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Expr-priority

Expr-priority-fix

Fixing function for expr-priority structures.

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

Definitions and Theorems

Function: expr-priority-fix$inline

(defun expr-priority-fix$inline (x)
  (declare (xargs :guard (expr-priorityp x)))
  (mbe :logic
       (case (expr-priority-kind x)
         (:primary (cons :primary (list)))
         (:postfix (cons :postfix (list)))
         (:unary (cons :unary (list)))
         (:cast (cons :cast (list)))
         (:mul (cons :mul (list)))
         (:add (cons :add (list)))
         (:sh (cons :sh (list)))
         (:rel (cons :rel (list)))
         (:eq (cons :eq (list)))
         (:and (cons :and (list)))
         (:xor (cons :xor (list)))
         (:ior (cons :ior (list)))
         (:logand (cons :logand (list)))
         (:logor (cons :logor (list)))
         (:cond (cons :cond (list)))
         (:asg (cons :asg (list)))
         (:expr (cons :expr (list))))
       :exec x))

Theorem: expr-priorityp-of-expr-priority-fix

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

Theorem: expr-priority-fix-when-expr-priorityp

(defthm expr-priority-fix-when-expr-priorityp
  (implies (expr-priorityp x)
           (equal (expr-priority-fix x) x)))

Function: expr-priority-equiv$inline

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

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

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

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

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

Theorem: expr-priority-fix-under-expr-priority-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: consp-of-expr-priority-fix

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

Theorem: expr-priority-fix$inline-of-expr-priority-fix-x

(defthm expr-priority-fix$inline-of-expr-priority-fix-x
  (equal (expr-priority-fix$inline (expr-priority-fix x))
         (expr-priority-fix$inline x)))

Theorem: expr-priority-fix$inline-expr-priority-equiv-congruence-on-x

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