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Equal-op

Equal-op-fix

Fixing function for equal-op structures.

Signature
(equal-op-fix x) → new-x
Arguments: x — Guard (equal-opp x).
Returns: new-x — Type (equal-opp new-x).

Definitions and Theorems

Function: equal-op-fix$inline

(defun equal-op-fix$inline (x)
  (declare (xargs :guard (equal-opp x)))
  (let ((__function__ 'equal-op-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (equal-op-kind x)
           (:is.eq (cons :is.eq (list)))
           (:is.neq (cons :is.neq (list))))
         :exec x)))

Theorem: equal-opp-of-equal-op-fix

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

Theorem: equal-op-fix-when-equal-opp

(defthm equal-op-fix-when-equal-opp
  (implies (equal-opp x)
           (equal (equal-op-fix x) x)))

Function: equal-op-equiv$inline

(defun equal-op-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (equal-opp acl2::x)
                              (equal-opp acl2::y))))
  (equal (equal-op-fix acl2::x)
         (equal-op-fix acl2::y)))

Theorem: equal-op-equiv-is-an-equivalence

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

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

(defthm equal-op-equiv-implies-equal-equal-op-fix-1
  (implies (equal-op-equiv acl2::x x-equiv)
           (equal (equal-op-fix acl2::x)
                  (equal-op-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: equal-op-fix-under-equal-op-equiv

(defthm equal-op-fix-under-equal-op-equiv
  (equal-op-equiv (equal-op-fix acl2::x)
                  acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

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

(defthm equal-of-equal-op-fix-1-forward-to-equal-op-equiv
  (implies (equal (equal-op-fix acl2::x) acl2::y)
           (equal-op-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

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

(defthm equal-of-equal-op-fix-2-forward-to-equal-op-equiv
  (implies (equal acl2::x (equal-op-fix acl2::y))
           (equal-op-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-op-equiv-of-equal-op-fix-1-forward

(defthm equal-op-equiv-of-equal-op-fix-1-forward
  (implies (equal-op-equiv (equal-op-fix acl2::x)
                           acl2::y)
           (equal-op-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-op-equiv-of-equal-op-fix-2-forward

(defthm equal-op-equiv-of-equal-op-fix-2-forward
  (implies (equal-op-equiv acl2::x (equal-op-fix acl2::y))
           (equal-op-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-op-kind$inline-of-equal-op-fix-x

(defthm equal-op-kind$inline-of-equal-op-fix-x
  (equal (equal-op-kind$inline (equal-op-fix x))
         (equal-op-kind$inline x)))

Theorem: equal-op-kind$inline-equal-op-equiv-congruence-on-x

(defthm equal-op-kind$inline-equal-op-equiv-congruence-on-x
  (implies (equal-op-equiv x x-equiv)
           (equal (equal-op-kind$inline x)
                  (equal-op-kind$inline x-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-equal-op-fix

(defthm consp-of-equal-op-fix
  (consp (equal-op-fix x))
  :rule-classes :type-prescription)