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H-char

H-char-equiv

Basic equivalence relation for h-char structures.

Definitions and Theorems

Function: h-char-equiv$inline

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

Theorem: h-char-equiv-is-an-equivalence

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

Theorem: h-char-equiv-implies-equal-h-char-fix-1

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

Theorem: h-char-fix-under-h-char-equiv

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

Theorem: equal-of-h-char-fix-1-forward-to-h-char-equiv

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

Theorem: equal-of-h-char-fix-2-forward-to-h-char-equiv

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

Theorem: h-char-equiv-of-h-char-fix-1-forward

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

Theorem: h-char-equiv-of-h-char-fix-2-forward

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