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Rewrite

Rewrite-fix

Fixing function for rewrite structures.

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

Definitions and Theorems

Function: rewrite-fix$inline

(defun rewrite-fix$inline (x)
  (declare (xargs :guard (rewrite-p x)))
  (let ((__function__ 'rewrite-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((hyps (pseudo-term-list-fix (std::da-nth 0 x)))
              (equiv (pseudo-fnsym-fix (std::da-nth 1 x)))
              (lhs (pseudo-term-fix (std::da-nth 2 x)))
              (rhs (pseudo-term-fix (std::da-nth 3 x))))
           (list hyps equiv lhs rhs))
         :exec x)))

Theorem: rewrite-p-of-rewrite-fix

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

Theorem: rewrite-fix-when-rewrite-p

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

Function: rewrite-equiv$inline

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

Theorem: rewrite-equiv-is-an-equivalence

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

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

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

Theorem: rewrite-fix-under-rewrite-equiv

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

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

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

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

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

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

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

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

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