		Search-engine friendly clone of the ACL2 documentation.

Axi-map

Axi-map-fix

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

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

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

Definitions and Theorems

Function: axi-map-fix$inline

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

Theorem: axi-map-p-of-axi-map-fix

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

Theorem: axi-map-fix-when-axi-map-p

(defthm axi-map-fix-when-axi-map-p
  (implies (axi-map-p x)
           (equal (axi-map-fix x) x)))

Function: axi-map-equiv$inline

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

Theorem: axi-map-equiv-is-an-equivalence

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

Theorem: axi-map-equiv-implies-equal-axi-map-fix-1

(defthm axi-map-equiv-implies-equal-axi-map-fix-1
  (implies (axi-map-equiv x x-equiv)
           (equal (axi-map-fix x)
                  (axi-map-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: axi-map-fix-under-axi-map-equiv

(defthm axi-map-fix-under-axi-map-equiv
  (axi-map-equiv (axi-map-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-axi-map-fix-1-forward-to-axi-map-equiv

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

Theorem: equal-of-axi-map-fix-2-forward-to-axi-map-equiv

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

Theorem: axi-map-equiv-of-axi-map-fix-1-forward

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

Theorem: axi-map-equiv-of-axi-map-fix-2-forward

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

Theorem: cons-of-axi-term-fix-k-under-axi-map-equiv

(defthm cons-of-axi-term-fix-k-under-axi-map-equiv
  (axi-map-equiv (cons (cons (axi-term-fix acl2::k) acl2::v)
                       x)
                 (cons (cons acl2::k acl2::v) x)))

Theorem: cons-axi-term-equiv-congruence-on-k-under-axi-map-equiv

(defthm cons-axi-term-equiv-congruence-on-k-under-axi-map-equiv
  (implies (axi-term-equiv acl2::k k-equiv)
           (axi-map-equiv (cons (cons acl2::k acl2::v) x)
                          (cons (cons k-equiv acl2::v) x)))
  :rule-classes :congruence)

Theorem: cons-of-axi-lit-fix-v-under-axi-map-equiv

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

Theorem: cons-axi-lit-equiv-congruence-on-v-under-axi-map-equiv

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

Theorem: cons-of-axi-map-fix-y-under-axi-map-equiv

(defthm cons-of-axi-map-fix-y-under-axi-map-equiv
  (axi-map-equiv (cons x (axi-map-fix acl2::y))
                 (cons x acl2::y)))

Theorem: cons-axi-map-equiv-congruence-on-y-under-axi-map-equiv

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

Theorem: axi-map-fix-of-acons

(defthm axi-map-fix-of-acons
  (equal (axi-map-fix (cons (cons acl2::a acl2::b) x))
         (cons (cons (axi-term-fix acl2::a)
                     (axi-lit-fix acl2::b))
               (axi-map-fix x))))

Theorem: axi-map-fix-of-append

(defthm axi-map-fix-of-append
  (equal (axi-map-fix (append std::a std::b))
         (append (axi-map-fix std::a)
                 (axi-map-fix std::b))))

Theorem: consp-car-of-axi-map-fix

(defthm consp-car-of-axi-map-fix
  (equal (consp (car (axi-map-fix x)))
         (consp (axi-map-fix x))))