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Unop

Unop-equiv

Basic equivalence relation for unop structures.

Definitions and Theorems

Function: unop-equiv$inline

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

Theorem: unop-equiv-is-an-equivalence

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

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

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

Theorem: unop-fix-under-unop-equiv

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

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

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

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

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

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

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

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

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