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• Pseudo-lambda

Pseudo-lambda-fix

Signature

(pseudo-lambda-fix x) → lambda

Arguments

x — Guard (pseudo-lambda-p x).

Returns

lambda — Type (pseudo-lambda-p lambda).

Definitions and Theorems

Function: pseudo-lambda-fix

(defun pseudo-lambda-fix (x)
  (declare (xargs :guard (pseudo-lambda-p x)))
  (let ((__function__ 'pseudo-lambda-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((formals (cadr x)) (body (caddr x)))
           (list 'lambda
                 (replace-non-symbols-with-nil formals)
                 (pseudo-term-fix body)))
         :exec x)))

Theorem: pseudo-lambda-p-of-pseudo-lambda-fix

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

Theorem: pseudo-lambda-fix-when-pseudo-lambda-p

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

Function: pseudo-lambda-equiv$inline

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

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

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

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

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

Theorem: pseudo-lambda-fix-under-pseudo-lambda-equiv

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

Theorem: pseudo-lambda-fix-of-pseudo-lambda-fix-x

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

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

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