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Fundef

Fundef-fix

Fixing function for fundef structures.

Signature
(fundef-fix x) → new-x
Arguments: x — Guard (fundefp x).
Returns: new-x — Type (fundefp new-x).

Definitions and Theorems

Function: fundef-fix$inline

(defun fundef-fix$inline (x)
  (declare (xargs :guard (fundefp x)))
  (mbe :logic
       (b* ((extension (bool-fix (car (car (car x)))))
            (specs (decl-spec-list-fix (cdr (car (car x)))))
            (declor (declor-fix (car (cdr (car x)))))
            (asm? (asm-name-spec-option-fix (cdr (cdr (car x)))))
            (attribs (attrib-spec-list-fix (car (car (cdr x)))))
            (declons (declon-list-fix (cdr (car (cdr x)))))
            (body (comp-stmt-fix (car (cdr (cdr x)))))
            (info (identity (cdr (cdr (cdr x))))))
         (cons (cons (cons extension specs)
                     (cons declor asm?))
               (cons (cons attribs declons)
                     (cons body info))))
       :exec x))

Theorem: fundefp-of-fundef-fix

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

Theorem: fundef-fix-when-fundefp

(defthm fundef-fix-when-fundefp
  (implies (fundefp x)
           (equal (fundef-fix x) x)))

Function: fundef-equiv$inline

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

Theorem: fundef-equiv-is-an-equivalence

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

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

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

Theorem: fundef-fix-under-fundef-equiv

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

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

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

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

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

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

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

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

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

Theorem: fundef-fix$inline-of-fundef-fix-x

(defthm fundef-fix$inline-of-fundef-fix-x
  (equal (fundef-fix$inline (fundef-fix x))
         (fundef-fix$inline x)))

Theorem: fundef-fix$inline-fundef-equiv-congruence-on-x

(defthm fundef-fix$inline-fundef-equiv-congruence-on-x
  (implies (fundef-equiv x x-equiv)
           (equal (fundef-fix$inline x)
                  (fundef-fix$inline x-equiv)))
  :rule-classes :congruence)