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Prose-val

Prose-val-fix

Fixing function for prose-val structures.

Signature
(prose-val-fix x) → new-x
Arguments: x — Guard (prose-val-p x).
Returns: new-x — Type (prose-val-p new-x).

Definitions and Theorems

Function: prose-val-fix$inline

(defun prose-val-fix$inline (x)
  (declare (xargs :guard (prose-val-p x)))
  (let ((__function__ 'prose-val-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((get (str-fix (std::da-nth 0 (cdr x)))))
           (cons :prose (list get)))
         :exec x)))

Theorem: prose-val-p-of-prose-val-fix

(defthm prose-val-p-of-prose-val-fix
  (b* ((new-x (prose-val-fix$inline x)))
    (prose-val-p new-x))
  :rule-classes :rewrite)

Theorem: prose-val-fix-when-prose-val-p

(defthm prose-val-fix-when-prose-val-p
  (implies (prose-val-p x)
           (equal (prose-val-fix x) x)))

Function: prose-val-equiv$inline

(defun prose-val-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (prose-val-p acl2::x)
                              (prose-val-p acl2::y))))
  (equal (prose-val-fix acl2::x)
         (prose-val-fix acl2::y)))

Theorem: prose-val-equiv-is-an-equivalence

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

Theorem: prose-val-equiv-implies-equal-prose-val-fix-1

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

Theorem: prose-val-fix-under-prose-val-equiv

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

Theorem: equal-of-prose-val-fix-1-forward-to-prose-val-equiv

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

Theorem: equal-of-prose-val-fix-2-forward-to-prose-val-equiv

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

Theorem: prose-val-equiv-of-prose-val-fix-1-forward

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

Theorem: prose-val-equiv-of-prose-val-fix-2-forward

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