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• Primitive-values

Numericx-value

Fixtype of Java numeric values [JLS14:4.2], including extended-exponent values [JLS14:4.2.3].

Definitions and Theorems

Function: numericx-valuep

(defun numericx-valuep (x)
  (declare (xargs :guard t))
  (let ((__function__ 'numericx-valuep))
    (declare (ignorable __function__))
    (or (numeric-valuep x)
        (floatx-valuep x)
        (doublex-valuep x))))

Theorem: booleanp-of-numericx-valuep

(defthm booleanp-of-numericx-valuep
  (b* ((yes/no (numericx-valuep x)))
    (booleanp yes/no))
  :rule-classes :rewrite)

Theorem: numericx-valuep-when-numeric-valuep

(defthm numericx-valuep-when-numeric-valuep
  (implies (numeric-valuep x)
           (numericx-valuep x)))

Theorem: numericx-valuep-when-floatx-valuep

(defthm numericx-valuep-when-floatx-valuep
  (implies (floatx-valuep x)
           (numericx-valuep x)))

Theorem: numericx-valuep-when-doublex-valuep

(defthm numericx-valuep-when-doublex-valuep
  (implies (doublex-valuep x)
           (numericx-valuep x)))

Function: numericx-value-fix

(defun numericx-value-fix (x)
  (declare (xargs :guard (numericx-valuep x)))
  (mbe :logic
       (if (numericx-valuep x)
           x
         (char-value 0))
       :exec x))

Theorem: numericx-valuep-of-numericx-value-fix

(defthm numericx-valuep-of-numericx-value-fix
  (b* ((fixed-x (numericx-value-fix x)))
    (numericx-valuep fixed-x))
  :rule-classes :rewrite)

Theorem: numericx-value-fix-when-numericx-valuep

(defthm numericx-value-fix-when-numericx-valuep
  (implies (numericx-valuep x)
           (equal (numericx-value-fix x) x)))

Function: numericx-value-equiv$inline

(defun numericx-value-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (numericx-valuep acl2::x)
                              (numericx-valuep acl2::y))))
  (equal (numericx-value-fix acl2::x)
         (numericx-value-fix acl2::y)))

Theorem: numericx-value-equiv-is-an-equivalence

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

Theorem: numericx-value-equiv-implies-equal-numericx-value-fix-1

(defthm numericx-value-equiv-implies-equal-numericx-value-fix-1
  (implies (numericx-value-equiv acl2::x x-equiv)
           (equal (numericx-value-fix acl2::x)
                  (numericx-value-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: numericx-value-fix-under-numericx-value-equiv

(defthm numericx-value-fix-under-numericx-value-equiv
  (numericx-value-equiv (numericx-value-fix acl2::x)
                        acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-numericx-value-fix-1-forward-to-numericx-value-equiv

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

Theorem: equal-of-numericx-value-fix-2-forward-to-numericx-value-equiv

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

Theorem: numericx-value-equiv-of-numericx-value-fix-1-forward

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

Theorem: numericx-value-equiv-of-numericx-value-fix-2-forward

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

Subtopics

Numericx-valuep

Recognizer for numericx-value.