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Fgl-rule

Fgl-rule-fix

Fixing function for fgl-rule structures.

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

Definitions and Theorems

Function: fgl-rule-fix$inline

(defun fgl-rule-fix$inline (x)
 (declare (xargs :guard (fgl-rule-p x)))
 (let ((__function__ 'fgl-rule-fix))
  (declare (ignorable __function__))
  (mbe
    :logic
    (case (fgl-rule-kind x)
      (:rewrite (b* ((rune (fgl-rune-fix (car (cdr x))))
                     (rule (cmr::rewrite-fix (cdr (cdr x)))))
                  (cons :rewrite (cons rune rule))))
      (:primitive
           (b* ((name (pseudo-fnsym-fix (std::da-nth 0 (cdr x)))))
             (cons :primitive (list name))))
      (:meta (b* ((name (pseudo-fnsym-fix (std::da-nth 0 (cdr x)))))
               (cons :meta (list name)))))
    :exec x)))

Theorem: fgl-rule-p-of-fgl-rule-fix

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

Theorem: fgl-rule-fix-when-fgl-rule-p

(defthm fgl-rule-fix-when-fgl-rule-p
  (implies (fgl-rule-p x)
           (equal (fgl-rule-fix x) x)))

Function: fgl-rule-equiv$inline

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

Theorem: fgl-rule-equiv-is-an-equivalence

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

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

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

Theorem: fgl-rule-fix-under-fgl-rule-equiv

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

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

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

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

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

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

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

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

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

Theorem: fgl-rule-kind$inline-of-fgl-rule-fix-x

(defthm fgl-rule-kind$inline-of-fgl-rule-fix-x
  (equal (fgl-rule-kind$inline (fgl-rule-fix x))
         (fgl-rule-kind$inline x)))

Theorem: fgl-rule-kind$inline-fgl-rule-equiv-congruence-on-x

(defthm fgl-rule-kind$inline-fgl-rule-equiv-congruence-on-x
  (implies (fgl-rule-equiv x x-equiv)
           (equal (fgl-rule-kind$inline x)
                  (fgl-rule-kind$inline x-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-fgl-rule-fix

(defthm consp-of-fgl-rule-fix
  (consp (fgl-rule-fix x))
  :rule-classes :type-prescription)