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Candidate-assign

Candidate-assign-fix

Fixing function for candidate-assign structures.

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

Definitions and Theorems

Function: candidate-assign-fix$inline

(defun candidate-assign-fix$inline (x)
  (declare (xargs :guard (candidate-assign-p x)))
  (let ((__function__ 'candidate-assign-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((edge (cgraph-edge-fix (std::prod-car x)))
              (val (std::prod-cdr x)))
           (std::prod-cons edge val))
         :exec x)))

Theorem: candidate-assign-p-of-candidate-assign-fix

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

Theorem: candidate-assign-fix-when-candidate-assign-p

(defthm candidate-assign-fix-when-candidate-assign-p
  (implies (candidate-assign-p x)
           (equal (candidate-assign-fix x) x)))

Function: candidate-assign-equiv$inline

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

Theorem: candidate-assign-equiv-is-an-equivalence

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

Theorem: candidate-assign-equiv-implies-equal-candidate-assign-fix-1

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

Theorem: candidate-assign-fix-under-candidate-assign-equiv

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

Theorem: equal-of-candidate-assign-fix-1-forward-to-candidate-assign-equiv

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

Theorem: equal-of-candidate-assign-fix-2-forward-to-candidate-assign-equiv

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

Theorem: candidate-assign-equiv-of-candidate-assign-fix-1-forward

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

Theorem: candidate-assign-equiv-of-candidate-assign-fix-2-forward

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