Section Lexicographic_Product.
Variables A B : Type.
Variable ltA : A -> A -> Prop.
Variable ltB : B -> B -> Prop.
Hypothesis wf_ltA : well_founded ltA.
Hypothesis wf_ltB : well_founded ltB.
Inductive lexprod : A*B -> A*B -> Prop :=
| lex_l : forall a a' b b', ltA a a' -> lexprod (a, b) (a', b')
| lex_r : forall a    b b', ltB b b' -> lexprod (a, b) (a,  b').

Lemma lexprod_trans :
(forall a1 a2 a3, ltA a1 a2 -> ltA a2 a3 -> ltA a1 a3) ->
(forall b1 b2 b3, ltB b1 b2 -> ltB b2 b3 -> ltB b1 b3) ->
forall x1 x2 x3, lexprod x1 x2 -> lexprod x2 x3 -> lexprod x1 x3.