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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))))