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• Valid-ord-info-option

Valid-ord-info-option-fix

Fixing function for valid-ord-info-option structures.

Signature

(valid-ord-info-option-fix x) → new-x

Arguments

x — Guard (valid-ord-info-optionp x).

Returns

new-x — Type (valid-ord-info-optionp new-x).

Definitions and Theorems

Function: valid-ord-info-option-fix$inline

(defun valid-ord-info-option-fix$inline (x)
  (declare (xargs :guard (valid-ord-info-optionp x)))
  (mbe :logic (cond ((not x) nil)
                    (t (b* ((fty::val (valid-ord-info-fix x)))
                         fty::val)))
       :exec x))

Theorem: valid-ord-info-optionp-of-valid-ord-info-option-fix

(defthm valid-ord-info-optionp-of-valid-ord-info-option-fix
  (b* ((new-x (valid-ord-info-option-fix$inline x)))
    (valid-ord-info-optionp new-x))
  :rule-classes :rewrite)

Theorem: valid-ord-info-option-fix-when-valid-ord-info-optionp

(defthm valid-ord-info-option-fix-when-valid-ord-info-optionp
  (implies (valid-ord-info-optionp x)
           (equal (valid-ord-info-option-fix x)
                  x)))

Function: valid-ord-info-option-equiv$inline

(defun valid-ord-info-option-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (valid-ord-info-optionp acl2::x)
                              (valid-ord-info-optionp acl2::y))))
  (equal (valid-ord-info-option-fix acl2::x)
         (valid-ord-info-option-fix acl2::y)))

Theorem: valid-ord-info-option-equiv-is-an-equivalence

(defthm valid-ord-info-option-equiv-is-an-equivalence
  (and (booleanp (valid-ord-info-option-equiv x y))
       (valid-ord-info-option-equiv x x)
       (implies (valid-ord-info-option-equiv x y)
                (valid-ord-info-option-equiv y x))
       (implies (and (valid-ord-info-option-equiv x y)
                     (valid-ord-info-option-equiv y z))
                (valid-ord-info-option-equiv x z)))
  :rule-classes (:equivalence))

Theorem: valid-ord-info-option-equiv-implies-equal-valid-ord-info-option-fix-1

(defthm
 valid-ord-info-option-equiv-implies-equal-valid-ord-info-option-fix-1
 (implies (valid-ord-info-option-equiv acl2::x x-equiv)
          (equal (valid-ord-info-option-fix acl2::x)
                 (valid-ord-info-option-fix x-equiv)))
 :rule-classes (:congruence))

Theorem: valid-ord-info-option-fix-under-valid-ord-info-option-equiv

(defthm valid-ord-info-option-fix-under-valid-ord-info-option-equiv
  (valid-ord-info-option-equiv (valid-ord-info-option-fix acl2::x)
                               acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-valid-ord-info-option-fix-1-forward-to-valid-ord-info-option-equiv

(defthm
 equal-of-valid-ord-info-option-fix-1-forward-to-valid-ord-info-option-equiv
 (implies (equal (valid-ord-info-option-fix acl2::x)
                 acl2::y)
          (valid-ord-info-option-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: equal-of-valid-ord-info-option-fix-2-forward-to-valid-ord-info-option-equiv

(defthm
 equal-of-valid-ord-info-option-fix-2-forward-to-valid-ord-info-option-equiv
 (implies (equal acl2::x
                 (valid-ord-info-option-fix acl2::y))
          (valid-ord-info-option-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: valid-ord-info-option-equiv-of-valid-ord-info-option-fix-1-forward

(defthm
 valid-ord-info-option-equiv-of-valid-ord-info-option-fix-1-forward
 (implies
    (valid-ord-info-option-equiv (valid-ord-info-option-fix acl2::x)
                                 acl2::y)
    (valid-ord-info-option-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: valid-ord-info-option-equiv-of-valid-ord-info-option-fix-2-forward

(defthm
 valid-ord-info-option-equiv-of-valid-ord-info-option-fix-2-forward
 (implies
   (valid-ord-info-option-equiv acl2::x
                                (valid-ord-info-option-fix acl2::y))
   (valid-ord-info-option-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: valid-ord-info-option-fix$inline-of-valid-ord-info-option-fix-x

(defthm
    valid-ord-info-option-fix$inline-of-valid-ord-info-option-fix-x
 (equal
    (valid-ord-info-option-fix$inline (valid-ord-info-option-fix x))
    (valid-ord-info-option-fix$inline x)))

Theorem: valid-ord-info-option-fix$inline-valid-ord-info-option-equiv-congruence-on-x

(defthm
 valid-ord-info-option-fix$inline-valid-ord-info-option-equiv-congruence-on-x
 (implies (valid-ord-info-option-equiv x x-equiv)
          (equal (valid-ord-info-option-fix$inline x)
                 (valid-ord-info-option-fix$inline x-equiv)))
 :rule-classes :congruence)