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Pflat

Pflat-fix

Fixing function for pflat structures.

Signature
(pflat-fix x) → new-x
Arguments: x — Guard (pflat-p x).
Returns: new-x — Type (pflat-p new-x).

Definitions and Theorems

Function: pflat-fix$inline

(defun pflat-fix$inline (x)
  (declare (xargs :guard (pflat-p x)))
  (let ((acl2::__function__ 'pflat-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (b* ((width (acl2::pos-fix (std::prod-car x)))
              (what (std::prod-cdr x)))
           (std::prod-cons width what))
         :exec x)))

Theorem: pflat-p-of-pflat-fix

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

Theorem: pflat-fix-when-pflat-p

(defthm pflat-fix-when-pflat-p
  (implies (pflat-p x)
           (equal (pflat-fix x) x)))

Function: pflat-equiv$inline

(defun pflat-equiv$inline (x y)
  (declare (xargs :guard (and (pflat-p x) (pflat-p y))))
  (equal (pflat-fix x) (pflat-fix y)))

Theorem: pflat-equiv-is-an-equivalence

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

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

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

Theorem: pflat-fix-under-pflat-equiv

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

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

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

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

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

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

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

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

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