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Swcase

Swcase-equiv

Basic equivalence relation for swcase structures.

Definitions and Theorems

Function: swcase-equiv$inline

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

Theorem: swcase-equiv-is-an-equivalence

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

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

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

Theorem: swcase-fix-under-swcase-equiv

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

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

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

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

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

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

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

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

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