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Lifetime-option

Lifetime-option-fix

Fixing function for lifetime-option structures.

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

Definitions and Theorems

Function: lifetime-option-fix$inline

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

Theorem: lifetime-optionp-of-lifetime-option-fix

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

Theorem: lifetime-option-fix-when-lifetime-optionp

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

Function: lifetime-option-equiv$inline

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

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

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

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

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

Theorem: lifetime-option-fix-under-lifetime-option-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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