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Value?+denv

Value?+denv-fix

Fixing function for value?+denv structures.

Signature
(value?+denv-fix x) → new-x
Arguments: x — Guard (value?+denv-p x).
Returns: new-x — Type (value?+denv-p new-x).

Definitions and Theorems

Function: value?+denv-fix$inline

(defun value?+denv-fix$inline (x)
 (declare (xargs :guard (value?+denv-p x)))
 (let ((__function__ 'value?+denv-fix))
  (declare (ignorable __function__))
  (mbe
      :logic
      (b* ((value? (value-option-fix (cdr (std::da-nth 0 (cdr x)))))
           (denv (denv-fix (cdr (std::da-nth 1 (cdr x))))))
        (cons :value?+denv (list (cons 'value? value?)
                                 (cons 'denv denv))))
      :exec x)))

Theorem: value?+denv-p-of-value?+denv-fix

(defthm value?+denv-p-of-value?+denv-fix
  (b* ((new-x (value?+denv-fix$inline x)))
    (value?+denv-p new-x))
  :rule-classes :rewrite)

Theorem: value?+denv-fix-when-value?+denv-p

(defthm value?+denv-fix-when-value?+denv-p
  (implies (value?+denv-p x)
           (equal (value?+denv-fix x) x)))

Function: value?+denv-equiv$inline

(defun value?+denv-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (value?+denv-p acl2::x)
                              (value?+denv-p acl2::y))))
  (equal (value?+denv-fix acl2::x)
         (value?+denv-fix acl2::y)))

Theorem: value?+denv-equiv-is-an-equivalence

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

Theorem: value?+denv-equiv-implies-equal-value?+denv-fix-1

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

Theorem: value?+denv-fix-under-value?+denv-equiv

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

Theorem: equal-of-value?+denv-fix-1-forward-to-value?+denv-equiv

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

Theorem: equal-of-value?+denv-fix-2-forward-to-value?+denv-equiv

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

Theorem: value?+denv-equiv-of-value?+denv-fix-1-forward

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

Theorem: value?+denv-equiv-of-value?+denv-fix-2-forward

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