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Expr-grade

Expr-grade-fix

Fixing function for expr-grade structures.

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

Definitions and Theorems

Function: expr-grade-fix$inline

(defun expr-grade-fix$inline (x)
  (declare (xargs :guard (expr-gradep x)))
  (let ((__function__ 'expr-grade-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (expr-grade-kind x)
           (:top (cons :top (list)))
           (:assignment (cons :assignment (list)))
           (:conditional (cons :conditional (list)))
           (:logical-or (cons :logical-or (list)))
           (:logical-and (cons :logical-and (list)))
           (:ior (cons :ior (list)))
           (:xor (cons :xor (list)))
           (:and (cons :and (list)))
           (:equality (cons :equality (list)))
           (:relational (cons :relational (list)))
           (:shift (cons :shift (list)))
           (:additive (cons :additive (list)))
           (:multiplicative (cons :multiplicative (list)))
           (:cast (cons :cast (list)))
           (:unary (cons :unary (list)))
           (:postfix (cons :postfix (list)))
           (:primary (cons :primary (list))))
         :exec x)))

Theorem: expr-gradep-of-expr-grade-fix

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

Theorem: expr-grade-fix-when-expr-gradep

(defthm expr-grade-fix-when-expr-gradep
  (implies (expr-gradep x)
           (equal (expr-grade-fix x) x)))

Function: expr-grade-equiv$inline

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

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

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

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

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

Theorem: expr-grade-fix-under-expr-grade-equiv

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

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

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

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

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

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

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

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

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

Theorem: expr-grade-kind$inline-of-expr-grade-fix-x

(defthm expr-grade-kind$inline-of-expr-grade-fix-x
  (equal (expr-grade-kind$inline (expr-grade-fix x))
         (expr-grade-kind$inline x)))

Theorem: expr-grade-kind$inline-expr-grade-equiv-congruence-on-x

(defthm expr-grade-kind$inline-expr-grade-equiv-congruence-on-x
  (implies (expr-grade-equiv x x-equiv)
           (equal (expr-grade-kind$inline x)
                  (expr-grade-kind$inline x-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-expr-grade-fix

(defthm consp-of-expr-grade-fix
  (consp (expr-grade-fix x))
  :rule-classes :type-prescription)