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Sign-option

Sign-option-fix

Fixing function for sign-option structures.

Signature
(sign-option-fix x) → new-x
Arguments: x — Guard (sign-optionp x).
Returns: new-x — Type (sign-optionp new-x).

Definitions and Theorems

Function: sign-option-fix$inline

(defun sign-option-fix$inline (x)
  (declare (xargs :guard (sign-optionp x)))
  (let ((__function__ 'sign-option-fix))
    (declare (ignorable __function__))
    (mbe :logic (cond ((not x) nil)
                      (t (b* ((fty::val (sign-fix x)))
                           fty::val)))
         :exec x)))

Theorem: sign-optionp-of-sign-option-fix

(defthm sign-optionp-of-sign-option-fix
  (b* ((new-x (sign-option-fix$inline x)))
    (sign-optionp new-x))
  :rule-classes :rewrite)

Theorem: sign-option-fix-when-sign-optionp

(defthm sign-option-fix-when-sign-optionp
  (implies (sign-optionp x)
           (equal (sign-option-fix x) x)))

Function: sign-option-equiv$inline

(defun sign-option-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (sign-optionp acl2::x)
                              (sign-optionp acl2::y))))
  (equal (sign-option-fix acl2::x)
         (sign-option-fix acl2::y)))

Theorem: sign-option-equiv-is-an-equivalence

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

Theorem: sign-option-equiv-implies-equal-sign-option-fix-1

(defthm sign-option-equiv-implies-equal-sign-option-fix-1
  (implies (sign-option-equiv acl2::x x-equiv)
           (equal (sign-option-fix acl2::x)
                  (sign-option-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: sign-option-fix-under-sign-option-equiv

(defthm sign-option-fix-under-sign-option-equiv
  (sign-option-equiv (sign-option-fix acl2::x)
                     acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-sign-option-fix-1-forward-to-sign-option-equiv

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

Theorem: equal-of-sign-option-fix-2-forward-to-sign-option-equiv

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

Theorem: sign-option-equiv-of-sign-option-fix-1-forward

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

Theorem: sign-option-equiv-of-sign-option-fix-2-forward

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