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Svtv-cyclephase

Svtv-cyclephase-fix

Fixing function for svtv-cyclephase structures.

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

Definitions and Theorems

Function: svtv-cyclephase-fix$inline

(defun svtv-cyclephase-fix$inline (x)
  (declare (xargs :guard (svtv-cyclephase-p x)))
  (let ((__function__ 'svtv-cyclephase-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((constants (svex-env-fix (car x)))
              (inputs-free (bool-fix (car (cdr x))))
              (outputs-captured (bool-fix (cdr (cdr x)))))
           (cons constants
                 (cons inputs-free outputs-captured)))
         :exec x)))

Theorem: svtv-cyclephase-p-of-svtv-cyclephase-fix

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

Theorem: svtv-cyclephase-fix-when-svtv-cyclephase-p

(defthm svtv-cyclephase-fix-when-svtv-cyclephase-p
  (implies (svtv-cyclephase-p x)
           (equal (svtv-cyclephase-fix x) x)))

Function: svtv-cyclephase-equiv$inline

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

Theorem: svtv-cyclephase-equiv-is-an-equivalence

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

Theorem: svtv-cyclephase-equiv-implies-equal-svtv-cyclephase-fix-1

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

Theorem: svtv-cyclephase-fix-under-svtv-cyclephase-equiv

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

Theorem: equal-of-svtv-cyclephase-fix-1-forward-to-svtv-cyclephase-equiv

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

Theorem: equal-of-svtv-cyclephase-fix-2-forward-to-svtv-cyclephase-equiv

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

Theorem: svtv-cyclephase-equiv-of-svtv-cyclephase-fix-1-forward

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

Theorem: svtv-cyclephase-equiv-of-svtv-cyclephase-fix-2-forward

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