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Any-table

Any-table-fix

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

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

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

Definitions and Theorems

Function: any-table-fix$inline

(defun any-table-fix$inline (x)
  (declare (xargs :guard (any-table-p x)))
  (let ((acl2::__function__ 'any-table-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (if (atom x)
             nil
           (if (consp (car x))
               (cons (cons (sig-path-fix (caar x))
                           (acl2::bool-fix (cdar x)))
                     (any-table-fix (cdr x)))
             (any-table-fix (cdr x))))
         :exec x)))

Theorem: any-table-p-of-any-table-fix

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

Theorem: any-table-fix-when-any-table-p

(defthm any-table-fix-when-any-table-p
  (implies (any-table-p x)
           (equal (any-table-fix x) x)))

Function: any-table-equiv$inline

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

Theorem: any-table-equiv-is-an-equivalence

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

Theorem: any-table-equiv-implies-equal-any-table-fix-1

(defthm any-table-equiv-implies-equal-any-table-fix-1
  (implies (any-table-equiv acl2::x x-equiv)
           (equal (any-table-fix acl2::x)
                  (any-table-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: any-table-fix-under-any-table-equiv

(defthm any-table-fix-under-any-table-equiv
  (any-table-equiv (any-table-fix acl2::x)
                   acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-any-table-fix-1-forward-to-any-table-equiv

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

Theorem: equal-of-any-table-fix-2-forward-to-any-table-equiv

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

Theorem: any-table-equiv-of-any-table-fix-1-forward

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

Theorem: any-table-equiv-of-any-table-fix-2-forward

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

Theorem: cons-of-sig-path-fix-k-under-any-table-equiv

(defthm cons-of-sig-path-fix-k-under-any-table-equiv
  (any-table-equiv (cons (cons (sig-path-fix acl2::k) acl2::v)
                         acl2::x)
                   (cons (cons acl2::k acl2::v) acl2::x)))

Theorem: cons-sig-path-equiv-congruence-on-k-under-any-table-equiv

(defthm cons-sig-path-equiv-congruence-on-k-under-any-table-equiv
  (implies (sig-path-equiv acl2::k k-equiv)
           (any-table-equiv (cons (cons acl2::k acl2::v) acl2::x)
                            (cons (cons k-equiv acl2::v) acl2::x)))
  :rule-classes :congruence)

Theorem: cons-of-bool-fix-v-under-any-table-equiv

(defthm cons-of-bool-fix-v-under-any-table-equiv
  (any-table-equiv (cons (cons acl2::k (acl2::bool-fix acl2::v))
                         acl2::x)
                   (cons (cons acl2::k acl2::v) acl2::x)))

Theorem: cons-iff-congruence-on-v-under-any-table-equiv

(defthm cons-iff-congruence-on-v-under-any-table-equiv
  (implies (iff acl2::v v-equiv)
           (any-table-equiv (cons (cons acl2::k acl2::v) acl2::x)
                            (cons (cons acl2::k v-equiv) acl2::x)))
  :rule-classes :congruence)

Theorem: cons-of-any-table-fix-y-under-any-table-equiv

(defthm cons-of-any-table-fix-y-under-any-table-equiv
  (any-table-equiv (cons acl2::x (any-table-fix acl2::y))
                   (cons acl2::x acl2::y)))

Theorem: cons-any-table-equiv-congruence-on-y-under-any-table-equiv

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

Theorem: any-table-fix-of-acons

(defthm any-table-fix-of-acons
  (equal (any-table-fix (cons (cons acl2::a acl2::b) x))
         (cons (cons (sig-path-fix acl2::a)
                     (acl2::bool-fix acl2::b))
               (any-table-fix x))))

Theorem: any-table-fix-of-append

(defthm any-table-fix-of-append
  (equal (any-table-fix (append std::a std::b))
         (append (any-table-fix std::a)
                 (any-table-fix std::b))))

Theorem: consp-car-of-any-table-fix

(defthm consp-car-of-any-table-fix
  (equal (consp (car (any-table-fix x)))
         (consp (any-table-fix x))))