We check for equivalency of two given logical formulas.

The basic outline of the program is below:

1. Use the Identity, Annihilation, and Inverse laws to simplify expressions involving ⊥ and ⊤. Note that this will either eliminate all ⊥ and ⊤’s, or one ends up with ⊥ or ⊤ as the final statement. If the resulting statement is ⊥, then the statement is in CDNF, and we can stop.
   We’ll also stop if the statement is ⊤, and A(ϕ, ψ) is empty. If the statement is ⊤, and A(ϕ, ψ) is not empty, then use Complement to replace with A ∨ ¬A where A ∈ A(ϕ, ψ), and proceed.
2. Put the statement into NNF by repeated applications of DeMorgan and Double Negation.
3. Put the statement into DNF by repeated applications of Distribution of ∧ over ∨.
4. ** Use Adjacency to ensure that every conjunction contains a literal A or ¬A for every A ∈ A(ϕ, ψ). That is: use Adjacency to ‘add’ literals that are ‘missing’ in any conjunction. E.g. if A(ϕ, ψ) ={P,Q, R}, then a conjunct such as P ∧ Q becomes (P ∧ Q ∧ R) ∨ (P ∧ Q ∧ ¬R)
5. ** Use Commutation to order the conjuncts in any conjunction according to O. E.g. A ∧ B ∧ A ∧ ¬A becomes A ∧ A ∧ ¬A ∧ B
6. Use Idempotence to eliminate duplicate conjuncts. E.g. A ∧ A ∧ ¬A ∧ B becomes A ∧ ¬A ∧ B
7. Use Complement and Annihilation to replace any conjunction containing a literal and its negation with a contradiction symbol. E.g. A ∧ ¬A ∧ B becomes ⊥ ∧ B becomes ⊥.
8. Use Identity to get rid of all disjuncts that are ⊥. If there is only one disjunct ⊥ left, then the statement is in CDNF and the process can stop.
9. Use Idempotence to eliminate any duplicate disjuncts. (If didn't stop at step 8)
10. Finally, use Commutation to get all disjuncts in the ‘right’ order. (If didn't stop at step 8)

It uses a functional format for inputting logical formulas. This is the base identity for inputs:

A
not(A)
and(A, B)
or(A, B)
if(A, B)
iff(A, B)

where A and B can either be atomic statements or a functional operator. All operators are either unary (not) or binary (and, or, if, iff) and there is no support for a generalized notation. This means that and(A, B, C) will thrown an error.