CNF_CONV

normalForms.CNF_CONV : conv

Converts a formula into Conjunctive Normal Form (CNF).

Given a formula consisting of truths, falsities, conjunctions, disjunctions, negations, equivalences, conditionals, and universal and existential quantifiers, CNF_CONV will convert it to the canonical form:

?a_1 ... a_k.
  (!v_1 ... v_m1. P_1 \/ ... \/ P_n1) /\
  ...                                 /\
  (!v_1 ... v_mp. P_1 \/ ... \/ P_np)

The P_ij are literals: possibly-negated atoms. In first-order logic an atom is a formula consisting of a top-level relation symbol applied to first-order terms: function symbols and variables. In higher-order logic there is no distinction between formulas and terms, so the concept of atom is not well-formed. Note also that the a_i existentially bound variables may be functions, as a result of Skolemization.

Failure

CNF_CONV should never fail.

Example

> normalForms.CNF_CONV ``!x. P x ==> ?y z. Q y \/ ~?z. P z /\ Q z``;
val it =
   ⊢ (∀x. P x ⇒ ∃y z. Q y ∨ ¬∃z. P z ∧ Q z) ⇔
     ∃y. ∀x z. ¬Q z ∨ ¬P z ∨ Q (y x) ∨ ¬P x: thm

Example

> normalForms.CNF_CONV ``~(~(x = y) = z) = ~(x = ~(y = z))``;
val it = ⊢ (((x ⇎ y) ⇎ z) ⇔ (x ⇎ (y ⇎ z))) ⇔ T: thm