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• Filepath-set

Filepath-set-fix

(filepath-set-fix x) is a usual ACL2::fty set fixing function.

Signature

(filepath-set-fix x) → *

Arguments

x — Guard (filepath-setp x).

In the logic, we apply filepath-fix to each member of the x. In the execution, none of that is actually necessary and this is just an inlined identity function.

Definitions and Theorems

Function: filepath-set-fix

(defun filepath-set-fix (x)
  (declare (xargs :guard (filepath-setp x)))
  (mbe :logic (if (filepath-setp x) x nil)
       :exec x))

Theorem: filepath-setp-of-filepath-set-fix

(defthm filepath-setp-of-filepath-set-fix
  (filepath-setp (filepath-set-fix x)))

Theorem: filepath-set-fix-when-filepath-setp

(defthm filepath-set-fix-when-filepath-setp
  (implies (filepath-setp x)
           (equal (filepath-set-fix x) x)))

Theorem: emptyp-filepath-set-fix

(defthm emptyp-filepath-set-fix
  (implies (or (emptyp x) (not (filepath-setp x)))
           (emptyp (filepath-set-fix x))))

Theorem: emptyp-of-filepath-set-fix

(defthm emptyp-of-filepath-set-fix
  (equal (emptyp (filepath-set-fix x))
         (or (not (filepath-setp x))
             (emptyp x))))

Function: filepath-set-equiv$inline

(defun filepath-set-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (filepath-setp acl2::x)
                              (filepath-setp acl2::y))))
  (equal (filepath-set-fix acl2::x)
         (filepath-set-fix acl2::y)))

Theorem: filepath-set-equiv-is-an-equivalence

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

Theorem: filepath-set-equiv-implies-equal-filepath-set-fix-1

(defthm filepath-set-equiv-implies-equal-filepath-set-fix-1
  (implies (filepath-set-equiv acl2::x x-equiv)
           (equal (filepath-set-fix acl2::x)
                  (filepath-set-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: filepath-set-fix-under-filepath-set-equiv

(defthm filepath-set-fix-under-filepath-set-equiv
  (filepath-set-equiv (filepath-set-fix acl2::x)
                      acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-filepath-set-fix-1-forward-to-filepath-set-equiv

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

Theorem: equal-of-filepath-set-fix-2-forward-to-filepath-set-equiv

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

Theorem: filepath-set-equiv-of-filepath-set-fix-1-forward

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

Theorem: filepath-set-equiv-of-filepath-set-fix-2-forward

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

Theorem: filepath-set-fix-of-filepath-set-fix-x

(defthm filepath-set-fix-of-filepath-set-fix-x
  (equal (filepath-set-fix (filepath-set-fix x))
         (filepath-set-fix x)))

Theorem: filepath-set-fix-filepath-set-equiv-congruence-on-x

(defthm filepath-set-fix-filepath-set-equiv-congruence-on-x
  (implies (filepath-set-equiv x x-equiv)
           (equal (filepath-set-fix x)
                  (filepath-set-fix x-equiv)))
  :rule-classes :congruence)