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Comp-db

Comp-db-fix

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

Signature
(comp-db-fix x) → fty::newx
Arguments: x — Guard (comp-dbp x).
Returns: fty::newx — Type (comp-dbp fty::newx).

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

Definitions and Theorems

Function: comp-db-fix$inline

(defun comp-db-fix$inline (x)
  (declare (xargs :guard (comp-dbp x)))
  (mbe :logic
       (if (atom x)
           nil
         (if (consp (car x))
             (cons (cons (str-fix (caar x))
                         (comp-db-entry-fix (cdar x)))
                   (comp-db-fix (cdr x)))
           (comp-db-fix (cdr x))))
       :exec x))

Theorem: comp-dbp-of-comp-db-fix

(defthm comp-dbp-of-comp-db-fix
  (b* ((fty::newx (comp-db-fix$inline x)))
    (comp-dbp fty::newx))
  :rule-classes :rewrite)

Theorem: comp-db-fix-when-comp-dbp

(defthm comp-db-fix-when-comp-dbp
  (implies (comp-dbp x)
           (equal (comp-db-fix x) x)))

Function: comp-db-equiv$inline

(defun comp-db-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (comp-dbp acl2::x)
                              (comp-dbp acl2::y))))
  (equal (comp-db-fix acl2::x)
         (comp-db-fix acl2::y)))

Theorem: comp-db-equiv-is-an-equivalence

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

Theorem: comp-db-equiv-implies-equal-comp-db-fix-1

(defthm comp-db-equiv-implies-equal-comp-db-fix-1
  (implies (comp-db-equiv acl2::x x-equiv)
           (equal (comp-db-fix acl2::x)
                  (comp-db-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: comp-db-fix-under-comp-db-equiv

(defthm comp-db-fix-under-comp-db-equiv
  (comp-db-equiv (comp-db-fix acl2::x)
                 acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-comp-db-fix-1-forward-to-comp-db-equiv

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

Theorem: equal-of-comp-db-fix-2-forward-to-comp-db-equiv

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

Theorem: comp-db-equiv-of-comp-db-fix-1-forward

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

Theorem: comp-db-equiv-of-comp-db-fix-2-forward

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

Theorem: cons-of-str-fix-k-under-comp-db-equiv

(defthm cons-of-str-fix-k-under-comp-db-equiv
  (comp-db-equiv (cons (cons (str-fix acl2::k) acl2::v)
                       acl2::x)
                 (cons (cons acl2::k acl2::v) acl2::x)))

Theorem: cons-streqv-congruence-on-k-under-comp-db-equiv

(defthm cons-streqv-congruence-on-k-under-comp-db-equiv
  (implies (acl2::streqv acl2::k k-equiv)
           (comp-db-equiv (cons (cons acl2::k acl2::v) acl2::x)
                          (cons (cons k-equiv acl2::v) acl2::x)))
  :rule-classes :congruence)

Theorem: cons-of-comp-db-entry-fix-v-under-comp-db-equiv

(defthm cons-of-comp-db-entry-fix-v-under-comp-db-equiv
  (comp-db-equiv (cons (cons acl2::k (comp-db-entry-fix acl2::v))
                       acl2::x)
                 (cons (cons acl2::k acl2::v) acl2::x)))

Theorem: cons-comp-db-entry-equiv-congruence-on-v-under-comp-db-equiv

(defthm cons-comp-db-entry-equiv-congruence-on-v-under-comp-db-equiv
  (implies (comp-db-entry-equiv acl2::v v-equiv)
           (comp-db-equiv (cons (cons acl2::k acl2::v) acl2::x)
                          (cons (cons acl2::k v-equiv) acl2::x)))
  :rule-classes :congruence)

Theorem: cons-of-comp-db-fix-y-under-comp-db-equiv

(defthm cons-of-comp-db-fix-y-under-comp-db-equiv
  (comp-db-equiv (cons acl2::x (comp-db-fix acl2::y))
                 (cons acl2::x acl2::y)))

Theorem: cons-comp-db-equiv-congruence-on-y-under-comp-db-equiv

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

Theorem: comp-db-fix-of-acons

(defthm comp-db-fix-of-acons
  (equal (comp-db-fix (cons (cons acl2::a acl2::b) x))
         (cons (cons (str-fix acl2::a)
                     (comp-db-entry-fix acl2::b))
               (comp-db-fix x))))

Theorem: comp-db-fix-of-append

(defthm comp-db-fix-of-append
  (equal (comp-db-fix (append std::a std::b))
         (append (comp-db-fix std::a)
                 (comp-db-fix std::b))))

Theorem: consp-car-of-comp-db-fix

(defthm consp-car-of-comp-db-fix
  (equal (consp (car (comp-db-fix x)))
         (consp (comp-db-fix x))))