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Expr-gin

Expr-gin-fix

Fixing function for expr-gin structures.

Signature
(expr-gin-fix x) → new-x
Arguments: x — Guard (expr-ginp x).
Returns: new-x — Type (expr-ginp new-x).

Definitions and Theorems

Function: expr-gin-fix$inline

(defun expr-gin-fix$inline (x)
 (declare (xargs :guard (expr-ginp x)))
 (let ((__function__ 'expr-gin-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (b*
    ((context (atc-context-fix (cdr (std::da-nth 0 x))))
     (inscope
        (atc-symbol-varinfo-alist-list-fix (cdr (std::da-nth 1 x))))
     (prec-tags
          (atc-string-taginfo-alist-fix (cdr (std::da-nth 2 x))))
     (fn (symbol-fix (cdr (std::da-nth 3 x))))
     (fn-guard (symbol-fix (cdr (std::da-nth 4 x))))
     (compst-var (symbol-fix (cdr (std::da-nth 5 x))))
     (thm-index (acl2::pos-fix (cdr (std::da-nth 6 x))))
     (names-to-avoid (symbol-list-fix (cdr (std::da-nth 7 x))))
     (proofs (acl2::bool-fix (cdr (std::da-nth 8 x)))))
    (list (cons 'context context)
          (cons 'inscope inscope)
          (cons 'prec-tags prec-tags)
          (cons 'fn fn)
          (cons 'fn-guard fn-guard)
          (cons 'compst-var compst-var)
          (cons 'thm-index thm-index)
          (cons 'names-to-avoid names-to-avoid)
          (cons 'proofs proofs)))
   :exec x)))

Theorem: expr-ginp-of-expr-gin-fix

(defthm expr-ginp-of-expr-gin-fix
  (b* ((new-x (expr-gin-fix$inline x)))
    (expr-ginp new-x))
  :rule-classes :rewrite)

Theorem: expr-gin-fix-when-expr-ginp

(defthm expr-gin-fix-when-expr-ginp
  (implies (expr-ginp x)
           (equal (expr-gin-fix x) x)))

Function: expr-gin-equiv$inline

(defun expr-gin-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (expr-ginp acl2::x)
                              (expr-ginp acl2::y))))
  (equal (expr-gin-fix acl2::x)
         (expr-gin-fix acl2::y)))

Theorem: expr-gin-equiv-is-an-equivalence

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

Theorem: expr-gin-equiv-implies-equal-expr-gin-fix-1

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

Theorem: expr-gin-fix-under-expr-gin-equiv

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

Theorem: equal-of-expr-gin-fix-1-forward-to-expr-gin-equiv

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

Theorem: equal-of-expr-gin-fix-2-forward-to-expr-gin-equiv

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

Theorem: expr-gin-equiv-of-expr-gin-fix-1-forward

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

Theorem: expr-gin-equiv-of-expr-gin-fix-2-forward

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