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Init-value

Init-value-fix

Fixing function for init-value structures.

Signature
(init-value-fix x) → new-x
Arguments: x — Guard (init-valuep x).
Returns: new-x — Type (init-valuep new-x).

Definitions and Theorems

Function: init-value-fix$inline

(defun init-value-fix$inline (x)
 (declare (xargs :guard (init-valuep x)))
 (let ((__function__ 'init-value-fix))
  (declare (ignorable __function__))
  (mbe :logic
       (case (init-value-kind x)
         (:single (b* ((get (value-fix (std::da-nth 0 (cdr x)))))
                    (cons :single (list get))))
         (:list (b* ((get (value-list-fix (std::da-nth 0 (cdr x)))))
                  (cons :list (list get)))))
       :exec x)))

Theorem: init-valuep-of-init-value-fix

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

Theorem: init-value-fix-when-init-valuep

(defthm init-value-fix-when-init-valuep
  (implies (init-valuep x)
           (equal (init-value-fix x) x)))

Function: init-value-equiv$inline

(defun init-value-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (init-valuep acl2::x)
                              (init-valuep acl2::y))))
  (equal (init-value-fix acl2::x)
         (init-value-fix acl2::y)))

Theorem: init-value-equiv-is-an-equivalence

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

Theorem: init-value-equiv-implies-equal-init-value-fix-1

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

Theorem: init-value-fix-under-init-value-equiv

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

Theorem: equal-of-init-value-fix-1-forward-to-init-value-equiv

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

Theorem: equal-of-init-value-fix-2-forward-to-init-value-equiv

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

Theorem: init-value-equiv-of-init-value-fix-1-forward

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

Theorem: init-value-equiv-of-init-value-fix-2-forward

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

Theorem: init-value-kind$inline-of-init-value-fix-x

(defthm init-value-kind$inline-of-init-value-fix-x
  (equal (init-value-kind$inline (init-value-fix x))
         (init-value-kind$inline x)))

Theorem: init-value-kind$inline-init-value-equiv-congruence-on-x

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

Theorem: consp-of-init-value-fix

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