Fixpoint weight (a : form) : nat :=
  match a with
  | Falsum => 1
  | Atom _ => 1
  | AndF a b => S (weight a + weight b)
  | OrF Falsum b => S (weight b)
  | OrF (Atom _) b => S (weight b)
  | OrF a b => S (S (weight b + weight a))
  | Imp a b => weight_neg a + weight b
  end
 
 with weight_neg (a : form) : nat :=
  match a with
  | Falsum => 0
  | Atom _ => 0
  | AndF a b => S (weight_neg a + weight_neg b)
  | OrF a b => S (S (S (weight_neg a + weight_neg b)))
  | Imp Falsum b => 1
  | Imp (Atom _) Falsum => 2
  | Imp (Atom _) (Atom _) => 1
  | Imp (Atom _) b => S (S (S (weight_neg b)))
  | Imp a b => S (S (S (S (weight_neg b + weight a))))
  end.Fixpoint weight_goal (a : form) : nat :=
  match a with
  | Falsum => 0
  | Atom _ => 0
  | AndF _ _ => 1
  | OrF _ _ => 1
  | Imp _ b => S (weight_goal b)
  end.Definition weight_gamma :=
  fold_right (fun (a : form) (n : nat) => weight a + n) 0.

Lemma weight_neg_le : forall (j : Int) (a : form), weight_neg (Imp (Atom j) a) <= S (S (S (weight_neg a))).