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• Boolean-value

Boolean-value-fix

Fixing function for boolean-value structures.

Signature

(boolean-value-fix x) → new-x

Arguments

x — Guard (boolean-valuep x).

Returns

new-x — Type (boolean-valuep new-x).

Definitions and Theorems

Function: boolean-value-fix$inline

(defun boolean-value-fix$inline (x)
  (declare (xargs :guard (boolean-valuep x)))
  (let ((__function__ 'boolean-value-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((bool (acl2::bool-fix (std::da-nth 0 (cdr x)))))
           (cons :boolean (list bool)))
         :exec x)))

Theorem: boolean-valuep-of-boolean-value-fix

(defthm boolean-valuep-of-boolean-value-fix
  (b* ((new-x (boolean-value-fix$inline x)))
    (boolean-valuep new-x))
  :rule-classes :rewrite)

Theorem: boolean-value-fix-when-boolean-valuep

(defthm boolean-value-fix-when-boolean-valuep
  (implies (boolean-valuep x)
           (equal (boolean-value-fix x) x)))

Function: boolean-value-equiv$inline

(defun boolean-value-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (boolean-valuep acl2::x)
                              (boolean-valuep acl2::y))))
  (equal (boolean-value-fix acl2::x)
         (boolean-value-fix acl2::y)))

Theorem: boolean-value-equiv-is-an-equivalence

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

Theorem: boolean-value-equiv-implies-equal-boolean-value-fix-1

(defthm boolean-value-equiv-implies-equal-boolean-value-fix-1
  (implies (boolean-value-equiv acl2::x x-equiv)
           (equal (boolean-value-fix acl2::x)
                  (boolean-value-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: boolean-value-fix-under-boolean-value-equiv

(defthm boolean-value-fix-under-boolean-value-equiv
  (boolean-value-equiv (boolean-value-fix acl2::x)
                       acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-boolean-value-fix-1-forward-to-boolean-value-equiv

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

Theorem: equal-of-boolean-value-fix-2-forward-to-boolean-value-equiv

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

Theorem: boolean-value-equiv-of-boolean-value-fix-1-forward

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

Theorem: boolean-value-equiv-of-boolean-value-fix-2-forward

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