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Q-char

Q-char-equiv

Basic equivalence relation for q-char structures.

Definitions and Theorems

Function: q-char-equiv$inline

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

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

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

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

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

Theorem: q-char-fix-under-q-char-equiv

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

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

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

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

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

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

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

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

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