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Const-fenv

Const-fenv-fix

(const-fenv-fix x) is a usual ACL2::fty omap fixing function.

Signature
(const-fenv-fix x) → *
Arguments: x — Guard (const-fenvp x).

Definitions and Theorems

Function: const-fenv-fix

(defun const-fenv-fix (x)
  (declare (xargs :guard (const-fenvp x)))
  (mbe :logic (if (const-fenvp x) x nil)
       :exec x))

Theorem: const-fenvp-of-const-fenv-fix

(defthm const-fenvp-of-const-fenv-fix
  (const-fenvp (const-fenv-fix x)))

Theorem: const-fenv-fix-when-const-fenvp

(defthm const-fenv-fix-when-const-fenvp
  (implies (const-fenvp x)
           (equal (const-fenv-fix x) x)))

Theorem: emptyp-const-fenv-fix

(defthm emptyp-const-fenv-fix
  (implies (or (omap::emptyp x)
               (not (const-fenvp x)))
           (omap::emptyp (const-fenv-fix x))))

Theorem: emptyp-of-const-fenv-fix-to-not-const-fenv-or-emptyp

(defthm emptyp-of-const-fenv-fix-to-not-const-fenv-or-emptyp
  (equal (omap::emptyp (const-fenv-fix x))
         (or (not (const-fenvp x))
             (omap::emptyp x))))

Function: const-fenv-equiv$inline

(defun const-fenv-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (const-fenvp acl2::x)
                              (const-fenvp acl2::y))))
  (equal (const-fenv-fix acl2::x)
         (const-fenv-fix acl2::y)))

Theorem: const-fenv-equiv-is-an-equivalence

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

Theorem: const-fenv-equiv-implies-equal-const-fenv-fix-1

(defthm const-fenv-equiv-implies-equal-const-fenv-fix-1
  (implies (const-fenv-equiv acl2::x x-equiv)
           (equal (const-fenv-fix acl2::x)
                  (const-fenv-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: const-fenv-fix-under-const-fenv-equiv

(defthm const-fenv-fix-under-const-fenv-equiv
  (const-fenv-equiv (const-fenv-fix acl2::x)
                    acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-const-fenv-fix-1-forward-to-const-fenv-equiv

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

Theorem: equal-of-const-fenv-fix-2-forward-to-const-fenv-equiv

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

Theorem: const-fenv-equiv-of-const-fenv-fix-1-forward

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

Theorem: const-fenv-equiv-of-const-fenv-fix-2-forward

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