Inductive lex_ord: A*B -> A*B -> Prop :=
  | lex_ord_left: forall a1 b1 a2 b2,
      ordA a1 a2 -> lex_ord (a1,b1) (a2,b2)
  | lex_ord_right: forall a b1 b2,
      ordB b1 b2 -> lex_ord (a,b1) (a,b2).Inductive lex_ord: A*B -> A*B -> Prop :=
  | lex_ord_left: forall a1 b1 a2 b2,
      ordA a1 a2 -> lex_ord (a1,b1) (a2,b2)
  | lex_ord_right: forall a b1 b2,
      ordB b1 b2 -> lex_ord (a,b1) (a,b2).Inductive lex_ord: A*B -> A*B -> Prop :=
  | lex_ord_left: forall a1 b1 a2 b2,
      ordA a1 a2 -> lex_ord (a1,b1) (a2,b2)
  | lex_ord_right: forall a b1 b2,
      ordB b1 b2 -> lex_ord (a,b1) (a,b2).

Lemma transitive_lex_ord: transitive _ ordA -> transitive _ ordB -> transitive _ lex_ord.