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Function-recursion

Function-recursion-fix

Fixing function for function-recursion structures.

Signature
(function-recursion-fix x) → new-x
Arguments: x — Guard (function-recursionp x).
Returns: new-x — Type (function-recursionp new-x).

Definitions and Theorems

Function: function-recursion-fix$inline

(defun function-recursion-fix$inline (x)
 (declare (xargs :guard (function-recursionp x)))
 (let ((__function__ 'function-recursion-fix))
  (declare (ignorable __function__))
  (mbe
    :logic
    (b*
      ((definitions
            (function-definition-list-fix (cdr (std::da-nth 0 x)))))
      (list (cons 'definitions definitions)))
    :exec x)))

Theorem: function-recursionp-of-function-recursion-fix

(defthm function-recursionp-of-function-recursion-fix
  (b* ((new-x (function-recursion-fix$inline x)))
    (function-recursionp new-x))
  :rule-classes :rewrite)

Theorem: function-recursion-fix-when-function-recursionp

(defthm function-recursion-fix-when-function-recursionp
  (implies (function-recursionp x)
           (equal (function-recursion-fix x) x)))

Function: function-recursion-equiv$inline

(defun function-recursion-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (function-recursionp acl2::x)
                              (function-recursionp acl2::y))))
  (equal (function-recursion-fix acl2::x)
         (function-recursion-fix acl2::y)))

Theorem: function-recursion-equiv-is-an-equivalence

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

Theorem: function-recursion-equiv-implies-equal-function-recursion-fix-1

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

Theorem: function-recursion-fix-under-function-recursion-equiv

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

Theorem: equal-of-function-recursion-fix-1-forward-to-function-recursion-equiv

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

Theorem: equal-of-function-recursion-fix-2-forward-to-function-recursion-equiv

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

Theorem: function-recursion-equiv-of-function-recursion-fix-1-forward

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

Theorem: function-recursion-equiv-of-function-recursion-fix-2-forward

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