]
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
]
]
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 ≝
#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