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Op-funct

Op-funct-fix

Fixing function for op-funct structures.

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

Definitions and Theorems

Function: op-funct-fix$inline

(defun op-funct-fix$inline (x)
  (declare (xargs :guard (op-funct-p x)))
  (let ((__function__ 'op-funct-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (op-funct-kind x)
           (:add (cons :add (list)))
           (:sub (cons :sub (list)))
           (:slt (cons :slt (list)))
           (:sltu (cons :sltu (list)))
           (:and (cons :and (list)))
           (:or (cons :or (list)))
           (:xor (cons :xor (list)))
           (:sll (cons :sll (list)))
           (:srl (cons :srl (list)))
           (:sra (cons :sra (list)))
           (:mul (cons :mul (list)))
           (:mulh (cons :mulh (list)))
           (:mulhu (cons :mulhu (list)))
           (:mulhsu (cons :mulhsu (list)))
           (:div (cons :div (list)))
           (:divu (cons :divu (list)))
           (:rem (cons :rem (list)))
           (:remu (cons :remu (list))))
         :exec x)))

Theorem: op-funct-p-of-op-funct-fix

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

Theorem: op-funct-fix-when-op-funct-p

(defthm op-funct-fix-when-op-funct-p
  (implies (op-funct-p x)
           (equal (op-funct-fix x) x)))

Function: op-funct-equiv$inline

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

Theorem: op-funct-equiv-is-an-equivalence

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

Theorem: op-funct-equiv-implies-equal-op-funct-fix-1

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

Theorem: op-funct-fix-under-op-funct-equiv

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

Theorem: equal-of-op-funct-fix-1-forward-to-op-funct-equiv

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

Theorem: equal-of-op-funct-fix-2-forward-to-op-funct-equiv

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

Theorem: op-funct-equiv-of-op-funct-fix-1-forward

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

Theorem: op-funct-equiv-of-op-funct-fix-2-forward

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

Theorem: op-funct-kind$inline-of-op-funct-fix-x

(defthm op-funct-kind$inline-of-op-funct-fix-x
  (equal (op-funct-kind$inline (op-funct-fix x))
         (op-funct-kind$inline x)))

Theorem: op-funct-kind$inline-op-funct-equiv-congruence-on-x

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

Theorem: consp-of-op-funct-fix

(defthm consp-of-op-funct-fix
  (consp (op-funct-fix x))
  :rule-classes :type-prescription)