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Pseudo-fn

Pseudo-fn-fix

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

Definitions and Theorems

Function: pseudo-fn-fix

(defun pseudo-fn-fix (x)
  (declare (xargs :guard (pseudo-fn-p x)))
  (let ((__function__ 'pseudo-fn-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (consp x)
             (pseudo-lambda-fix x)
           (pseudo-fnsym-fix x))
         :exec x)))

Theorem: pseudo-fn-p-of-pseudo-fn-fix

(defthm pseudo-fn-p-of-pseudo-fn-fix
  (b* ((new-x (pseudo-fn-fix x)))
    (pseudo-fn-p new-x))
  :rule-classes :rewrite)

Theorem: pseudo-fn-fix-when-pseudo-fn-p

(defthm pseudo-fn-fix-when-pseudo-fn-p
  (implies (pseudo-fn-p x)
           (equal (pseudo-fn-fix x) x)))

Theorem: pseudo-fn-fix-when-consp

(defthm pseudo-fn-fix-when-consp
  (implies (consp x)
           (equal (pseudo-fn-fix x)
                  (pseudo-lambda-fix x))))

Theorem: pseudo-fn-fix-when-not-consp

(defthm pseudo-fn-fix-when-not-consp
  (implies (not (consp x))
           (equal (pseudo-fn-fix x)
                  (pseudo-fnsym-fix x))))

Function: pseudo-fn-equiv$inline

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

Theorem: pseudo-fn-equiv-is-an-equivalence

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

Theorem: pseudo-fn-equiv-implies-equal-pseudo-fn-fix-1

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

Theorem: pseudo-fn-fix-under-pseudo-fn-equiv

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

Theorem: pseudo-fn-fix-of-pseudo-fn-fix-x

(defthm pseudo-fn-fix-of-pseudo-fn-fix-x
  (equal (pseudo-fn-fix (pseudo-fn-fix x))
         (pseudo-fn-fix x)))

Theorem: pseudo-fn-fix-pseudo-fn-equiv-congruence-on-x

(defthm pseudo-fn-fix-pseudo-fn-equiv-congruence-on-x
  (implies (pseudo-fn-equiv x x-equiv)
           (equal (pseudo-fn-fix x)
                  (pseudo-fn-fix x-equiv)))
  :rule-classes :congruence)