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Quantifier

Quantifier-fix

Fixing function for quantifier structures.

Signature
(quantifier-fix x) → new-x
Arguments: x — Guard (quantifierp x).
Returns: new-x — Type (quantifierp new-x).

Definitions and Theorems

Function: quantifier-fix$inline

(defun quantifier-fix$inline (x)
  (declare (xargs :guard (quantifierp x)))
  (let ((__function__ 'quantifier-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (quantifier-kind x)
           (:forall (cons :forall (list)))
           (:exists (cons :exists (list))))
         :exec x)))

Theorem: quantifierp-of-quantifier-fix

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

Theorem: quantifier-fix-when-quantifierp

(defthm quantifier-fix-when-quantifierp
  (implies (quantifierp x)
           (equal (quantifier-fix x) x)))

Function: quantifier-equiv$inline

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

Theorem: quantifier-equiv-is-an-equivalence

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

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

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

Theorem: quantifier-fix-under-quantifier-equiv

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

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

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

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

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

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

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

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

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

Theorem: quantifier-kind$inline-of-quantifier-fix-x

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

Theorem: quantifier-kind$inline-quantifier-equiv-congruence-on-x

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

Theorem: consp-of-quantifier-fix

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