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State64

State64-equiv

Basic equivalence relation for state64 structures.

Definitions and Theorems

Function: state64-equiv$inline

(defun state64-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (state64p acl2::x)
                              (state64p acl2::y))))
  (equal (state64-fix acl2::x)
         (state64-fix acl2::y)))

Theorem: state64-equiv-is-an-equivalence

(defthm state64-equiv-is-an-equivalence
  (and (booleanp (state64-equiv x y))
       (state64-equiv x x)
       (implies (state64-equiv x y)
                (state64-equiv y x))
       (implies (and (state64-equiv x y)
                     (state64-equiv y z))
                (state64-equiv x z)))
  :rule-classes (:equivalence))

Theorem: state64-equiv-implies-equal-state64-fix-1

(defthm state64-equiv-implies-equal-state64-fix-1
  (implies (state64-equiv acl2::x x-equiv)
           (equal (state64-fix acl2::x)
                  (state64-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: state64-fix-under-state64-equiv

(defthm state64-fix-under-state64-equiv
  (state64-equiv (state64-fix acl2::x)
                 acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-state64-fix-1-forward-to-state64-equiv

(defthm equal-of-state64-fix-1-forward-to-state64-equiv
  (implies (equal (state64-fix acl2::x) acl2::y)
           (state64-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-state64-fix-2-forward-to-state64-equiv

(defthm equal-of-state64-fix-2-forward-to-state64-equiv
  (implies (equal acl2::x (state64-fix acl2::y))
           (state64-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: state64-equiv-of-state64-fix-1-forward

(defthm state64-equiv-of-state64-fix-1-forward
  (implies (state64-equiv (state64-fix acl2::x)
                          acl2::y)
           (state64-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: state64-equiv-of-state64-fix-2-forward

(defthm state64-equiv-of-state64-fix-2-forward
  (implies (state64-equiv acl2::x (state64-fix acl2::y))
           (state64-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)