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Validators-state

Validators-state-fix

(validators-state-fix x) is a usual ACL2::fty omap fixing function.

Signature
(validators-state-fix x) → *
Arguments: x — Guard (validators-statep x).

Definitions and Theorems

Function: validators-state-fix

(defun validators-state-fix (x)
  (declare (xargs :guard (validators-statep x)))
  (mbe :logic (if (validators-statep x) x nil)
       :exec x))

Theorem: validators-statep-of-validators-state-fix

(defthm validators-statep-of-validators-state-fix
  (validators-statep (validators-state-fix x)))

Theorem: validators-state-fix-when-validators-statep

(defthm validators-state-fix-when-validators-statep
  (implies (validators-statep x)
           (equal (validators-state-fix x) x)))

Theorem: emptyp-validators-state-fix

(defthm emptyp-validators-state-fix
  (implies (or (omap::emptyp x)
               (not (validators-statep x)))
           (omap::emptyp (validators-state-fix x))))

Theorem: emptyp-of-validators-state-fix-to-not-validators-state-or-emptyp

(defthm
   emptyp-of-validators-state-fix-to-not-validators-state-or-emptyp
  (equal (omap::emptyp (validators-state-fix x))
         (or (not (validators-statep x))
             (omap::emptyp x))))

Function: validators-state-equiv$inline

(defun validators-state-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (validators-statep acl2::x)
                              (validators-statep acl2::y))))
  (equal (validators-state-fix acl2::x)
         (validators-state-fix acl2::y)))

Theorem: validators-state-equiv-is-an-equivalence

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

Theorem: validators-state-equiv-implies-equal-validators-state-fix-1

(defthm validators-state-equiv-implies-equal-validators-state-fix-1
  (implies (validators-state-equiv acl2::x x-equiv)
           (equal (validators-state-fix acl2::x)
                  (validators-state-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: validators-state-fix-under-validators-state-equiv

(defthm validators-state-fix-under-validators-state-equiv
  (validators-state-equiv (validators-state-fix acl2::x)
                          acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-validators-state-fix-1-forward-to-validators-state-equiv

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

Theorem: equal-of-validators-state-fix-2-forward-to-validators-state-equiv

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

Theorem: validators-state-equiv-of-validators-state-fix-1-forward

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

Theorem: validators-state-equiv-of-validators-state-fix-2-forward

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