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Rule

Rule-fix

Fixing function for rule structures.

Signature
(rule-fix x) → new-x
Arguments: x — Guard (rulep x).
Returns: new-x — Type (rulep new-x).

Definitions and Theorems

Function: rule-fix$inline

(defun rule-fix$inline (x)
  (declare (xargs :guard (rulep x)))
  (let ((__function__ 'rule-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((name (rulename-fix (std::da-nth 0 (cdr x))))
              (incremental (acl2::bool-fix (std::da-nth 1 (cdr x))))
              (definiens (alternation-fix (std::da-nth 2 (cdr x)))))
           (cons :rule (list name incremental definiens)))
         :exec x)))

Theorem: rulep-of-rule-fix

(defthm rulep-of-rule-fix
  (b* ((new-x (rule-fix$inline x)))
    (rulep new-x))
  :rule-classes :rewrite)

Theorem: rule-fix-when-rulep

(defthm rule-fix-when-rulep
  (implies (rulep x)
           (equal (rule-fix x) x)))

Function: rule-equiv$inline

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

Theorem: rule-equiv-is-an-equivalence

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

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

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

Theorem: rule-fix-under-rule-equiv

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

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

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

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

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

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

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

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

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