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Jcinitializer

Jcinitializer-equiv

Basic equivalence relation for jcinitializer structures.

Definitions and Theorems

Function: jcinitializer-equiv$inline

(defun jcinitializer-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (jcinitializerp acl2::x)
                              (jcinitializerp acl2::y))))
  (equal (jcinitializer-fix acl2::x)
         (jcinitializer-fix acl2::y)))

Theorem: jcinitializer-equiv-is-an-equivalence

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

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

(defthm jcinitializer-equiv-implies-equal-jcinitializer-fix-1
  (implies (jcinitializer-equiv acl2::x x-equiv)
           (equal (jcinitializer-fix acl2::x)
                  (jcinitializer-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: jcinitializer-fix-under-jcinitializer-equiv

(defthm jcinitializer-fix-under-jcinitializer-equiv
  (jcinitializer-equiv (jcinitializer-fix acl2::x)
                       acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-jcinitializer-fix-1-forward-to-jcinitializer-equiv

(defthm equal-of-jcinitializer-fix-1-forward-to-jcinitializer-equiv
  (implies (equal (jcinitializer-fix acl2::x)
                  acl2::y)
           (jcinitializer-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-jcinitializer-fix-2-forward-to-jcinitializer-equiv

(defthm equal-of-jcinitializer-fix-2-forward-to-jcinitializer-equiv
  (implies (equal acl2::x (jcinitializer-fix acl2::y))
           (jcinitializer-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: jcinitializer-equiv-of-jcinitializer-fix-1-forward

(defthm jcinitializer-equiv-of-jcinitializer-fix-1-forward
  (implies (jcinitializer-equiv (jcinitializer-fix acl2::x)
                                acl2::y)
           (jcinitializer-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: jcinitializer-equiv-of-jcinitializer-fix-2-forward

(defthm jcinitializer-equiv-of-jcinitializer-fix-2-forward
  (implies (jcinitializer-equiv acl2::x (jcinitializer-fix acl2::y))
           (jcinitializer-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)