hind mem_tl) #l1 * #l2 #eqtl
63 @(ex_intro … (b::l1)) @(ex_intro … l2) >eqtl //
67 lemma not_memb_to_not_eq: ∀S,a,b,l.
68 memb S a l = false → memb S b l = true → a==b = false.
69 #S #a #b #l cases (true_or_false (a==b)) //
70 #eqab >(proj1 … (eqb_true …) eqab) #H >H #abs @False_ind /2/
73 lemma memb_map: ∀S1,S2,f,a,l. memb S1 a l= true →
74 memb S2 (f a) (map … f l) = true.
75 #S1 #S2 #f #a #l elim l normalize [//]
76 #x #tl #memba cases (true_or_false (x==a))
77 [#eqx >eqx >(proj1 … (eqb_true …) eqx)
78 >(proj2 … (eqb_true …) (refl … (f a))) normalize //
79 |#eqx >eqx cases (f x==f a) normalize /2/
83 lemma memb_compose: ∀S1,S2,S3,op,a1,a2,l1,l2.
84 memb S1 a1 l1 = true → memb S2 a2 l2 = true →
85 memb S3 (op a1 a2) (compose S1 S2 S3 op l1 l2) = true.
86 #S1 #S2 #S3 #op #a1 #a2 #l1 elim l1 [normalize //]
87 #x #tl #Hind #l2 #memba1 #memba2 cases (orb_true_l … memba1)
88 [#eqa1 >(proj1 … (eqb_true …) eqa1) @memb_append_l1 @memb_map //
89 |#membtl @memb_append_l2 @Hind //
93 (**************** unicity test *****************)
95 let rec uniqueb (S:DeqSet) l on l : bool ≝
98 | cons a tl ⇒ notb (memb S a tl) ∧ uniqueb S tl
101 (* unique_append l1 l2 add l1 in fornt of l2, but preserving unicity *)
103 let rec unique_append (S:DeqSet) (l1,l2: list S) on l1 ≝
107 let r ≝ unique_append S tl l2 in
108 if memb S a r then r else a::r
111 axiom unique_append_elim: ∀S:DeqSet.∀P: S → Prop.∀l1,l2.
112 (∀x. memb S x l1 = true → P x) → (∀x. memb S x l2 = true → P x) →
113 ∀x. memb S x (unique_append S l1 l2) = true → P x.
115 lemma unique_append_unique: ∀S,l1,l2. uniqueb S l2 = true →
116 uniqueb S (unique_append S l1 l2) = true.
117 #S #l1 elim l1 normalize // #a #tl #Hind #l2 #uniquel2
118 cases (true_or_false … (memb S a (unique_append S tl l2)))
119 #H >H normalize [@Hind //] >H normalize @Hind //
122 (******************* sublist *******************)
124 λS,l1,l2.∀a. memb S a l1 = true → memb S a l2 = true.
126 lemma sublist_length: ∀S,l1,l2.
127 uniqueb S l1 = true → sublist S l1 l2 → |l1| ≤ |l2|.
129 #a #tl #Hind #l2 #unique #sub
130 cut (∃l3,l4.l2=l3@(a::l4)) [@memb_exists @sub //]
131 * #l3 * #l4 #eql2 >eql2 >length_append normalize
132 applyS le_S_S <length_append @Hind [@(andb_true_r … unique)]
133 >eql2 in sub; #sub #x #membx
134 cases (memb_append … (sub x (orb_true_r2 … membx)))
135 [#membxl3 @memb_append_l1 //
136 |#membxal4 cases (orb_true_l … membxal4)
137 [#eqax @False_ind lapply (andb_true_l … unique)
138 >(proj1 … (eqb_true …) eqax) >membx normalize /2/
139 |#membxl4 @memb_append_l2 //
144 lemma sublist_unique_append_l1:
145 ∀S,l1,l2. sublist S l1 (unique_append S l1 l2).
146 #S #l1 elim l1 normalize [#l2 #S #abs @False_ind /2/]
148 normalize cases (true_or_false … (x==a)) #eqxa >eqxa
149 [>(proj1 … (eqb_true …) eqxa) cases (true_or_false (memb S a (unique_append S tl l2)))
150 [#H >H normalize // | #H >H normalize >(proj2 … (eqb_true …) (refl … a)) //]
151 |cases (memb S x (unique_append S tl l2)) normalize
152 [/2/ |>eqxa normalize /2/]
156 lemma sublist_unique_append_l2:
157 ∀S,l1,l2. sublist S l2 (unique_append S l1 l2).
158 #S #l1 elim l1 [normalize //] #x #tl #Hind normalize
159 #l2 #a cases (memb S x (unique_append S tl l2)) normalize
160 [@Hind | cases (x==a) normalize // @Hind]
163 (********************* filtering *****************)
165 lemma filter_true: ∀S,f,a,l.
166 memb S a (filter S f l) = true → f a = true.
167 #S #f #a #l elim l [normalize #H @False_ind /2/]
168 #b #tl #Hind cases (true_or_false (f b)) #H
169 normalize >H normalize [2:@Hind]
170 cases (true_or_false (b==a)) #eqab
171 [#_ <(proj1 … (eqb_true …) eqab) // | >eqab normalize @Hind]
174 lemma memb_filter_memb: ∀S,f,a,l.
175 memb S a (filter S f l) = true → memb S a l = true.
176 #S #f #a #l elim l [normalize //]
177 #b #tl #Hind normalize (cases (f b)) normalize
178 cases (b==a) normalize // @Hind
181 lemma memb_filter: ∀S,f,l,x. memb S x (filter ? f l) = true →
182 memb S x l = true ∧ (f x = true).
185 lemma memb_filter_l: ∀S,f,x,l. (f x = true) → memb S x l = true →
186 memb S x (filter ? f l) = true.
187 #S #f #x #l #fx elim l normalize //
188 #b #tl #Hind cases (true_or_false (b==x)) #eqbx
189 [>(proj1 … (eqb_true … ) eqbx) >(proj2 … (eqb_true …) (refl … x))
190 >fx normalize >(proj2 … (eqb_true …) (refl … x)) normalize //
191 |>eqbx cases (f b) normalize [>eqbx normalize @Hind| @Hind]
195 (********************* exists *****************)
197 let rec exists (A:Type[0]) (p:A → bool) (l:list A) on l : bool ≝
200 | cons h t ⇒ orb (p h) (exists A p t)
203 lemma Exists_exists : ∀A,P,l.
206 #A #P #l elim l [ * | #hd #tl #IH * [ #H %{hd} @H | @IH ]