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• Trans-state

Trans-state-equiv

Basic equivalence relation for trans-state structures.

Definitions and Theorems

Function: trans-state-equiv$inline

(defun trans-state-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (trans-statep acl2::x)
                              (trans-statep acl2::y))))
  (equal (trans-state-fix acl2::x)
         (trans-state-fix acl2::y)))

Theorem: trans-state-equiv-is-an-equivalence

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

Theorem: trans-state-equiv-implies-equal-trans-state-fix-1

(defthm trans-state-equiv-implies-equal-trans-state-fix-1
  (implies (trans-state-equiv acl2::x x-equiv)
           (equal (trans-state-fix acl2::x)
                  (trans-state-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: trans-state-fix-under-trans-state-equiv

(defthm trans-state-fix-under-trans-state-equiv
  (trans-state-equiv (trans-state-fix acl2::x)
                     acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-trans-state-fix-1-forward-to-trans-state-equiv

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

Theorem: equal-of-trans-state-fix-2-forward-to-trans-state-equiv

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

Theorem: trans-state-equiv-of-trans-state-fix-1-forward

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

Theorem: trans-state-equiv-of-trans-state-fix-2-forward

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