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• Pseudo-fnsym

Pseudo-fnsym-fix

Signature

(pseudo-fnsym-fix x) → new-x

Arguments

x — Guard (pseudo-fnsym-p x).

Returns

new-x — Type (pseudo-fnsym-p new-x).

Definitions and Theorems

Function: pseudo-fnsym-fix

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

Theorem: pseudo-fnsym-p-of-pseudo-fnsym-fix

(defthm pseudo-fnsym-p-of-pseudo-fnsym-fix
  (b* ((new-x (pseudo-fnsym-fix x)))
    (pseudo-fnsym-p new-x))
  :rule-classes
  (:rewrite (:type-prescription :typed-term (pseudo-fnsym-fix x))))

Theorem: pseudo-fnsym-fix-when-pseudo-fnsym-p

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

Function: pseudo-fnsym-equiv$inline

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

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

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

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

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

Theorem: pseudo-fnsym-fix-under-pseudo-fnsym-equiv

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

Theorem: not-quote-of-pseudo-fnsym-fix

(defthm not-quote-of-pseudo-fnsym-fix
  (not (equal (pseudo-fnsym-fix x) 'quote)))

Theorem: pseudo-fnsym-fix-of-pseudo-fnsym-fix-x

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

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

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