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Use-set

Use-set-fix

Fixing function for use-set structures.

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

Definitions and Theorems

Function: use-set-fix$inline

(defun use-set-fix$inline (x)
  (declare (xargs :guard (use-set-p x)))
  (let ((__function__ 'use-set-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((uses (rangemap-fix (std::prod-car x)))
              (sets (rangemap-fix (std::prod-cdr x))))
           (std::prod-cons uses sets))
         :exec x)))

Theorem: use-set-p-of-use-set-fix

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

Theorem: use-set-fix-when-use-set-p

(defthm use-set-fix-when-use-set-p
  (implies (use-set-p x)
           (equal (use-set-fix x) x)))

Function: use-set-equiv$inline

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

Theorem: use-set-equiv-is-an-equivalence

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

Theorem: use-set-equiv-implies-equal-use-set-fix-1

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

Theorem: use-set-fix-under-use-set-equiv

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

Theorem: equal-of-use-set-fix-1-forward-to-use-set-equiv

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

Theorem: equal-of-use-set-fix-2-forward-to-use-set-equiv

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

Theorem: use-set-equiv-of-use-set-fix-1-forward

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

Theorem: use-set-equiv-of-use-set-fix-2-forward

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