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Fun-sinfo

Fun-sinfo-fix

Fixing function for fun-sinfo structures.

Signature
(fun-sinfo-fix x) → new-x
Arguments: x — Guard (fun-sinfop x).
Returns: new-x — Type (fun-sinfop new-x).

Definitions and Theorems

Function: fun-sinfo-fix$inline

(defun fun-sinfo-fix$inline (x)
  (declare (xargs :guard (fun-sinfop x)))
  (let ((__function__ 'fun-sinfo-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((inputs (type-list-fix (cdr (std::da-nth 0 x))))
              (output (type-fix (cdr (std::da-nth 1 x))))
              (definedp (acl2::bool-fix (cdr (std::da-nth 2 x)))))
           (list (cons 'inputs inputs)
                 (cons 'output output)
                 (cons 'definedp definedp)))
         :exec x)))

Theorem: fun-sinfop-of-fun-sinfo-fix

(defthm fun-sinfop-of-fun-sinfo-fix
  (b* ((new-x (fun-sinfo-fix$inline x)))
    (fun-sinfop new-x))
  :rule-classes :rewrite)

Theorem: fun-sinfo-fix-when-fun-sinfop

(defthm fun-sinfo-fix-when-fun-sinfop
  (implies (fun-sinfop x)
           (equal (fun-sinfo-fix x) x)))

Function: fun-sinfo-equiv$inline

(defun fun-sinfo-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (fun-sinfop acl2::x)
                              (fun-sinfop acl2::y))))
  (equal (fun-sinfo-fix acl2::x)
         (fun-sinfo-fix acl2::y)))

Theorem: fun-sinfo-equiv-is-an-equivalence

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

Theorem: fun-sinfo-equiv-implies-equal-fun-sinfo-fix-1

(defthm fun-sinfo-equiv-implies-equal-fun-sinfo-fix-1
  (implies (fun-sinfo-equiv acl2::x x-equiv)
           (equal (fun-sinfo-fix acl2::x)
                  (fun-sinfo-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: fun-sinfo-fix-under-fun-sinfo-equiv

(defthm fun-sinfo-fix-under-fun-sinfo-equiv
  (fun-sinfo-equiv (fun-sinfo-fix acl2::x)
                   acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-fun-sinfo-fix-1-forward-to-fun-sinfo-equiv

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

Theorem: equal-of-fun-sinfo-fix-2-forward-to-fun-sinfo-equiv

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

Theorem: fun-sinfo-equiv-of-fun-sinfo-fix-1-forward

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

Theorem: fun-sinfo-equiv-of-fun-sinfo-fix-2-forward

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