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Fhg-args

Fhg-args-fix

Fixing function for fhg-args structures.

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

Definitions and Theorems

Function: fhg-args-fix$inline

(defun fhg-args-fix$inline (x)
 (declare (xargs :guard (fhg-args-p x)))
 (let ((acl2::__function__ 'fhg-args-fix))
  (declare (ignorable acl2::__function__))
  (mbe
   :logic
   (b*
    ((term-lst (pseudo-term-list-fix (cdr (std::da-nth 0 x))))
     (fn-lst (func-alist-fix (cdr (std::da-nth 1 x))))
     (fn-returns-hint-acc
          (hint-pair-list-fix (cdr (std::da-nth 2 x))))
     (fn-more-returns-hint-acc
          (hint-pair-list-fix (cdr (std::da-nth 3 x))))
     (lambda-acc (lambda-binding-list-fix (cdr (std::da-nth 4 x)))))
    (list (cons 'term-lst term-lst)
          (cons 'fn-lst fn-lst)
          (cons 'fn-returns-hint-acc
                fn-returns-hint-acc)
          (cons 'fn-more-returns-hint-acc
                fn-more-returns-hint-acc)
          (cons 'lambda-acc lambda-acc)))
   :exec x)))

Theorem: fhg-args-p-of-fhg-args-fix

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

Theorem: fhg-args-fix-when-fhg-args-p

(defthm fhg-args-fix-when-fhg-args-p
  (implies (fhg-args-p x)
           (equal (fhg-args-fix x) x)))

Function: fhg-args-equiv$inline

(defun fhg-args-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (fhg-args-p acl2::x)
                              (fhg-args-p acl2::y))))
  (equal (fhg-args-fix acl2::x)
         (fhg-args-fix acl2::y)))

Theorem: fhg-args-equiv-is-an-equivalence

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

Theorem: fhg-args-equiv-implies-equal-fhg-args-fix-1

(defthm fhg-args-equiv-implies-equal-fhg-args-fix-1
  (implies (fhg-args-equiv acl2::x x-equiv)
           (equal (fhg-args-fix acl2::x)
                  (fhg-args-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: fhg-args-fix-under-fhg-args-equiv

(defthm fhg-args-fix-under-fhg-args-equiv
  (fhg-args-equiv (fhg-args-fix acl2::x)
                  acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-fhg-args-fix-1-forward-to-fhg-args-equiv

(defthm equal-of-fhg-args-fix-1-forward-to-fhg-args-equiv
  (implies (equal (fhg-args-fix acl2::x) acl2::y)
           (fhg-args-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-fhg-args-fix-2-forward-to-fhg-args-equiv

(defthm equal-of-fhg-args-fix-2-forward-to-fhg-args-equiv
  (implies (equal acl2::x (fhg-args-fix acl2::y))
           (fhg-args-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: fhg-args-equiv-of-fhg-args-fix-1-forward

(defthm fhg-args-equiv-of-fhg-args-fix-1-forward
  (implies (fhg-args-equiv (fhg-args-fix acl2::x)
                           acl2::y)
           (fhg-args-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: fhg-args-equiv-of-fhg-args-fix-2-forward

(defthm fhg-args-equiv-of-fhg-args-fix-2-forward
  (implies (fhg-args-equiv acl2::x (fhg-args-fix acl2::y))
           (fhg-args-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)