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Bexp

Bexp-equiv

Basic equivalence relation for bexp structures.

Definitions and Theorems

Function: bexp-equiv$inline

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

Theorem: bexp-equiv-is-an-equivalence

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

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

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

Theorem: bexp-fix-under-bexp-equiv

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

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

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

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

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

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

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

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

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