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Isuffix

Isuffix-equiv

Basic equivalence relation for isuffix structures.

Definitions and Theorems

Function: isuffix-equiv$inline

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

Theorem: isuffix-equiv-is-an-equivalence

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

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

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

Theorem: isuffix-fix-under-isuffix-equiv

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

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

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

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

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

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

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

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

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