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Sign

Sign-fix

Fixing function for sign structures.

Signature
(sign-fix x) → new-x
Arguments: x — Guard (signp x).
Returns: new-x — Type (signp new-x).

Definitions and Theorems

Function: sign-fix$inline

(defun sign-fix$inline (x)
  (declare (xargs :guard (signp x)))
  (let ((__function__ 'sign-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (sign-kind x)
           (:plus (cons :plus (list)))
           (:minus (cons :minus (list))))
         :exec x)))

Theorem: signp-of-sign-fix

(defthm signp-of-sign-fix
  (b* ((new-x (sign-fix$inline x)))
    (signp new-x))
  :rule-classes :rewrite)

Theorem: sign-fix-when-signp

(defthm sign-fix-when-signp
  (implies (signp x)
           (equal (sign-fix x) x)))

Function: sign-equiv$inline

(defun sign-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (signp acl2::x) (signp acl2::y))))
  (equal (sign-fix acl2::x)
         (sign-fix acl2::y)))

Theorem: sign-equiv-is-an-equivalence

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

Theorem: sign-equiv-implies-equal-sign-fix-1

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

Theorem: sign-fix-under-sign-equiv

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

Theorem: equal-of-sign-fix-1-forward-to-sign-equiv

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

Theorem: equal-of-sign-fix-2-forward-to-sign-equiv

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

Theorem: sign-equiv-of-sign-fix-1-forward

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

Theorem: sign-equiv-of-sign-fix-2-forward

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

Theorem: sign-kind$inline-of-sign-fix-x

(defthm sign-kind$inline-of-sign-fix-x
  (equal (sign-kind$inline (sign-fix x))
         (sign-kind$inline x)))

Theorem: sign-kind$inline-sign-equiv-congruence-on-x

(defthm sign-kind$inline-sign-equiv-congruence-on-x
  (implies (sign-equiv x x-equiv)
           (equal (sign-kind$inline x)
                  (sign-kind$inline x-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-sign-fix

(defthm consp-of-sign-fix
  (consp (sign-fix x))
  :rule-classes :type-prescription)