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    • Expr-gout

    Expr-gout-fix

    Fixing function for expr-gout structures.

    Signature
    (expr-gout-fix x) → new-x
    Arguments
    x — Guard (expr-goutp x).
    Returns
    new-x — Type (expr-goutp new-x).

    Definitions and Theorems

    Function: expr-gout-fix$inline

    (defun expr-gout-fix$inline (x)
     (declare (xargs :guard (expr-goutp x)))
     (let ((__function__ 'expr-gout-fix))
      (declare (ignorable __function__))
      (mbe
       :logic
       (b*
        ((expr (expr-fix (cdr (std::da-nth 0 x))))
         (type (type-fix (cdr (std::da-nth 1 x))))
         (term (pseudo-term-fix (cdr (std::da-nth 2 x))))
         (events
             (acl2::pseudo-event-form-list-fix (cdr (std::da-nth 3 x))))
         (thm-name (symbol-fix (cdr (std::da-nth 4 x))))
         (thm-index (pos-fix (cdr (std::da-nth 5 x))))
         (names-to-avoid (symbol-list-fix (cdr (std::da-nth 6 x)))))
        (list (cons 'expr expr)
              (cons 'type type)
              (cons 'term term)
              (cons 'events events)
              (cons 'thm-name thm-name)
              (cons 'thm-index thm-index)
              (cons 'names-to-avoid names-to-avoid)))
       :exec x)))

    Theorem: expr-goutp-of-expr-gout-fix

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

    Theorem: expr-gout-fix-when-expr-goutp

    (defthm expr-gout-fix-when-expr-goutp
      (implies (expr-goutp x)
               (equal (expr-gout-fix x) x)))

    Function: expr-gout-equiv$inline

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

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

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

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

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

    Theorem: expr-gout-fix-under-expr-gout-equiv

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

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

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

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

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

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

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

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

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