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Lstate

Lstate-equiv

Basic equivalence relation for lstate structures.

Definitions and Theorems

Function: lstate-equiv$inline

(defun lstate-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (lstatep acl2::x)
                              (lstatep acl2::y))))
  (equal (lstate-fix acl2::x)
         (lstate-fix acl2::y)))

Theorem: lstate-equiv-is-an-equivalence

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

Theorem: lstate-equiv-implies-equal-lstate-fix-1

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

Theorem: lstate-fix-under-lstate-equiv

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

Theorem: equal-of-lstate-fix-1-forward-to-lstate-equiv

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

Theorem: equal-of-lstate-fix-2-forward-to-lstate-equiv

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

Theorem: lstate-equiv-of-lstate-fix-1-forward

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

Theorem: lstate-equiv-of-lstate-fix-2-forward

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