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Lsuffix

Lsuffix-equiv

Basic equivalence relation for lsuffix structures.

Definitions and Theorems

Function: lsuffix-equiv$inline

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

Theorem: lsuffix-equiv-is-an-equivalence

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

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

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

Theorem: lsuffix-fix-under-lsuffix-equiv

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

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

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

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

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

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

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

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

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