(*  The order on the lists is not lexicographic! It's Every element is smaller or equal. ALSO : surprisingly, we don't need transitivity.  *)
Inductive list_ord {A} (ord: A -> A -> Prop): list A -> list A -> Prop:=
| lt_head: forall x y ls, ord x y -> list_ord ord (x::ls) (y::ls)
| lt_tail: forall x ls ls', list_ord ord ls ls' ->
list_ord ord (x::ls) (x::ls').
Inductive list_ord_part {A}
(list_ord: list A -> list A -> Prop)
(ord: A -> A -> Prop): list A -> list A -> Prop:=
| lt_head_part: forall x y ls, ord x y ->
list_ord_part list_ord ord (x::ls) (y::ls)
| lt_tail_part: forall x ls ls', list_ord ls ls' ->
list_ord_part list_ord ord (x::ls) (x::ls').
Inductive list_ord_trans {A} (ord: A -> A -> Prop): list A -> list A -> Prop:=
| lt_head_trans: forall x y ls, ord x y -> list_ord_trans ord (x::ls) (y::ls)
| lt_tail_trans: forall x ls ls', list_ord_trans ord ls ls' ->
list_ord_trans ord (x::ls) (x::ls')
| lt_all: forall x y ls ls', ord x y ->
list_ord_trans ord ls ls' ->
list_ord_trans ord (x::ls) (y::ls').
Instance list_ord_is_trans {A} (ord: relation A):
Transitive ord -> Transitive (list_ord_trans ord).
Definition trans_symprod {A B} ltA ltB : relation (A * B):=
Relation_Operators.clos_trans _ (@Relation_Operators.symprod A B ltA ltB).
(* List order annotated with length. *)
Definition ln_ord {A} (ord: (list A) -> (list A) -> Prop) n:=
fun ls1 ls2 =>
length ls1 = n /\ length ls2 = n /\ ord ls1 ls2.

Lemma ln_ord_inversion:
forall {A} (ord: relation A) n y x,
length x = n ->
list_ord ord y x ->
ln_ord (list_ord ord) n y x.