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• Aexp

Aexp-equiv

Basic equivalence relation for aexp structures.

Definitions and Theorems

Function: aexp-equiv$inline

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

Theorem: aexp-equiv-is-an-equivalence

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

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

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

Theorem: aexp-fix-under-aexp-equiv

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

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

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

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

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

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

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

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

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