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• Qualified-ident-option-set

Qualified-ident-option-set-fix

(qualified-ident-option-set-fix x) is a usual ACL2::fty set fixing function.

Signature

(qualified-ident-option-set-fix x) → *

Arguments

x — Guard (qualified-ident-option-setp x).

In the logic, we apply qualified-ident-option-fix to each member of the x. In the execution, none of that is actually necessary and this is just an inlined identity function.

Definitions and Theorems

Function: qualified-ident-option-set-fix

(defun qualified-ident-option-set-fix (x)
  (declare (xargs :guard (qualified-ident-option-setp x)))
  (mbe :logic
       (if (qualified-ident-option-setp x)
           x
         nil)
       :exec x))

Theorem: qualified-ident-option-setp-of-qualified-ident-option-set-fix

(defthm
      qualified-ident-option-setp-of-qualified-ident-option-set-fix
  (qualified-ident-option-setp (qualified-ident-option-set-fix x)))

Theorem: qualified-ident-option-set-fix-when-qualified-ident-option-setp

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

Theorem: emptyp-qualified-ident-option-set-fix

(defthm emptyp-qualified-ident-option-set-fix
  (implies (or (emptyp x)
               (not (qualified-ident-option-setp x)))
           (emptyp (qualified-ident-option-set-fix x))))

Theorem: emptyp-of-qualified-ident-option-set-fix

(defthm emptyp-of-qualified-ident-option-set-fix
  (equal (emptyp (qualified-ident-option-set-fix x))
         (or (not (qualified-ident-option-setp x))
             (emptyp x))))

Function: qualified-ident-option-set-equiv$inline

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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