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Function-definer

Function-definer-fix

Fixing function for function-definer structures.

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

Definitions and Theorems

Function: function-definer-fix$inline

(defun function-definer-fix$inline (x)
 (declare (xargs :guard (function-definerp x)))
 (let ((__function__ 'function-definer-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (case (function-definer-kind x)
    (:regular
      (b* ((body (expression-fix (std::da-nth 0 (cdr x))))
           (measure (maybe-expression-fix (std::da-nth 1 (cdr x)))))
        (cons :regular (list body measure))))
    (:quantified
     (b*
      ((quantifier (quantifier-fix (std::da-nth 0 (cdr x))))
       (variables (typed-variable-list-fix (std::da-nth 1 (cdr x))))
       (matrix (expression-fix (std::da-nth 2 (cdr x)))))
      (cons :quantified (list quantifier variables matrix)))))
   :exec x)))

Theorem: function-definerp-of-function-definer-fix

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

Theorem: function-definer-fix-when-function-definerp

(defthm function-definer-fix-when-function-definerp
  (implies (function-definerp x)
           (equal (function-definer-fix x) x)))

Function: function-definer-equiv$inline

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

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

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

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

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

Theorem: function-definer-fix-under-function-definer-equiv

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

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

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

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

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

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

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

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

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

Theorem: function-definer-kind$inline-of-function-definer-fix-x

(defthm function-definer-kind$inline-of-function-definer-fix-x
  (equal (function-definer-kind$inline (function-definer-fix x))
         (function-definer-kind$inline x)))

Theorem: function-definer-kind$inline-function-definer-equiv-congruence-on-x

(defthm
 function-definer-kind$inline-function-definer-equiv-congruence-on-x
 (implies (function-definer-equiv x x-equiv)
          (equal (function-definer-kind$inline x)
                 (function-definer-kind$inline x-equiv)))
 :rule-classes :congruence)

Theorem: consp-of-function-definer-fix

(defthm consp-of-function-definer-fix
  (consp (function-definer-fix x))
  :rule-classes :type-prescription)