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Valid-defstatus

Valid-defstatus-equiv

Basic equivalence relation for valid-defstatus structures.

Definitions and Theorems

Function: valid-defstatus-equiv$inline

(defun valid-defstatus-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (valid-defstatusp acl2::x)
                              (valid-defstatusp acl2::y))))
  (equal (valid-defstatus-fix acl2::x)
         (valid-defstatus-fix acl2::y)))

Theorem: valid-defstatus-equiv-is-an-equivalence

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

Theorem: valid-defstatus-equiv-implies-equal-valid-defstatus-fix-1

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

Theorem: valid-defstatus-fix-under-valid-defstatus-equiv

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

Theorem: equal-of-valid-defstatus-fix-1-forward-to-valid-defstatus-equiv

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

Theorem: equal-of-valid-defstatus-fix-2-forward-to-valid-defstatus-equiv

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

Theorem: valid-defstatus-equiv-of-valid-defstatus-fix-1-forward

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

Theorem: valid-defstatus-equiv-of-valid-defstatus-fix-2-forward

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