Definition lex_prod_dec: 
  (forall a1 a2 : A, {a1 = a2} + {a1 <> a2}) ->
  (forall a1 a2, {ltA a1 a2} + {~ltA a1 a2}) ->
  (forall b1 b2, {ltB b1 b2} + {~ltB b1 b2}) ->
  (forall x y, {lexprod x y} + {~ lexprod x y}).

Lemma beq_eq_dec {A : Type} {beq : A -> A -> bool} (Hbeq : forall x y, beq x y = true <-> x = y) : forall (x y : A), {x = y} + {~ x = y}.