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    • Valid-ord-scope

    Valid-ord-scope-fix

    (valid-ord-scope-fix x) is an ACL2::fty alist fixing function that follows the fix-keys strategy.

    Signature
    (valid-ord-scope-fix x) → fty::newx
    Arguments
    x — Guard (valid-ord-scopep x).
    Returns
    fty::newx — Type (valid-ord-scopep fty::newx).

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

    Definitions and Theorems

    Function: valid-ord-scope-fix$inline

    (defun valid-ord-scope-fix$inline (x)
      (declare (xargs :guard (valid-ord-scopep x)))
      (let ((__function__ 'valid-ord-scope-fix))
        (declare (ignorable __function__))
        (mbe :logic
             (if (atom x)
                 nil
               (if (consp (car x))
                   (cons (cons (ident-fix (caar x))
                               (valid-ord-info-fix (cdar x)))
                         (valid-ord-scope-fix (cdr x)))
                 (valid-ord-scope-fix (cdr x))))
             :exec x)))

    Theorem: valid-ord-scopep-of-valid-ord-scope-fix

    (defthm valid-ord-scopep-of-valid-ord-scope-fix
      (b* ((fty::newx (valid-ord-scope-fix$inline x)))
        (valid-ord-scopep fty::newx))
      :rule-classes :rewrite)

    Theorem: valid-ord-scope-fix-when-valid-ord-scopep

    (defthm valid-ord-scope-fix-when-valid-ord-scopep
      (implies (valid-ord-scopep x)
               (equal (valid-ord-scope-fix x) x)))

    Function: valid-ord-scope-equiv$inline

    (defun valid-ord-scope-equiv$inline (acl2::x acl2::y)
      (declare (xargs :guard (and (valid-ord-scopep acl2::x)
                                  (valid-ord-scopep acl2::y))))
      (equal (valid-ord-scope-fix acl2::x)
             (valid-ord-scope-fix acl2::y)))

    Theorem: valid-ord-scope-equiv-is-an-equivalence

    (defthm valid-ord-scope-equiv-is-an-equivalence
      (and (booleanp (valid-ord-scope-equiv x y))
           (valid-ord-scope-equiv x x)
           (implies (valid-ord-scope-equiv x y)
                    (valid-ord-scope-equiv y x))
           (implies (and (valid-ord-scope-equiv x y)
                         (valid-ord-scope-equiv y z))
                    (valid-ord-scope-equiv x z)))
      :rule-classes (:equivalence))

    Theorem: valid-ord-scope-equiv-implies-equal-valid-ord-scope-fix-1

    (defthm valid-ord-scope-equiv-implies-equal-valid-ord-scope-fix-1
      (implies (valid-ord-scope-equiv acl2::x x-equiv)
               (equal (valid-ord-scope-fix acl2::x)
                      (valid-ord-scope-fix x-equiv)))
      :rule-classes (:congruence))

    Theorem: valid-ord-scope-fix-under-valid-ord-scope-equiv

    (defthm valid-ord-scope-fix-under-valid-ord-scope-equiv
      (valid-ord-scope-equiv (valid-ord-scope-fix acl2::x)
                             acl2::x)
      :rule-classes (:rewrite :rewrite-quoted-constant))

    Theorem: equal-of-valid-ord-scope-fix-1-forward-to-valid-ord-scope-equiv

    (defthm
        equal-of-valid-ord-scope-fix-1-forward-to-valid-ord-scope-equiv
      (implies (equal (valid-ord-scope-fix acl2::x)
                      acl2::y)
               (valid-ord-scope-equiv acl2::x acl2::y))
      :rule-classes :forward-chaining)

    Theorem: equal-of-valid-ord-scope-fix-2-forward-to-valid-ord-scope-equiv

    (defthm
        equal-of-valid-ord-scope-fix-2-forward-to-valid-ord-scope-equiv
      (implies (equal acl2::x (valid-ord-scope-fix acl2::y))
               (valid-ord-scope-equiv acl2::x acl2::y))
      :rule-classes :forward-chaining)

    Theorem: valid-ord-scope-equiv-of-valid-ord-scope-fix-1-forward

    (defthm valid-ord-scope-equiv-of-valid-ord-scope-fix-1-forward
      (implies (valid-ord-scope-equiv (valid-ord-scope-fix acl2::x)
                                      acl2::y)
               (valid-ord-scope-equiv acl2::x acl2::y))
      :rule-classes :forward-chaining)

    Theorem: valid-ord-scope-equiv-of-valid-ord-scope-fix-2-forward

    (defthm valid-ord-scope-equiv-of-valid-ord-scope-fix-2-forward
      (implies
           (valid-ord-scope-equiv acl2::x (valid-ord-scope-fix acl2::y))
           (valid-ord-scope-equiv acl2::x acl2::y))
      :rule-classes :forward-chaining)

    Theorem: cons-of-ident-fix-k-under-valid-ord-scope-equiv

    (defthm cons-of-ident-fix-k-under-valid-ord-scope-equiv
      (valid-ord-scope-equiv (cons (cons (ident-fix acl2::k) acl2::v)
                                   acl2::x)
                             (cons (cons acl2::k acl2::v) acl2::x)))

    Theorem: cons-ident-equiv-congruence-on-k-under-valid-ord-scope-equiv

    (defthm cons-ident-equiv-congruence-on-k-under-valid-ord-scope-equiv
     (implies
          (ident-equiv acl2::k k-equiv)
          (valid-ord-scope-equiv (cons (cons acl2::k acl2::v) acl2::x)
                                 (cons (cons k-equiv acl2::v) acl2::x)))
     :rule-classes :congruence)

    Theorem: cons-of-valid-ord-info-fix-v-under-valid-ord-scope-equiv

    (defthm cons-of-valid-ord-info-fix-v-under-valid-ord-scope-equiv
      (valid-ord-scope-equiv
           (cons (cons acl2::k (valid-ord-info-fix acl2::v))
                 acl2::x)
           (cons (cons acl2::k acl2::v) acl2::x)))

    Theorem: cons-valid-ord-info-equiv-congruence-on-v-under-valid-ord-scope-equiv

    (defthm
     cons-valid-ord-info-equiv-congruence-on-v-under-valid-ord-scope-equiv
     (implies
          (valid-ord-info-equiv acl2::v v-equiv)
          (valid-ord-scope-equiv (cons (cons acl2::k acl2::v) acl2::x)
                                 (cons (cons acl2::k v-equiv) acl2::x)))
     :rule-classes :congruence)

    Theorem: cons-of-valid-ord-scope-fix-y-under-valid-ord-scope-equiv

    (defthm cons-of-valid-ord-scope-fix-y-under-valid-ord-scope-equiv
     (valid-ord-scope-equiv (cons acl2::x (valid-ord-scope-fix acl2::y))
                            (cons acl2::x acl2::y)))

    Theorem: cons-valid-ord-scope-equiv-congruence-on-y-under-valid-ord-scope-equiv

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

    Theorem: valid-ord-scope-fix-of-acons

    (defthm valid-ord-scope-fix-of-acons
      (equal (valid-ord-scope-fix (cons (cons acl2::a acl2::b) x))
             (cons (cons (ident-fix acl2::a)
                         (valid-ord-info-fix acl2::b))
                   (valid-ord-scope-fix x))))

    Theorem: valid-ord-scope-fix-of-append

    (defthm valid-ord-scope-fix-of-append
      (equal (valid-ord-scope-fix (append std::a std::b))
             (append (valid-ord-scope-fix std::a)
                     (valid-ord-scope-fix std::b))))

    Theorem: consp-car-of-valid-ord-scope-fix

    (defthm consp-car-of-valid-ord-scope-fix
      (equal (consp (car (valid-ord-scope-fix x)))
             (consp (valid-ord-scope-fix x))))