Greatest value in natural number list - coq

Question

I have defined a function,which finds greatest value in the list of natural numbers and head of the list save this value. I want to prove that all the elements in the list are less or equal to natural number present at head of the list.I have problem in proving the lemma. I have written two lemmas,I want to know,which would be helpful in solving the problem. Thanks for help and
support .
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.
Definition change_variable (n: nat) (l: list nat) : list nat:=
match l with
| nil => l
| h::t => if n <=? h then l else n::t
end.
Fixpoint largest_value (numbers: nat) (l: list nat) {struct numbers}: nat:=
match l with
| nil => 0
| cons b nil => b
| cons h l => match numbers with
| O => h
| S numbers' => largest_value numbers' (change_variable h l)
end
end.
Theorem all_values_less :forall (n c :nat)(l:list nat),
(largest_value (length (c :: l))
(change_variable n (c :: l)) <= n).
First in this way,
Inductive changing : l -> Prop :=
| change_nil : changing nil
| change_1 n : changing (cons n nil)
| change_head n h l :
n <= h ->
changing (cons h l) ->
changing (cons n l).
Lemma head_is_gt l a:
changing l -> forall n, In n l -> n <= hd a l.
Proof.
induction 1. intros k H'.
now exfalso; apply in_nil in H'.
Admitted.
Secondly ,
Definition head_is_greater (l: list nat): nil <> l -> nat.
intros.
destruct l.
destruct (H (#erefl (list nat) nil)).
apply : (largest_value s l).
Defined.
Theorem values_les_n : forall l (H : [] <> l) n, In n l -> n <=
head_is_greater H.

Answer 1

The theorem does not hold, unfortunately. Here is a counterexample:
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Require Import Lia.
Import ListNotations.
Definition change_variable (n: nat) (l: list nat) : list nat:=
match l with
| nil => l
| h::t => if n <=? h then l else n::t
end.
Fixpoint largest_value (numbers: nat) (l: list nat) {struct numbers}: nat:=
match l with
| nil => 0
| cons b nil => b
| cons h l => match numbers with
| O => h
| S numbers' => largest_value numbers' (change_variable h l)
end
end.
Hypothesis all_values_less :forall (n c :nat)(l:list nat),
(largest_value (length (c :: l))
(change_variable n (c :: l)) <= n).
Theorem contra : False.
Proof.
pose proof (all_values_less 0 1 []).
simpl in *. lia.
Qed.
You probably need to add more hypothesis to your statement to rule out such cases.