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• Assertion

Assertion-equiv

Basic equivalence relation for assertion structures.

Definitions and Theorems

Function: assertion-equiv$inline

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

Theorem: assertion-equiv-is-an-equivalence

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

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

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

Theorem: assertion-fix-under-assertion-equiv

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

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

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

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

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

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

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

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

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