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Constraint

Constraint-fix

Fixing function for constraint structures.

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

Definitions and Theorems

Function: constraint-fix$inline

(defun constraint-fix$inline (x)
  (declare (xargs :guard (constraint-p x)))
  (let ((__function__ 'constraint-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((name (std::prod-car x))
              (cond (svex-fix (std::prod-cdr x))))
           (std::prod-cons name cond))
         :exec x)))

Theorem: constraint-p-of-constraint-fix

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

Theorem: constraint-fix-when-constraint-p

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

Function: constraint-equiv$inline

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

Theorem: constraint-equiv-is-an-equivalence

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

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

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

Theorem: constraint-fix-under-constraint-equiv

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

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

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

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

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

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

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

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

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