(* * Maximum element in a list *)
From Undecidability Require Export TM.Util.Prelim TM.Util.ArithPrelim.
(* ** Basic lemmas about upper bounds *)
(* ** Tail Recursive Definition *)
(* Compute the maximum of a list and a start-value tail-recursively *)
Fixpoint max_list_rec (s : nat) (xs : list nat) { struct xs } : nat :=
match xs with
| nil => s
| x :: xs' => max_list_rec (max x s) xs'
end.

(* ... and also w.r.t. the lists *)
Lemma max_list_rec_monotone' (xs1 xs2 : list nat) (s0 s1 : nat) :
(Forall2 le xs1 xs2) ->
s0 <= s1 ->
max_list_rec s0 xs1 <= max_list_rec s1 xs2.