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Assert-op

Assert-op-fix

Fixing function for assert-op structures.

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

Definitions and Theorems

Function: assert-op-fix$inline

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

Theorem: assert-opp-of-assert-op-fix

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

Theorem: assert-op-fix-when-assert-opp

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

Function: assert-op-equiv$inline

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

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

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

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

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

Theorem: assert-op-fix-under-assert-op-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: consp-of-assert-op-fix

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