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• Iconst-option

Iconst-option-equiv

Basic equivalence relation for iconst-option structures.

Definitions and Theorems

Function: iconst-option-equiv$inline

(defun iconst-option-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (iconst-optionp acl2::x)
                              (iconst-optionp acl2::y))))
  (equal (iconst-option-fix acl2::x)
         (iconst-option-fix acl2::y)))

Theorem: iconst-option-equiv-is-an-equivalence

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

Theorem: iconst-option-equiv-implies-equal-iconst-option-fix-1

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

Theorem: iconst-option-fix-under-iconst-option-equiv

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

Theorem: equal-of-iconst-option-fix-1-forward-to-iconst-option-equiv

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

Theorem: equal-of-iconst-option-fix-2-forward-to-iconst-option-equiv

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

Theorem: iconst-option-equiv-of-iconst-option-fix-1-forward

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

Theorem: iconst-option-equiv-of-iconst-option-fix-2-forward

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