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• Identifier-identifier-alist

Identifier-identifier-alist-fix

(identifier-identifier-alist-fix x) is an ACL2::fty alist fixing function that follows the fix-keys strategy.

Signature

(identifier-identifier-alist-fix x) → fty::newx

Arguments

x — Guard (identifier-identifier-alistp x).

Returns

fty::newx — Type (identifier-identifier-alistp fty::newx).

Note that in the execution this is just an inline identity function.

Definitions and Theorems

Function: identifier-identifier-alist-fix$inline

(defun identifier-identifier-alist-fix$inline (x)
  (declare (xargs :guard (identifier-identifier-alistp x)))
  (let ((__function__ 'identifier-identifier-alist-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (if (consp (car x))
               (cons (cons (identifier-fix (caar x))
                           (identifier-fix (cdar x)))
                     (identifier-identifier-alist-fix (cdr x)))
             (identifier-identifier-alist-fix (cdr x))))
         :exec x)))

Theorem: identifier-identifier-alistp-of-identifier-identifier-alist-fix

(defthm
    identifier-identifier-alistp-of-identifier-identifier-alist-fix
  (b* ((fty::newx (identifier-identifier-alist-fix$inline x)))
    (identifier-identifier-alistp fty::newx))
  :rule-classes :rewrite)

Theorem: identifier-identifier-alist-fix-when-identifier-identifier-alistp

(defthm
  identifier-identifier-alist-fix-when-identifier-identifier-alistp
  (implies (identifier-identifier-alistp x)
           (equal (identifier-identifier-alist-fix x)
                  x)))

Function: identifier-identifier-alist-equiv$inline

(defun identifier-identifier-alist-equiv$inline (acl2::x acl2::y)
  (declare
       (xargs :guard (and (identifier-identifier-alistp acl2::x)
                          (identifier-identifier-alistp acl2::y))))
  (equal (identifier-identifier-alist-fix acl2::x)
         (identifier-identifier-alist-fix acl2::y)))

Theorem: identifier-identifier-alist-equiv-is-an-equivalence

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

Theorem: identifier-identifier-alist-equiv-implies-equal-identifier-identifier-alist-fix-1

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

Theorem: identifier-identifier-alist-fix-under-identifier-identifier-alist-equiv

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

Theorem: equal-of-identifier-identifier-alist-fix-1-forward-to-identifier-identifier-alist-equiv

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

Theorem: equal-of-identifier-identifier-alist-fix-2-forward-to-identifier-identifier-alist-equiv

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

Theorem: identifier-identifier-alist-equiv-of-identifier-identifier-alist-fix-1-forward

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

Theorem: identifier-identifier-alist-equiv-of-identifier-identifier-alist-fix-2-forward

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

Theorem: cons-of-identifier-fix-k-under-identifier-identifier-alist-equiv

(defthm
   cons-of-identifier-fix-k-under-identifier-identifier-alist-equiv
  (identifier-identifier-alist-equiv
       (cons (cons (identifier-fix acl2::k) acl2::v)
             acl2::x)
       (cons (cons acl2::k acl2::v) acl2::x)))

Theorem: cons-identifier-equiv-congruence-on-k-under-identifier-identifier-alist-equiv

(defthm
 cons-identifier-equiv-congruence-on-k-under-identifier-identifier-alist-equiv
 (implies (identifier-equiv acl2::k k-equiv)
          (identifier-identifier-alist-equiv
               (cons (cons acl2::k acl2::v) acl2::x)
               (cons (cons k-equiv acl2::v) acl2::x)))
 :rule-classes :congruence)

Theorem: cons-of-identifier-fix-v-under-identifier-identifier-alist-equiv

(defthm
   cons-of-identifier-fix-v-under-identifier-identifier-alist-equiv
  (identifier-identifier-alist-equiv
       (cons (cons acl2::k (identifier-fix acl2::v))
             acl2::x)
       (cons (cons acl2::k acl2::v) acl2::x)))

Theorem: cons-identifier-equiv-congruence-on-v-under-identifier-identifier-alist-equiv

(defthm
 cons-identifier-equiv-congruence-on-v-under-identifier-identifier-alist-equiv
 (implies (identifier-equiv acl2::v v-equiv)
          (identifier-identifier-alist-equiv
               (cons (cons acl2::k acl2::v) acl2::x)
               (cons (cons acl2::k v-equiv) acl2::x)))
 :rule-classes :congruence)

Theorem: cons-of-identifier-identifier-alist-fix-y-under-identifier-identifier-alist-equiv

(defthm
 cons-of-identifier-identifier-alist-fix-y-under-identifier-identifier-alist-equiv
 (identifier-identifier-alist-equiv
      (cons acl2::x
            (identifier-identifier-alist-fix acl2::y))
      (cons acl2::x acl2::y)))

Theorem: cons-identifier-identifier-alist-equiv-congruence-on-y-under-identifier-identifier-alist-equiv

(defthm
 cons-identifier-identifier-alist-equiv-congruence-on-y-under-identifier-identifier-alist-equiv
 (implies
      (identifier-identifier-alist-equiv acl2::y y-equiv)
      (identifier-identifier-alist-equiv (cons acl2::x acl2::y)
                                         (cons acl2::x y-equiv)))
 :rule-classes :congruence)

Theorem: identifier-identifier-alist-fix-of-acons

(defthm identifier-identifier-alist-fix-of-acons
 (equal
   (identifier-identifier-alist-fix (cons (cons acl2::a acl2::b) x))
   (cons (cons (identifier-fix acl2::a)
               (identifier-fix acl2::b))
         (identifier-identifier-alist-fix x))))

Theorem: identifier-identifier-alist-fix-of-append

(defthm identifier-identifier-alist-fix-of-append
  (equal (identifier-identifier-alist-fix (append std::a std::b))
         (append (identifier-identifier-alist-fix std::a)
                 (identifier-identifier-alist-fix std::b))))

Theorem: consp-car-of-identifier-identifier-alist-fix

(defthm consp-car-of-identifier-identifier-alist-fix
  (equal (consp (car (identifier-identifier-alist-fix x)))
         (consp (identifier-identifier-alist-fix x))))