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Expr-option

Expr-option-fix

Fixing function for expr-option structures.

Signature
(expr-option-fix x) → new-x
Arguments: x — Guard (expr-optionp x).
Returns: new-x — Type (expr-optionp new-x).

Definitions and Theorems

Function: expr-option-fix$inline

(defun expr-option-fix$inline (x)
  (declare (xargs :guard (expr-optionp x)))
  (let ((__function__ 'expr-option-fix))
    (declare (ignorable __function__))
    (mbe :logic (cond ((not x) nil)
                      (t (b* ((fty::val (expr-fix x)))
                           fty::val)))
         :exec x)))

Theorem: expr-optionp-of-expr-option-fix

(defthm expr-optionp-of-expr-option-fix
  (b* ((new-x (expr-option-fix$inline x)))
    (expr-optionp new-x))
  :rule-classes :rewrite)

Theorem: expr-option-fix-when-expr-optionp

(defthm expr-option-fix-when-expr-optionp
  (implies (expr-optionp x)
           (equal (expr-option-fix x) x)))

Function: expr-option-equiv$inline

(defun expr-option-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (expr-optionp acl2::x)
                              (expr-optionp acl2::y))))
  (equal (expr-option-fix acl2::x)
         (expr-option-fix acl2::y)))

Theorem: expr-option-equiv-is-an-equivalence

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

Theorem: expr-option-equiv-implies-equal-expr-option-fix-1

(defthm expr-option-equiv-implies-equal-expr-option-fix-1
  (implies (expr-option-equiv acl2::x x-equiv)
           (equal (expr-option-fix acl2::x)
                  (expr-option-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: expr-option-fix-under-expr-option-equiv

(defthm expr-option-fix-under-expr-option-equiv
  (expr-option-equiv (expr-option-fix acl2::x)
                     acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-expr-option-fix-1-forward-to-expr-option-equiv

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

Theorem: equal-of-expr-option-fix-2-forward-to-expr-option-equiv

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

Theorem: expr-option-equiv-of-expr-option-fix-1-forward

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

Theorem: expr-option-equiv-of-expr-option-fix-2-forward

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