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• Param

Param-equiv

Basic equivalence relation for param structures.

Definitions and Theorems

Function: param-equiv$inline

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

Theorem: param-equiv-is-an-equivalence

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

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

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

Theorem: param-fix-under-param-equiv

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

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

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

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

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

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

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

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

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