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• Atj-test

Atj-test-equiv

Basic equivalence relation for atj-test structures.

Definitions and Theorems

Function: atj-test-equiv$inline

(defun atj-test-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (atj-testp acl2::x)
                              (atj-testp acl2::y))))
  (equal (atj-test-fix acl2::x)
         (atj-test-fix acl2::y)))

Theorem: atj-test-equiv-is-an-equivalence

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

Theorem: atj-test-equiv-implies-equal-atj-test-fix-1

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

Theorem: atj-test-fix-under-atj-test-equiv

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

Theorem: equal-of-atj-test-fix-1-forward-to-atj-test-equiv

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

Theorem: equal-of-atj-test-fix-2-forward-to-atj-test-equiv

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

Theorem: atj-test-equiv-of-atj-test-fix-1-forward

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

Theorem: atj-test-equiv-of-atj-test-fix-2-forward

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