Section before_func.
Variable A : Type.
Variables f g : A -> bool.
Definition before_func_dec : forall l, {before_func f g l} + {~ before_func f g l}.
intros; induction l; simpl in *.
- intuition.
- destruct (f a); destruct (g a); intuition congruence.
Defined.
Proof using.
intros. induction l; simpl in *; intuition.
Qed.
Proof using.
induction l; simpl; intuition.
- eauto.
- congruence.
- congruence.
Qed.
Proof using.
induction l'; intros; simpl in *; intuition.
Qed.

Lemma before_func_before :
forall l,
before_func f g l ->
forall y,
g y = true ->
exists x : A,
f x = true /\
before x y l.