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Inverter

Inverter-fix

Fixing function for inverter structures.

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

Definitions and Theorems

Function: inverter-fix$inline

(defun inverter-fix$inline (x)
  (declare (xargs :guard (inverter-p x)))
  (let ((acl2::__function__ 'inverter-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (b* ((input (sig-path-fix (cdr (std::da-nth 0 x))))
              (output (sig-path-fix (cdr (std::da-nth 1 x)))))
           (list (cons 'input input)
                 (cons 'output output)))
         :exec x)))

Theorem: inverter-p-of-inverter-fix

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

Theorem: inverter-fix-when-inverter-p

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

Function: inverter-equiv$inline

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

Theorem: inverter-equiv-is-an-equivalence

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

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

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

Theorem: inverter-fix-under-inverter-equiv

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

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

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

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

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

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

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

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

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