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Enum-spec

Enum-spec-equiv

Basic equivalence relation for enum-spec structures.

Definitions and Theorems

Function: enum-spec-equiv$inline

(defun enum-spec-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (enum-specp acl2::x)
                              (enum-specp acl2::y))))
  (equal (enum-spec-fix acl2::x)
         (enum-spec-fix acl2::y)))

Theorem: enum-spec-equiv-is-an-equivalence

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

Theorem: enum-spec-equiv-implies-equal-enum-spec-fix-1

(defthm enum-spec-equiv-implies-equal-enum-spec-fix-1
  (implies (enum-spec-equiv acl2::x x-equiv)
           (equal (enum-spec-fix acl2::x)
                  (enum-spec-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: enum-spec-fix-under-enum-spec-equiv

(defthm enum-spec-fix-under-enum-spec-equiv
  (enum-spec-equiv (enum-spec-fix acl2::x)
                   acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-enum-spec-fix-1-forward-to-enum-spec-equiv

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

Theorem: equal-of-enum-spec-fix-2-forward-to-enum-spec-equiv

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

Theorem: enum-spec-equiv-of-enum-spec-fix-1-forward

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

Theorem: enum-spec-equiv-of-enum-spec-fix-2-forward

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