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Sig-fix

Fixing function for sig structures.

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

Definitions and Theorems

Function: sig-fix$inline

(defun sig-fix$inline (x)
  (declare (xargs :guard (sig-p x)))
  (let ((acl2::__function__ 'sig-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (b* ((module (symbol-fix (cdr (std::da-nth 0 x))))
              (index (ifix (cdr (std::da-nth 1 x)))))
           (list (cons 'module module)
                 (cons 'index index)))
         :exec x)))

Theorem: sig-p-of-sig-fix

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

Theorem: sig-fix-when-sig-p

(defthm sig-fix-when-sig-p
  (implies (sig-p x)
           (equal (sig-fix x) x)))

Function: sig-equiv$inline

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

Theorem: sig-equiv-is-an-equivalence

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

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

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

Theorem: sig-fix-under-sig-equiv

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

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

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

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

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

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

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

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

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