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Named-lit-list-map

Named-lit-list-map-fix

(named-lit-list-map-fix x) is an ACL2::fty alist fixing function that follows the fix-keys strategy.

Signature
(named-lit-list-map-fix x) → fty::newx
Arguments: x — Guard (named-lit-list-map-p x).
Returns: fty::newx — Type (named-lit-list-map-p fty::newx).

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

Definitions and Theorems

Function: named-lit-list-map-fix$inline

(defun named-lit-list-map-fix$inline (x)
  (declare (xargs :guard (named-lit-list-map-p x)))
  (let ((__function__ 'named-lit-list-map-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (if (consp (car x))
               (cons (cons (acl2::symbol-fix (caar x))
                           (lit-list-fix (cdar x)))
                     (named-lit-list-map-fix (cdr x)))
             (named-lit-list-map-fix (cdr x))))
         :exec x)))

Theorem: named-lit-list-map-p-of-named-lit-list-map-fix

(defthm named-lit-list-map-p-of-named-lit-list-map-fix
  (b* ((fty::newx (named-lit-list-map-fix$inline x)))
    (named-lit-list-map-p fty::newx))
  :rule-classes :rewrite)

Theorem: named-lit-list-map-fix-when-named-lit-list-map-p

(defthm named-lit-list-map-fix-when-named-lit-list-map-p
  (implies (named-lit-list-map-p x)
           (equal (named-lit-list-map-fix x) x)))

Function: named-lit-list-map-equiv$inline

(defun named-lit-list-map-equiv$inline (x acl2::y)
  (declare (xargs :guard (and (named-lit-list-map-p x)
                              (named-lit-list-map-p acl2::y))))
  (equal (named-lit-list-map-fix x)
         (named-lit-list-map-fix acl2::y)))

Theorem: named-lit-list-map-equiv-is-an-equivalence

(defthm named-lit-list-map-equiv-is-an-equivalence
  (and (booleanp (named-lit-list-map-equiv x y))
       (named-lit-list-map-equiv x x)
       (implies (named-lit-list-map-equiv x y)
                (named-lit-list-map-equiv y x))
       (implies (and (named-lit-list-map-equiv x y)
                     (named-lit-list-map-equiv y z))
                (named-lit-list-map-equiv x z)))
  :rule-classes (:equivalence))

Theorem: named-lit-list-map-equiv-implies-equal-named-lit-list-map-fix-1

(defthm
    named-lit-list-map-equiv-implies-equal-named-lit-list-map-fix-1
  (implies (named-lit-list-map-equiv x x-equiv)
           (equal (named-lit-list-map-fix x)
                  (named-lit-list-map-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: named-lit-list-map-fix-under-named-lit-list-map-equiv

(defthm named-lit-list-map-fix-under-named-lit-list-map-equiv
  (named-lit-list-map-equiv (named-lit-list-map-fix x)
                            x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-named-lit-list-map-fix-1-forward-to-named-lit-list-map-equiv

(defthm
 equal-of-named-lit-list-map-fix-1-forward-to-named-lit-list-map-equiv
 (implies (equal (named-lit-list-map-fix x)
                 acl2::y)
          (named-lit-list-map-equiv x acl2::y))
 :rule-classes :forward-chaining)

Theorem: equal-of-named-lit-list-map-fix-2-forward-to-named-lit-list-map-equiv

(defthm
 equal-of-named-lit-list-map-fix-2-forward-to-named-lit-list-map-equiv
 (implies (equal x (named-lit-list-map-fix acl2::y))
          (named-lit-list-map-equiv x acl2::y))
 :rule-classes :forward-chaining)

Theorem: named-lit-list-map-equiv-of-named-lit-list-map-fix-1-forward

(defthm named-lit-list-map-equiv-of-named-lit-list-map-fix-1-forward
  (implies (named-lit-list-map-equiv (named-lit-list-map-fix x)
                                     acl2::y)
           (named-lit-list-map-equiv x acl2::y))
  :rule-classes :forward-chaining)

Theorem: named-lit-list-map-equiv-of-named-lit-list-map-fix-2-forward

(defthm named-lit-list-map-equiv-of-named-lit-list-map-fix-2-forward
  (implies
       (named-lit-list-map-equiv x (named-lit-list-map-fix acl2::y))
       (named-lit-list-map-equiv x acl2::y))
  :rule-classes :forward-chaining)

Theorem: cons-of-symbol-fix-k-under-named-lit-list-map-equiv

(defthm cons-of-symbol-fix-k-under-named-lit-list-map-equiv
  (named-lit-list-map-equiv (cons (cons (acl2::symbol-fix acl2::k)
                                        acl2::v)
                                  x)
                            (cons (cons acl2::k acl2::v) x)))

Theorem: cons-symbol-equiv-congruence-on-k-under-named-lit-list-map-equiv

(defthm
   cons-symbol-equiv-congruence-on-k-under-named-lit-list-map-equiv
  (implies
       (acl2::symbol-equiv acl2::k k-equiv)
       (named-lit-list-map-equiv (cons (cons acl2::k acl2::v) x)
                                 (cons (cons k-equiv acl2::v) x)))
  :rule-classes :congruence)

Theorem: cons-of-lit-list-fix-v-under-named-lit-list-map-equiv

(defthm cons-of-lit-list-fix-v-under-named-lit-list-map-equiv
  (named-lit-list-map-equiv
       (cons (cons acl2::k (lit-list-fix acl2::v))
             x)
       (cons (cons acl2::k acl2::v) x)))

Theorem: cons-lit-list-equiv-congruence-on-v-under-named-lit-list-map-equiv

(defthm
 cons-lit-list-equiv-congruence-on-v-under-named-lit-list-map-equiv
 (implies
      (satlink::lit-list-equiv acl2::v v-equiv)
      (named-lit-list-map-equiv (cons (cons acl2::k acl2::v) x)
                                (cons (cons acl2::k v-equiv) x)))
 :rule-classes :congruence)

Theorem: cons-of-named-lit-list-map-fix-y-under-named-lit-list-map-equiv

(defthm
    cons-of-named-lit-list-map-fix-y-under-named-lit-list-map-equiv
 (named-lit-list-map-equiv (cons x (named-lit-list-map-fix acl2::y))
                           (cons x acl2::y)))

Theorem: cons-named-lit-list-map-equiv-congruence-on-y-under-named-lit-list-map-equiv

(defthm
 cons-named-lit-list-map-equiv-congruence-on-y-under-named-lit-list-map-equiv
 (implies (named-lit-list-map-equiv acl2::y y-equiv)
          (named-lit-list-map-equiv (cons x acl2::y)
                                    (cons x y-equiv)))
 :rule-classes :congruence)

Theorem: named-lit-list-map-fix-of-acons

(defthm named-lit-list-map-fix-of-acons
  (equal (named-lit-list-map-fix (cons (cons acl2::a acl2::b) x))
         (cons (cons (acl2::symbol-fix acl2::a)
                     (lit-list-fix acl2::b))
               (named-lit-list-map-fix x))))

Theorem: named-lit-list-map-fix-of-append

(defthm named-lit-list-map-fix-of-append
  (equal (named-lit-list-map-fix (append std::a std::b))
         (append (named-lit-list-map-fix std::a)
                 (named-lit-list-map-fix std::b))))

Theorem: consp-car-of-named-lit-list-map-fix

(defthm consp-car-of-named-lit-list-map-fix
  (equal (consp (car (named-lit-list-map-fix x)))
         (consp (named-lit-list-map-fix x))))