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Programdecl

Programdecl-fix

Fixing function for programdecl structures.

Signature
(programdecl-fix x) → new-x
Arguments: x — Guard (programdeclp x).
Returns: new-x — Type (programdeclp new-x).

Definitions and Theorems

Function: programdecl-fix$inline

(defun programdecl-fix$inline (x)
 (declare (xargs :guard (programdeclp x)))
 (let ((__function__ 'programdecl-fix))
  (declare (ignorable __function__))
  (mbe
      :logic
      (b* ((id (programid-fix (cdr (std::da-nth 0 (cdr x)))))
           (items (topdecl-list-fix (cdr (std::da-nth 1 (cdr x))))))
        (cons :programdecl (list (cons 'id id)
                                 (cons 'items items))))
      :exec x)))

Theorem: programdeclp-of-programdecl-fix

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

Theorem: programdecl-fix-when-programdeclp

(defthm programdecl-fix-when-programdeclp
  (implies (programdeclp x)
           (equal (programdecl-fix x) x)))

Function: programdecl-equiv$inline

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

Theorem: programdecl-equiv-is-an-equivalence

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

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

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

Theorem: programdecl-fix-under-programdecl-equiv

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

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

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

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

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

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

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

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

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