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• Sinteger-bit-role

Sinteger-bit-role-equiv

Basic equivalence relation for sinteger-bit-role structures.

Definitions and Theorems

Function: sinteger-bit-role-equiv$inline

(defun sinteger-bit-role-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (sinteger-bit-rolep acl2::x)
                              (sinteger-bit-rolep acl2::y))))
  (equal (sinteger-bit-role-fix acl2::x)
         (sinteger-bit-role-fix acl2::y)))

Theorem: sinteger-bit-role-equiv-is-an-equivalence

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

Theorem: sinteger-bit-role-equiv-implies-equal-sinteger-bit-role-fix-1

(defthm
      sinteger-bit-role-equiv-implies-equal-sinteger-bit-role-fix-1
  (implies (sinteger-bit-role-equiv acl2::x x-equiv)
           (equal (sinteger-bit-role-fix acl2::x)
                  (sinteger-bit-role-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: sinteger-bit-role-fix-under-sinteger-bit-role-equiv

(defthm sinteger-bit-role-fix-under-sinteger-bit-role-equiv
  (sinteger-bit-role-equiv (sinteger-bit-role-fix acl2::x)
                           acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-sinteger-bit-role-fix-1-forward-to-sinteger-bit-role-equiv

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

Theorem: equal-of-sinteger-bit-role-fix-2-forward-to-sinteger-bit-role-equiv

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

Theorem: sinteger-bit-role-equiv-of-sinteger-bit-role-fix-1-forward

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

Theorem: sinteger-bit-role-equiv-of-sinteger-bit-role-fix-2-forward

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