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Variable-substitution

Variable-substitution-fix

(variable-substitution-fix x) is a usual ACL2::fty omap fixing function.

Signature
(variable-substitution-fix x) → *
Arguments: x — Guard (variable-substitutionp x).

Definitions and Theorems

Function: variable-substitution-fix

(defun variable-substitution-fix (x)
  (declare (xargs :guard (variable-substitutionp x)))
  (mbe :logic (if (variable-substitutionp x) x nil)
       :exec x))

Theorem: variable-substitutionp-of-variable-substitution-fix

(defthm variable-substitutionp-of-variable-substitution-fix
  (variable-substitutionp (variable-substitution-fix x)))

Theorem: variable-substitution-fix-when-variable-substitutionp

(defthm variable-substitution-fix-when-variable-substitutionp
  (implies (variable-substitutionp x)
           (equal (variable-substitution-fix x)
                  x)))

Theorem: emptyp-variable-substitution-fix

(defthm emptyp-variable-substitution-fix
  (implies (or (omap::emptyp x)
               (not (variable-substitutionp x)))
           (omap::emptyp (variable-substitution-fix x))))

Theorem: emptyp-of-variable-substitution-fix-to-not-variable-substitution-or-emptyp

(defthm
 emptyp-of-variable-substitution-fix-to-not-variable-substitution-or-emptyp
 (equal (omap::emptyp (variable-substitution-fix x))
        (or (not (variable-substitutionp x))
            (omap::emptyp x))))

Function: variable-substitution-equiv$inline

(defun variable-substitution-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (variable-substitutionp acl2::x)
                              (variable-substitutionp acl2::y))))
  (equal (variable-substitution-fix acl2::x)
         (variable-substitution-fix acl2::y)))

Theorem: variable-substitution-equiv-is-an-equivalence

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

Theorem: variable-substitution-equiv-implies-equal-variable-substitution-fix-1

(defthm
 variable-substitution-equiv-implies-equal-variable-substitution-fix-1
 (implies (variable-substitution-equiv acl2::x x-equiv)
          (equal (variable-substitution-fix acl2::x)
                 (variable-substitution-fix x-equiv)))
 :rule-classes (:congruence))

Theorem: variable-substitution-fix-under-variable-substitution-equiv

(defthm variable-substitution-fix-under-variable-substitution-equiv
  (variable-substitution-equiv (variable-substitution-fix acl2::x)
                               acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-variable-substitution-fix-1-forward-to-variable-substitution-equiv

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

Theorem: equal-of-variable-substitution-fix-2-forward-to-variable-substitution-equiv

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

Theorem: variable-substitution-equiv-of-variable-substitution-fix-1-forward

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

Theorem: variable-substitution-equiv-of-variable-substitution-fix-2-forward

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