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Valid-scope

Valid-scope-fix

Fixing function for valid-scope structures.

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

Definitions and Theorems

Function: valid-scope-fix$inline

(defun valid-scope-fix$inline (x)
  (declare (xargs :guard (valid-scopep x)))
  (mbe :logic
       (b* ((ord (valid-ord-scope-fix (cdr (std::da-nth 0 x)))))
         (list (cons 'ord ord)))
       :exec x))

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

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

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

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

Function: valid-scope-equiv$inline

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: valid-scope-fix$inline-of-valid-scope-fix-x

(defthm valid-scope-fix$inline-of-valid-scope-fix-x
  (equal (valid-scope-fix$inline (valid-scope-fix x))
         (valid-scope-fix$inline x)))

Theorem: valid-scope-fix$inline-valid-scope-equiv-congruence-on-x

(defthm valid-scope-fix$inline-valid-scope-equiv-congruence-on-x
  (implies (valid-scope-equiv x x-equiv)
           (equal (valid-scope-fix$inline x)
                  (valid-scope-fix$inline x-equiv)))
  :rule-classes :congruence)