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• Litp

Lit-equiv

Basic equivalence relation for literals.

Signature

(lit-equiv x y) → *

Arguments

x — Guard (litp x).

y — Guard (litp y).

Definitions and Theorems

Function: lit-equiv

(defun lit-equiv (x y)
  (declare (xargs :guard (and (litp x) (litp y))))
  (let ((__function__ 'lit-equiv))
    (declare (ignorable __function__))
    (int= (lit-fix x) (lit-fix y))))

Theorem: lit-equiv-is-an-equivalence

(defthm lit-equiv-is-an-equivalence
  (and (booleanp (lit-equiv x y))
       (lit-equiv x x)
       (implies (lit-equiv x y)
                (lit-equiv y x))
       (implies (and (lit-equiv x y) (lit-equiv y z))
                (lit-equiv x z)))
  :rule-classes (:equivalence))

Theorem: lit-equiv-implies-equal-lit-fix-1

(defthm lit-equiv-implies-equal-lit-fix-1
  (implies (lit-equiv x x-equiv)
           (equal (lit-fix x) (lit-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: lit-equiv-of-lit-fix

(defthm lit-equiv-of-lit-fix
  (lit-equiv (lit-fix lit) lit))

Theorem: __deffixtype-lit-equiv-means-equal-of-lit-fix

(defthm acl2::__deffixtype-lit-equiv-means-equal-of-lit-fix
  (implies (lit-equiv acl2::x acl2::y)
           (equal (lit-fix acl2::x)
                  (lit-fix acl2::y)))
  :rule-classes nil)

Theorem: lit-equiv-of-lit-fix-x

(defthm lit-equiv-of-lit-fix-x
  (equal (lit-equiv (lit-fix x) y)
         (lit-equiv x y)))

Theorem: lit-equiv-lit-equiv-congruence-on-x

(defthm lit-equiv-lit-equiv-congruence-on-x
  (implies (lit-equiv x x-equiv)
           (equal (lit-equiv x y)
                  (lit-equiv x-equiv y)))
  :rule-classes :congruence)

Theorem: lit-equiv-of-lit-fix-y

(defthm lit-equiv-of-lit-fix-y
  (equal (lit-equiv x (lit-fix y))
         (lit-equiv x y)))

Theorem: lit-equiv-lit-equiv-congruence-on-y

(defthm lit-equiv-lit-equiv-congruence-on-y
  (implies (lit-equiv y y-equiv)
           (equal (lit-equiv x y)
                  (lit-equiv x y-equiv)))
  :rule-classes :congruence)