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Cgraph-equivnode

Cgraph-equivnode-fix

Fixing function for cgraph-equivnode structures.

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

Definitions and Theorems

Function: cgraph-equivnode-fix$inline

(defun cgraph-equivnode-fix$inline (x)
  (declare (xargs :guard (cgraph-equivnode-p x)))
  (let ((__function__ 'cgraph-equivnode-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((equivalence (fgl-object-fix (cdr (std::da-nth 0 x))))
              (other (fgl-object-fix (cdr (std::da-nth 1 x)))))
           (list (cons 'equivalence equivalence)
                 (cons 'other other)))
         :exec x)))

Theorem: cgraph-equivnode-p-of-cgraph-equivnode-fix

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

Theorem: cgraph-equivnode-fix-when-cgraph-equivnode-p

(defthm cgraph-equivnode-fix-when-cgraph-equivnode-p
  (implies (cgraph-equivnode-p x)
           (equal (cgraph-equivnode-fix x) x)))

Function: cgraph-equivnode-equiv$inline

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

Theorem: cgraph-equivnode-equiv-is-an-equivalence

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

Theorem: cgraph-equivnode-equiv-implies-equal-cgraph-equivnode-fix-1

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

Theorem: cgraph-equivnode-fix-under-cgraph-equivnode-equiv

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

Theorem: equal-of-cgraph-equivnode-fix-1-forward-to-cgraph-equivnode-equiv

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

Theorem: equal-of-cgraph-equivnode-fix-2-forward-to-cgraph-equivnode-equiv

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

Theorem: cgraph-equivnode-equiv-of-cgraph-equivnode-fix-1-forward

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

Theorem: cgraph-equivnode-equiv-of-cgraph-equivnode-fix-2-forward

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