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Lhatom

Lhatom-fix

Fixing function for lhatom structures.

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

Definitions and Theorems

Function: lhatom-fix$inline

(defun lhatom-fix$inline (x)
 (declare (xargs :guard (lhatom-p x)))
 (let ((__function__ 'lhatom-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (case (lhatom-kind x)
    (:z ':z)
    (:var
     (b*
      ((name
          (svar-fix (if (or (atom x)
                            (and (eq (car x) :var) (consp (cdr x))))
                        x
                      (car x))))
       (rsh (nfix (if (or (atom x)
                          (and (eq (car x) :var) (consp (cdr x))))
                      0
                    (cdr x)))))
      (if (and (eql 0 rsh) (not (eq name :z)))
          name
        (cons name rsh)))))
   :exec x)))

Theorem: lhatom-p-of-lhatom-fix

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

Theorem: lhatom-fix-when-lhatom-p

(defthm lhatom-fix-when-lhatom-p
  (implies (lhatom-p x)
           (equal (lhatom-fix x) x)))

Function: lhatom-equiv$inline

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

Theorem: lhatom-equiv-is-an-equivalence

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

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

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

Theorem: lhatom-fix-under-lhatom-equiv

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

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

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

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

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

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

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

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

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

Theorem: lhatom-kind$inline-of-lhatom-fix-x

(defthm lhatom-kind$inline-of-lhatom-fix-x
  (equal (lhatom-kind$inline (lhatom-fix x))
         (lhatom-kind$inline x)))

Theorem: lhatom-kind$inline-lhatom-equiv-congruence-on-x

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