End ListFun2.
Existing Instance proper_fold_left.
Section ListFun1.
Context {A: Type}.
Context {RA: relation A}.
Context {EqRA: Equivalence RA}.

Lemma monoid_fold_left_head: forall {f} {Proper_f: Proper (equiv ==> equiv ==> equiv) f} (e: A) a l,
(forall x, f e x === x) ->
(forall x, f x e === x) ->
(forall x y z, f (f x y) z === f x (f y z)) ->
fold_left f (a :: l) e === f a (fold_left f l e).