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Program

Program-fix

Fixing function for program structures.

Signature
(program-fix x) → new-x
Arguments: x — Guard (programp x).
Returns: new-x — Type (programp new-x).

Definitions and Theorems

Function: program-fix$inline

(defun program-fix$inline (x)
 (declare (xargs :guard (programp x)))
 (let ((__function__ 'program-fix))
  (declare (ignorable __function__))
  (mbe
    :logic
    (b* ((name (identifier-fix (cdr (std::da-nth 0 (cdr x)))))
         (imports (import-list-fix (cdr (std::da-nth 1 (cdr x)))))
         (defs (programdef-list-fix (cdr (std::da-nth 2 (cdr x))))))
      (cons :program (list (cons 'name name)
                           (cons 'imports imports)
                           (cons 'defs defs))))
    :exec x)))

Theorem: programp-of-program-fix

(defthm programp-of-program-fix
  (b* ((new-x (program-fix$inline x)))
    (programp new-x))
  :rule-classes :rewrite)

Theorem: program-fix-when-programp

(defthm program-fix-when-programp
  (implies (programp x)
           (equal (program-fix x) x)))

Function: program-equiv$inline

(defun program-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (programp acl2::x)
                              (programp acl2::y))))
  (equal (program-fix acl2::x)
         (program-fix acl2::y)))

Theorem: program-equiv-is-an-equivalence

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

Theorem: program-equiv-implies-equal-program-fix-1

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

Theorem: program-fix-under-program-equiv

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

Theorem: equal-of-program-fix-1-forward-to-program-equiv

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

Theorem: equal-of-program-fix-2-forward-to-program-equiv

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

Theorem: program-equiv-of-program-fix-1-forward

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

Theorem: program-equiv-of-program-fix-2-forward

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