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 last_in (A:Type) (l:list A) x: (last l x = x) \/ (In (last l x) l).