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• Qualified-ident-option

Qualified-ident-option-fix

Fixing function for qualified-ident-option structures.

Signature

(qualified-ident-option-fix x) → new-x

Arguments

x — Guard (qualified-ident-optionp x).

Returns

new-x — Type (qualified-ident-optionp new-x).

Definitions and Theorems

Function: qualified-ident-option-fix$inline

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

Theorem: qualified-ident-optionp-of-qualified-ident-option-fix

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

Theorem: qualified-ident-option-fix-when-qualified-ident-optionp

(defthm qualified-ident-option-fix-when-qualified-ident-optionp
  (implies (qualified-ident-optionp x)
           (equal (qualified-ident-option-fix x)
                  x)))

Function: qualified-ident-option-equiv$inline

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

Theorem: qualified-ident-option-equiv-is-an-equivalence

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

Theorem: qualified-ident-option-equiv-implies-equal-qualified-ident-option-fix-1

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

Theorem: qualified-ident-option-fix-under-qualified-ident-option-equiv

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

Theorem: equal-of-qualified-ident-option-fix-1-forward-to-qualified-ident-option-equiv

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

Theorem: equal-of-qualified-ident-option-fix-2-forward-to-qualified-ident-option-equiv

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

Theorem: qualified-ident-option-equiv-of-qualified-ident-option-fix-1-forward

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

Theorem: qualified-ident-option-equiv-of-qualified-ident-option-fix-2-forward

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