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• Mach-o-section-headers

Mach-o-section-headers-fix

(mach-o-section-headers-fix x) is a usual ACL2::fty list fixing function.

Signature

(mach-o-section-headers-fix x) → fty::newx

Arguments

x — Guard (mach-o-section-headers-p x).

Returns

fty::newx — Type (mach-o-section-headers-p fty::newx).

In the logic, we apply mach-o-section-header-fix to each member of the x. In the execution, none of that is actually necessary and this is just an inlined identity function.

Definitions and Theorems

Function: mach-o-section-headers-fix$inline

(defun mach-o-section-headers-fix$inline (x)
  (declare (xargs :guard (mach-o-section-headers-p x)))
  (let ((__function__ 'mach-o-section-headers-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (mach-o-section-header-fix (car x))
                 (mach-o-section-headers-fix (cdr x))))
         :exec x)))

Theorem: mach-o-section-headers-p-of-mach-o-section-headers-fix

(defthm mach-o-section-headers-p-of-mach-o-section-headers-fix
  (b* ((fty::newx (mach-o-section-headers-fix$inline x)))
    (mach-o-section-headers-p fty::newx))
  :rule-classes :rewrite)

Theorem: mach-o-section-headers-fix-when-mach-o-section-headers-p

(defthm mach-o-section-headers-fix-when-mach-o-section-headers-p
  (implies (mach-o-section-headers-p x)
           (equal (mach-o-section-headers-fix x)
                  x)))

Function: mach-o-section-headers-equiv$inline

(defun mach-o-section-headers-equiv$inline (x y)
  (declare (xargs :guard (and (mach-o-section-headers-p x)
                              (mach-o-section-headers-p y))))
  (equal (mach-o-section-headers-fix x)
         (mach-o-section-headers-fix y)))

Theorem: mach-o-section-headers-equiv-is-an-equivalence

(defthm mach-o-section-headers-equiv-is-an-equivalence
  (and (booleanp (mach-o-section-headers-equiv x y))
       (mach-o-section-headers-equiv x x)
       (implies (mach-o-section-headers-equiv x y)
                (mach-o-section-headers-equiv y x))
       (implies (and (mach-o-section-headers-equiv x y)
                     (mach-o-section-headers-equiv y z))
                (mach-o-section-headers-equiv x z)))
  :rule-classes (:equivalence))

Theorem: mach-o-section-headers-equiv-implies-equal-mach-o-section-headers-fix-1

(defthm
 mach-o-section-headers-equiv-implies-equal-mach-o-section-headers-fix-1
 (implies (mach-o-section-headers-equiv x x-equiv)
          (equal (mach-o-section-headers-fix x)
                 (mach-o-section-headers-fix x-equiv)))
 :rule-classes (:congruence))

Theorem: mach-o-section-headers-fix-under-mach-o-section-headers-equiv

(defthm
      mach-o-section-headers-fix-under-mach-o-section-headers-equiv
  (mach-o-section-headers-equiv (mach-o-section-headers-fix x)
                                x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-mach-o-section-headers-fix-1-forward-to-mach-o-section-headers-equiv

(defthm
 equal-of-mach-o-section-headers-fix-1-forward-to-mach-o-section-headers-equiv
 (implies (equal (mach-o-section-headers-fix x) y)
          (mach-o-section-headers-equiv x y))
 :rule-classes :forward-chaining)

Theorem: equal-of-mach-o-section-headers-fix-2-forward-to-mach-o-section-headers-equiv

(defthm
 equal-of-mach-o-section-headers-fix-2-forward-to-mach-o-section-headers-equiv
 (implies (equal x (mach-o-section-headers-fix y))
          (mach-o-section-headers-equiv x y))
 :rule-classes :forward-chaining)

Theorem: mach-o-section-headers-equiv-of-mach-o-section-headers-fix-1-forward

(defthm
 mach-o-section-headers-equiv-of-mach-o-section-headers-fix-1-forward
 (implies
      (mach-o-section-headers-equiv (mach-o-section-headers-fix x)
                                    y)
      (mach-o-section-headers-equiv x y))
 :rule-classes :forward-chaining)

Theorem: mach-o-section-headers-equiv-of-mach-o-section-headers-fix-2-forward

(defthm
 mach-o-section-headers-equiv-of-mach-o-section-headers-fix-2-forward
 (implies
     (mach-o-section-headers-equiv x (mach-o-section-headers-fix y))
     (mach-o-section-headers-equiv x y))
 :rule-classes :forward-chaining)

Theorem: car-of-mach-o-section-headers-fix-x-under-mach-o-section-header-equiv

(defthm
 car-of-mach-o-section-headers-fix-x-under-mach-o-section-header-equiv
 (mach-o-section-header-equiv (car (mach-o-section-headers-fix x))
                              (car x)))

Theorem: car-mach-o-section-headers-equiv-congruence-on-x-under-mach-o-section-header-equiv

(defthm
 car-mach-o-section-headers-equiv-congruence-on-x-under-mach-o-section-header-equiv
 (implies (mach-o-section-headers-equiv x x-equiv)
          (mach-o-section-header-equiv (car x)
                                       (car x-equiv)))
 :rule-classes :congruence)

Theorem: cdr-of-mach-o-section-headers-fix-x-under-mach-o-section-headers-equiv

(defthm
 cdr-of-mach-o-section-headers-fix-x-under-mach-o-section-headers-equiv
 (mach-o-section-headers-equiv (cdr (mach-o-section-headers-fix x))
                               (cdr x)))

Theorem: cdr-mach-o-section-headers-equiv-congruence-on-x-under-mach-o-section-headers-equiv

(defthm
 cdr-mach-o-section-headers-equiv-congruence-on-x-under-mach-o-section-headers-equiv
 (implies (mach-o-section-headers-equiv x x-equiv)
          (mach-o-section-headers-equiv (cdr x)
                                        (cdr x-equiv)))
 :rule-classes :congruence)

Theorem: cons-of-mach-o-section-header-fix-x-under-mach-o-section-headers-equiv

(defthm
 cons-of-mach-o-section-header-fix-x-under-mach-o-section-headers-equiv
 (mach-o-section-headers-equiv
      (cons (mach-o-section-header-fix x) y)
      (cons x y)))

Theorem: cons-mach-o-section-header-equiv-congruence-on-x-under-mach-o-section-headers-equiv

(defthm
 cons-mach-o-section-header-equiv-congruence-on-x-under-mach-o-section-headers-equiv
 (implies (mach-o-section-header-equiv x x-equiv)
          (mach-o-section-headers-equiv (cons x y)
                                        (cons x-equiv y)))
 :rule-classes :congruence)

Theorem: cons-of-mach-o-section-headers-fix-y-under-mach-o-section-headers-equiv

(defthm
 cons-of-mach-o-section-headers-fix-y-under-mach-o-section-headers-equiv
 (mach-o-section-headers-equiv
      (cons x (mach-o-section-headers-fix y))
      (cons x y)))

Theorem: cons-mach-o-section-headers-equiv-congruence-on-y-under-mach-o-section-headers-equiv

(defthm
 cons-mach-o-section-headers-equiv-congruence-on-y-under-mach-o-section-headers-equiv
 (implies (mach-o-section-headers-equiv y y-equiv)
          (mach-o-section-headers-equiv (cons x y)
                                        (cons x y-equiv)))
 :rule-classes :congruence)

Theorem: consp-of-mach-o-section-headers-fix

(defthm consp-of-mach-o-section-headers-fix
  (equal (consp (mach-o-section-headers-fix x))
         (consp x)))

Theorem: mach-o-section-headers-fix-under-iff

(defthm mach-o-section-headers-fix-under-iff
  (iff (mach-o-section-headers-fix x)
       (consp x)))

Theorem: mach-o-section-headers-fix-of-cons

(defthm mach-o-section-headers-fix-of-cons
  (equal (mach-o-section-headers-fix (cons a x))
         (cons (mach-o-section-header-fix a)
               (mach-o-section-headers-fix x))))

Theorem: len-of-mach-o-section-headers-fix

(defthm len-of-mach-o-section-headers-fix
  (equal (len (mach-o-section-headers-fix x))
         (len x)))

Theorem: mach-o-section-headers-fix-of-append

(defthm mach-o-section-headers-fix-of-append
  (equal (mach-o-section-headers-fix (append std::a std::b))
         (append (mach-o-section-headers-fix std::a)
                 (mach-o-section-headers-fix std::b))))

Theorem: mach-o-section-headers-fix-of-repeat

(defthm mach-o-section-headers-fix-of-repeat
  (equal (mach-o-section-headers-fix (acl2::repeat n x))
         (acl2::repeat n (mach-o-section-header-fix x))))

Theorem: list-equiv-refines-mach-o-section-headers-equiv

(defthm list-equiv-refines-mach-o-section-headers-equiv
  (implies (acl2::list-equiv x y)
           (mach-o-section-headers-equiv x y))
  :rule-classes :refinement)

Theorem: nth-of-mach-o-section-headers-fix

(defthm nth-of-mach-o-section-headers-fix
  (equal (nth n (mach-o-section-headers-fix x))
         (if (< (nfix n) (len x))
             (mach-o-section-header-fix (nth n x))
           nil)))

Theorem: mach-o-section-headers-equiv-implies-mach-o-section-headers-equiv-append-1

(defthm
 mach-o-section-headers-equiv-implies-mach-o-section-headers-equiv-append-1
 (implies (mach-o-section-headers-equiv x fty::x-equiv)
          (mach-o-section-headers-equiv (append x y)
                                        (append fty::x-equiv y)))
 :rule-classes (:congruence))

Theorem: mach-o-section-headers-equiv-implies-mach-o-section-headers-equiv-append-2

(defthm
 mach-o-section-headers-equiv-implies-mach-o-section-headers-equiv-append-2
 (implies (mach-o-section-headers-equiv y fty::y-equiv)
          (mach-o-section-headers-equiv (append x y)
                                        (append x fty::y-equiv)))
 :rule-classes (:congruence))

Theorem: mach-o-section-headers-equiv-implies-mach-o-section-headers-equiv-nthcdr-2

(defthm
 mach-o-section-headers-equiv-implies-mach-o-section-headers-equiv-nthcdr-2
 (implies (mach-o-section-headers-equiv l l-equiv)
          (mach-o-section-headers-equiv (nthcdr n l)
                                        (nthcdr n l-equiv)))
 :rule-classes (:congruence))

Theorem: mach-o-section-headers-equiv-implies-mach-o-section-headers-equiv-take-2

(defthm
 mach-o-section-headers-equiv-implies-mach-o-section-headers-equiv-take-2
 (implies (mach-o-section-headers-equiv l l-equiv)
          (mach-o-section-headers-equiv (take n l)
                                        (take n l-equiv)))
 :rule-classes (:congruence))