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Var-table-scope

Var-table-scope-fix

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

Signature
(var-table-scope-fix x) → *
Arguments: x — Guard (var-table-scopep x).

Definitions and Theorems

Function: var-table-scope-fix

(defun var-table-scope-fix (x)
  (declare (xargs :guard (var-table-scopep x)))
  (mbe :logic (if (var-table-scopep x) x nil)
       :exec x))

Theorem: var-table-scopep-of-var-table-scope-fix

(defthm var-table-scopep-of-var-table-scope-fix
  (var-table-scopep (var-table-scope-fix x)))

Theorem: var-table-scope-fix-when-var-table-scopep

(defthm var-table-scope-fix-when-var-table-scopep
  (implies (var-table-scopep x)
           (equal (var-table-scope-fix x) x)))

Theorem: emptyp-var-table-scope-fix

(defthm emptyp-var-table-scope-fix
  (implies (or (omap::emptyp x)
               (not (var-table-scopep x)))
           (omap::emptyp (var-table-scope-fix x))))

Theorem: emptyp-of-var-table-scope-fix-to-not-var-table-scope-or-emptyp

(defthm
     emptyp-of-var-table-scope-fix-to-not-var-table-scope-or-emptyp
  (equal (omap::emptyp (var-table-scope-fix x))
         (or (not (var-table-scopep x))
             (omap::emptyp x))))

Function: var-table-scope-equiv$inline

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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