		Search-engine friendly clone of the ACL2 documentation.

Var-table

Var-table-fix

(var-table-fix x) is a usual ACL2::fty list fixing function.

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

In the logic, we apply var-table-scope-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: var-table-fix$inline

(defun var-table-fix$inline (x)
  (declare (xargs :guard (var-tablep x)))
  (let ((__function__ 'var-table-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (consp (cdr x))
             (cons (var-table-scope-fix (car x))
                   (var-table-fix (cdr x)))
           (cons (var-table-scope-fix (car x))
                 nil))
         :exec x)))

Theorem: var-tablep-of-var-table-fix

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

Theorem: var-table-fix-when-var-tablep

(defthm var-table-fix-when-var-tablep
  (implies (var-tablep x)
           (equal (var-table-fix x) x)))

Function: var-table-equiv$inline

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

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

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

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

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

Theorem: var-table-fix-under-var-table-equiv

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

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

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

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

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

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

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

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

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

Theorem: car-of-var-table-fix-x-under-var-table-scope-equiv

(defthm car-of-var-table-fix-x-under-var-table-scope-equiv
  (var-table-scope-equiv (car (var-table-fix acl2::x))
                         (car acl2::x)))

Theorem: car-var-table-equiv-congruence-on-x-under-var-table-scope-equiv

(defthm
    car-var-table-equiv-congruence-on-x-under-var-table-scope-equiv
  (implies (var-table-equiv acl2::x x-equiv)
           (var-table-scope-equiv (car acl2::x)
                                  (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-var-table-fix-x-under-var-table-equiv

(defthm cdr-of-var-table-fix-x-under-var-table-equiv
  (var-table-equiv (cdr (var-table-fix acl2::x))
                   (cdr acl2::x)))

Theorem: cdr-var-table-equiv-congruence-on-x-under-var-table-equiv

(defthm cdr-var-table-equiv-congruence-on-x-under-var-table-equiv
  (implies (var-table-equiv acl2::x x-equiv)
           (var-table-equiv (cdr acl2::x)
                            (cdr x-equiv)))
  :rule-classes :congruence)

Theorem: cons-of-var-table-scope-fix-x-under-var-table-equiv

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

Theorem: cons-var-table-scope-equiv-congruence-on-x-under-var-table-equiv

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

Theorem: consp-of-var-table-fix

(defthm consp-of-var-table-fix
  (consp (var-table-fix acl2::x)))

Theorem: consp-cdr-of-var-table-fix

(defthm consp-cdr-of-var-table-fix
  (equal (consp (cdr (var-table-fix acl2::x)))
         (consp (cdr acl2::x))))

Theorem: car-of-var-table-fix

(defthm car-of-var-table-fix
  (equal (car (var-table-fix acl2::x))
         (var-table-scope-fix (car acl2::x))))

Theorem: var-table-fix-under-iff

(defthm var-table-fix-under-iff
  (var-table-fix acl2::x))

Theorem: var-table-fix-of-cons

(defthm var-table-fix-of-cons
  (equal (var-table-fix (cons a x))
         (cons (var-table-scope-fix a)
               (and (consp x) (var-table-fix x)))))

Theorem: len-of-var-table-fix

(defthm len-of-var-table-fix
  (equal (len (var-table-fix acl2::x))
         (max 1 (len acl2::x))))