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').
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).