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Declon/stmt

Declon/stmt-fix

Fixing function for declon/stmt structures.

Signature
(declon/stmt-fix x) → new-x
Arguments: x — Guard (declon/stmt-p x).
Returns: new-x — Type (declon/stmt-p new-x).

Definitions and Theorems

Function: declon/stmt-fix$inline

(defun declon/stmt-fix$inline (x)
  (declare (xargs :guard (declon/stmt-p x)))
  (mbe :logic
       (case (declon/stmt-kind x)
         (:declon (b* ((declon (declon-fix (cdr x))))
                    (cons :declon declon)))
         (:stmt (b* ((expr (expr-fix (cdr x))))
                  (cons :stmt expr))))
       :exec x))

Theorem: declon/stmt-p-of-declon/stmt-fix

(defthm declon/stmt-p-of-declon/stmt-fix
  (b* ((new-x (declon/stmt-fix$inline x)))
    (declon/stmt-p new-x))
  :rule-classes :rewrite)

Theorem: declon/stmt-fix-when-declon/stmt-p

(defthm declon/stmt-fix-when-declon/stmt-p
  (implies (declon/stmt-p x)
           (equal (declon/stmt-fix x) x)))

Function: declon/stmt-equiv$inline

(defun declon/stmt-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (declon/stmt-p acl2::x)
                              (declon/stmt-p acl2::y))))
  (equal (declon/stmt-fix acl2::x)
         (declon/stmt-fix acl2::y)))

Theorem: declon/stmt-equiv-is-an-equivalence

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

Theorem: declon/stmt-equiv-implies-equal-declon/stmt-fix-1

(defthm declon/stmt-equiv-implies-equal-declon/stmt-fix-1
  (implies (declon/stmt-equiv acl2::x x-equiv)
           (equal (declon/stmt-fix acl2::x)
                  (declon/stmt-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: declon/stmt-fix-under-declon/stmt-equiv

(defthm declon/stmt-fix-under-declon/stmt-equiv
  (declon/stmt-equiv (declon/stmt-fix acl2::x)
                     acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-declon/stmt-fix-1-forward-to-declon/stmt-equiv

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

Theorem: equal-of-declon/stmt-fix-2-forward-to-declon/stmt-equiv

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

Theorem: declon/stmt-equiv-of-declon/stmt-fix-1-forward

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

Theorem: declon/stmt-equiv-of-declon/stmt-fix-2-forward

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

Theorem: declon/stmt-kind$inline-of-declon/stmt-fix-x

(defthm declon/stmt-kind$inline-of-declon/stmt-fix-x
  (equal (declon/stmt-kind$inline (declon/stmt-fix x))
         (declon/stmt-kind$inline x)))

Theorem: declon/stmt-kind$inline-declon/stmt-equiv-congruence-on-x

(defthm declon/stmt-kind$inline-declon/stmt-equiv-congruence-on-x
  (implies (declon/stmt-equiv x x-equiv)
           (equal (declon/stmt-kind$inline x)
                  (declon/stmt-kind$inline x-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-declon/stmt-fix

(defthm consp-of-declon/stmt-fix
  (consp (declon/stmt-fix x))
  :rule-classes :type-prescription)

Theorem: declon/stmt-fix$inline-of-declon/stmt-fix-x

(defthm declon/stmt-fix$inline-of-declon/stmt-fix-x
  (equal (declon/stmt-fix$inline (declon/stmt-fix x))
         (declon/stmt-fix$inline x)))

Theorem: declon/stmt-fix$inline-declon/stmt-equiv-congruence-on-x

(defthm declon/stmt-fix$inline-declon/stmt-equiv-congruence-on-x
  (implies (declon/stmt-equiv x x-equiv)
           (equal (declon/stmt-fix$inline x)
                  (declon/stmt-fix$inline x-equiv)))
  :rule-classes :congruence)