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".
17 include "basics/deqsets.ma".
19 (********* search *********)
21 let rec memb (S:DeqSet) (x:S) (l: list S) on l ≝
24 | cons a tl ⇒ (a == x) ∨ memb S x tl
27 notation < "\memb x l" non associative with precedence 90 for @{'memb $x $l}.
28 interpretation "boolean membership" 'memb a l = (memb ? a l).
30 lemma memb_hd: ∀S,a,l. memb S a (a::l) = true.
31 #S #a #l normalize >(proj2 … (eqb_true S …) (refl S a)) //
34 lemma memb_cons: ∀S,a,b,l.
35 memb S a l = true → memb S a (b::l) = true.
36 #S #a #b #l normalize cases (b==a) normalize //
39 lemma memb_append: ∀S,a,l1,l2.
40 memb S a (l1@l2) = true →
41 memb S a l1= true ∨ memb S a l2 = true.
42 #S #a #l1 elim l1 normalize [#l2 #H %2 //]
43 #b #tl #Hind #l2 cases (b==a) normalize /2/
46 lemma memb_append_l1: ∀S,a,l1,l2.
47 memb S a l1= true → memb S a (l1@l2) = true.
48 #S #a #l1 elim l1 normalize
49 [normalize #le #abs @False_ind /2/
50 |#b #tl #Hind #l2 cases (b==a) normalize /2/
54 lemma memb_append_l2: ∀S,a,l1,l2.
55 memb S a l2= true → memb S a (l1@l2) = true.
56 #S #a #l1 elim l1 normalize //
57 #b #tl #Hind #l2 cases (b==a) normalize /2/
60 lemma memb_exists: ∀S,a,l.memb S a l = true →
62 #S #a #l elim l [normalize #abs @False_ind /2/]
63 #b #tl #Hind #H cases (orb_true_l … H)
64 [#eqba @(ex_intro … (nil S)) @(ex_intro … tl)
65 >(proj1 … (eqb_true …) eqba) //
66 |#mem_tl cases (Hind mem_tl) #l1 * #l2 #eqtl
67 @(ex_intro … (b::l1)) @(ex_intro … l2) >eqtl //
71 lemma not_memb_to_not_eq: ∀S,a,b,l.
72 memb S a l = false → memb S b l = true → a==b = false.
73 #S #a #b #l cases (true_or_false (a==b)) //
74 #eqab >(proj1 … (eqb_true …) eqab) #H >H #abs @False_ind /2/
77 lemma memb_map: ∀S1,S2,f,a,l. memb S1 a l= true →
78 memb S2 (f a) (map … f l) = true.
79 #S1 #S2 #f #a #l elim l normalize [//]
80 #x #tl #memba cases (true_or_false (x==a))
81 [#eqx >eqx >(proj1 … (eqb_true …) eqx)
82 >(proj2 … (eqb_true …) (refl … (f a))) normalize //
83 |#eqx >eqx cases (f x==f a) normalize /2/
87 lemma memb_compose: ∀S1,S2,S3,op,a1,a2,l1,l2.
88 memb S1 a1 l1 = true → memb S2 a2 l2 = true →
89 memb S3 (op a1 a2) (compose S1 S2 S3 op l1 l2) = true.
90 #S1 #S2 #S3 #op #a1 #a2 #l1 elim l1 [normalize //]
91 #x #tl #Hind #l2 #memba1 #memba2 cases (orb_true_l … memba1)
92 [#eqa1 >(proj1 … (eqb_true …) eqa1) @memb_append_l1 @memb_map //
93 |#membtl @memb_append_l2 @Hind //
97 (**************** unicity test *****************)
99 let rec uniqueb (S:DeqSet) l on l : bool ≝
102 | cons a tl ⇒ notb (memb S a tl) ∧ uniqueb S tl
105 (* unique_append l1 l2 add l1 in fornt of l2, but preserving unicity *)
107 let rec unique_append (S:DeqSet) (l1,l2: list S) on l1 ≝
111 let r ≝ unique_append S tl l2 in
112 if memb S a r then r else a::r
115 axiom unique_append_elim: ∀S:DeqSet.∀P: S → Prop.∀l1,l2.
116 (∀x. memb S x l1 = true → P x) → (∀x. memb S x l2 = true → P x) →
117 ∀x. memb S x (unique_append S l1 l2) = true → P x.
119 lemma unique_append_unique: ∀S,l1,l2. uniqueb S l2 = true →
120 uniqueb S (unique_append S l1 l2) = true.
121 #S #l1 elim l1 normalize // #a #tl #Hind #l2 #uniquel2
122 cases (true_or_false … (memb S a (unique_append S tl l2)))
123 #H >H normalize [@Hind //] >H normalize @Hind //
126 (******************* sublist *******************)
128 λS,l1,l2.∀a. memb S a l1 = true → memb S a l2 = true.
130 lemma sublist_length: ∀S,l1,l2.
131 uniqueb S l1 = true → sublist S l1 l2 → |l1| ≤ |l2|.
133 #a #tl #Hind #l2 #unique #sub
134 cut (∃l3,l4.l2=l3@(a::l4)) [@memb_exists @sub //]
135 * #l3 * #l4 #eql2 >eql2 >length_append normalize
136 applyS le_S_S <length_append @Hind [@(andb_true_r … unique)]
137 >eql2 in sub; #sub #x #membx
138 cases (memb_append … (sub x (orb_true_r2 … membx)))
139 [#membxl3 @memb_append_l1 //
140 |#membxal4 cases (orb_true_l … membxal4)
141 [#eqax @False_ind lapply (andb_true_l … unique)
142 >(proj1 … (eqb_true …) eqax) >membx normalize /2/
143 |#membxl4 @memb_append_l2 //
148 lemma sublist_unique_append_l1:
149 ∀S,l1,l2. sublist S l1 (unique_append S l1 l2).
150 #S #l1 elim l1 normalize [#l2 #S #abs @False_ind /2/]
152 normalize cases (true_or_false … (x==a)) #eqxa >eqxa
153 [>(proj1 … (eqb_true …) eqxa) cases (true_or_false (memb S a (unique_append S tl l2)))
154 [#H >H normalize // | #H >H normalize >(proj2 … (eqb_true …) (refl … a)) //]
155 |cases (memb S x (unique_append S tl l2)) normalize
156 [/2/ |>eqxa normalize /2/]
160 lemma sublist_unique_append_l2:
161 ∀S,l1,l2. sublist S l2 (unique_append S l1 l2).
162 #S #l1 elim l1 [normalize //] #x #tl #Hind normalize
163 #l2 #a cases (memb S x (unique_append S tl l2)) normalize
164 [@Hind | cases (x==a) normalize // @Hind]
167 (********************* filtering *****************)
169 lemma filter_true: ∀S,f,a,l.
170 memb S a (filter S f l) = true → f a = true.
171 #S #f #a #l elim l [normalize #H @False_ind /2/]
172 #b #tl #Hind cases (true_or_false (f b)) #H
173 normalize >H normalize [2:@Hind]
174 cases (true_or_false (b==a)) #eqab
175 [#_ <(proj1 … (eqb_true …) eqab) // | >eqab normalize @Hind]
178 lemma memb_filter_memb: ∀S,f,a,l.
179 memb S a (filter S f l) = true → memb S a l = true.
180 #S #f #a #l elim l [normalize //]
181 #b #tl #Hind normalize (cases (f b)) normalize
182 cases (b==a) normalize // @Hind
185 lemma memb_filter: ∀S,f,l,x. memb S x (filter ? f l) = true →
186 memb S x l = true ∧ (f x = true).
189 lemma memb_filter_l: ∀S,f,x,l. (f x = true) → memb S x l = true →
190 memb S x (filter ? f l) = true.
191 #S #f #x #l #fx elim l normalize //
192 #b #tl #Hind cases (true_or_false (b==x)) #eqbx
193 [>(proj1 … (eqb_true … ) eqbx) >(proj2 … (eqb_true …) (refl … x))
194 >fx normalize >(proj2 … (eqb_true …) (refl … x)) normalize //
195 |>eqbx cases (f b) normalize [>eqbx normalize @Hind| @Hind]
199 (********************* exists *****************)
201 let rec exists (A:Type[0]) (p:A → bool) (l:list A) on l : bool ≝
204 | cons h t ⇒ orb (p h) (exists A p t)
207 lemma Exists_exists : ∀A,P,l.
210 #A #P #l elim l [ * | #hd #tl #IH * [ #H %{hd} @H | @IH ]