		Search-engine friendly clone of the ACL2 documentation.

Func

Func-fix

Fixing function for func structures.

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

Definitions and Theorems

Function: func-fix$inline

(defun func-fix$inline (x)
 (declare (xargs :guard (func-p x)))
 (let ((acl2::__function__ 'func-fix))
  (declare (ignorable acl2::__function__))
  (mbe
    :logic
    (b*
     ((name (symbol-fix (cdr (std::da-nth 0 x))))
      (formals (decl-list-fix (cdr (std::da-nth 1 x))))
      (guard (hint-pair-fix (cdr (std::da-nth 2 x))))
      (returns (decl-list-fix (cdr (std::da-nth 3 x))))
      (more-returns (hint-pair-list-fix (cdr (std::da-nth 4 x))))
      (expansion-depth (nfix (cdr (std::da-nth 5 x))))
      (flattened-formals (symbol-list-fix (cdr (std::da-nth 6 x))))
      (flattened-returns (symbol-list-fix (cdr (std::da-nth 7 x)))))
     (list (cons 'name name)
           (cons 'formals formals)
           (cons 'guard guard)
           (cons 'returns returns)
           (cons 'more-returns more-returns)
           (cons 'expansion-depth expansion-depth)
           (cons 'flattened-formals
                 flattened-formals)
           (cons 'flattened-returns
                 flattened-returns)))
    :exec x)))

Theorem: func-p-of-func-fix

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

Theorem: func-fix-when-func-p

(defthm func-fix-when-func-p
  (implies (func-p x)
           (equal (func-fix x) x)))

Function: func-equiv$inline

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

Theorem: func-equiv-is-an-equivalence

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

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

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

Theorem: func-fix-under-func-equiv

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

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

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

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

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

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

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

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

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