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Ringosc3

Ringosc3-equiv

Basic equivalence relation for ringosc3 structures.

Definitions and Theorems

Function: ringosc3-equiv$inline

(defun ringosc3-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (ringosc3-p acl2::x)
                              (ringosc3-p acl2::y))))
  (equal (ringosc3-fix acl2::x)
         (ringosc3-fix acl2::y)))

Theorem: ringosc3-equiv-is-an-equivalence

(defthm ringosc3-equiv-is-an-equivalence
  (and (booleanp (ringosc3-equiv x y))
       (ringosc3-equiv x x)
       (implies (ringosc3-equiv x y)
                (ringosc3-equiv y x))
       (implies (and (ringosc3-equiv x y)
                     (ringosc3-equiv y z))
                (ringosc3-equiv x z)))
  :rule-classes (:equivalence))

Theorem: ringosc3-equiv-implies-equal-ringosc3-fix-1

(defthm ringosc3-equiv-implies-equal-ringosc3-fix-1
  (implies (ringosc3-equiv acl2::x x-equiv)
           (equal (ringosc3-fix acl2::x)
                  (ringosc3-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: ringosc3-fix-under-ringosc3-equiv

(defthm ringosc3-fix-under-ringosc3-equiv
  (ringosc3-equiv (ringosc3-fix acl2::x)
                  acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-ringosc3-fix-1-forward-to-ringosc3-equiv

(defthm equal-of-ringosc3-fix-1-forward-to-ringosc3-equiv
  (implies (equal (ringosc3-fix acl2::x) acl2::y)
           (ringosc3-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-ringosc3-fix-2-forward-to-ringosc3-equiv

(defthm equal-of-ringosc3-fix-2-forward-to-ringosc3-equiv
  (implies (equal acl2::x (ringosc3-fix acl2::y))
           (ringosc3-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: ringosc3-equiv-of-ringosc3-fix-1-forward

(defthm ringosc3-equiv-of-ringosc3-fix-1-forward
  (implies (ringosc3-equiv (ringosc3-fix acl2::x)
                           acl2::y)
           (ringosc3-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: ringosc3-equiv-of-ringosc3-fix-2-forward

(defthm ringosc3-equiv-of-ringosc3-fix-2-forward
  (implies (ringosc3-equiv acl2::x (ringosc3-fix acl2::y))
           (ringosc3-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)