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Hint-pair

Hint-pair-fix

Fixing function for hint-pair structures.

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

Definitions and Theorems

Function: hint-pair-fix$inline

(defun hint-pair-fix$inline (x)
  (declare (xargs :guard (hint-pair-p x)))
  (let ((acl2::__function__ 'hint-pair-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (b* ((thm (pseudo-term-fix (cdr (std::da-nth 0 x))))
              (hints (true-list-fix (cdr (std::da-nth 1 x)))))
           (list (cons 'thm thm)
                 (cons 'hints hints)))
         :exec x)))

Theorem: hint-pair-p-of-hint-pair-fix

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

Theorem: hint-pair-fix-when-hint-pair-p

(defthm hint-pair-fix-when-hint-pair-p
  (implies (hint-pair-p x)
           (equal (hint-pair-fix x) x)))

Function: hint-pair-equiv$inline

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

Theorem: hint-pair-equiv-is-an-equivalence

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

Theorem: hint-pair-equiv-implies-equal-hint-pair-fix-1

(defthm hint-pair-equiv-implies-equal-hint-pair-fix-1
  (implies (hint-pair-equiv acl2::x x-equiv)
           (equal (hint-pair-fix acl2::x)
                  (hint-pair-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: hint-pair-fix-under-hint-pair-equiv

(defthm hint-pair-fix-under-hint-pair-equiv
  (hint-pair-equiv (hint-pair-fix acl2::x)
                   acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-hint-pair-fix-1-forward-to-hint-pair-equiv

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

Theorem: equal-of-hint-pair-fix-2-forward-to-hint-pair-equiv

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

Theorem: hint-pair-equiv-of-hint-pair-fix-1-forward

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

Theorem: hint-pair-equiv-of-hint-pair-fix-2-forward

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