simp only [add_le_add_iff_right, projection_length_leq]
f : Formula
S : Finset Formula
f_not_in_S : f ∉ S
IH : lengthOfSet (projection S) ≤ lengthOfSet S
insert_comm_proj : ∀ (X : Finset Formula) (f : Formula), projection (insert f X) = projection {f} ∪ projection X
⊢ (projection {f}).sum lengthOfFormula + (projection S).sum lengthOfFormula ≤
    lengthOfFormula f + (projection S).sum lengthOfFormula
no goals