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• Unary-op

Unary-op-equiv

Basic equivalence relation for unary-op structures.

Definitions and Theorems

Function: unary-op-equiv$inline

(defun unary-op-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (unary-opp acl2::x)
                              (unary-opp acl2::y))))
  (equal (unary-op-fix acl2::x)
         (unary-op-fix acl2::y)))

Theorem: unary-op-equiv-is-an-equivalence

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

Theorem: unary-op-equiv-implies-equal-unary-op-fix-1

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

Theorem: unary-op-fix-under-unary-op-equiv

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

Theorem: equal-of-unary-op-fix-1-forward-to-unary-op-equiv

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

Theorem: equal-of-unary-op-fix-2-forward-to-unary-op-equiv

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

Theorem: unary-op-equiv-of-unary-op-fix-1-forward

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

Theorem: unary-op-equiv-of-unary-op-fix-2-forward

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