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Type-option

Type-option-fix

Fixing function for type-option structures.

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

Definitions and Theorems

Function: type-option-fix$inline

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

Theorem: type-optionp-of-type-option-fix

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

Theorem: type-option-fix-when-type-optionp

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

Function: type-option-equiv$inline

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

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

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

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

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

Theorem: type-option-fix-under-type-option-equiv

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

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

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

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

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

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

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

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

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

Theorem: type-option-fix$inline-of-type-option-fix-x

(defthm type-option-fix$inline-of-type-option-fix-x
  (equal (type-option-fix$inline (type-option-fix x))
         (type-option-fix$inline x)))

Theorem: type-option-fix$inline-type-option-equiv-congruence-on-x

(defthm type-option-fix$inline-type-option-equiv-congruence-on-x
  (implies (type-option-equiv x x-equiv)
           (equal (type-option-fix$inline x)
                  (type-option-fix$inline x-equiv)))
  :rule-classes :congruence)