Fact if_eqA_else : forall a a' (B:Type)(b b':B),
 ~eqA a a' -> (if eqA_dec a a' then b else b') = b'.Fact if_eqA_refl : forall a (B:Type)(b b':B),
 (if eqA_dec a a then b else b') = b.(** PL: Inutilisable dans un rewrite sans un change prealable. *)

Global Instance if_eqA (B:Type)(b b':B) :
 Proper (eqA==>eqA==>@eq _) (fun x y => if eqA_dec x y then b else b').Fact if_eqA_rewrite_l : forall a1 a1' a2 (B:Type)(b b':B),
 eqA a1 a1' -> (if eqA_dec a1 a2 then b else b') =
               (if eqA_dec a1' a2 then b else b').Fact if_eqA_rewrite_r : forall a1 a2 a2' (B:Type)(b b':B),
 eqA a2 a2' -> (if eqA_dec a1 a2 then b else b') =
               (if eqA_dec a1 a2' then b else b').Global Instance multiplicity_eqA (l:list A) :
 Proper (eqA==>@eq _) (multiplicity (list_contents l)).

Lemma multiplicity_InA_O : forall l a, ~ InA eqA a l -> multiplicity (list_contents l) a = 0.