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• Schar-format

Schar-format-equiv

Basic equivalence relation for schar-format structures.

Definitions and Theorems

Function: schar-format-equiv$inline

(defun schar-format-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (schar-formatp acl2::x)
                              (schar-formatp acl2::y))))
  (equal (schar-format-fix acl2::x)
         (schar-format-fix acl2::y)))

Theorem: schar-format-equiv-is-an-equivalence

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

Theorem: schar-format-equiv-implies-equal-schar-format-fix-1

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

Theorem: schar-format-fix-under-schar-format-equiv

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

Theorem: equal-of-schar-format-fix-1-forward-to-schar-format-equiv

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

Theorem: equal-of-schar-format-fix-2-forward-to-schar-format-equiv

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

Theorem: schar-format-equiv-of-schar-format-fix-1-forward

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

Theorem: schar-format-equiv-of-schar-format-fix-2-forward

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