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• Vars+modes

Vars+modes-equiv

Basic equivalence relation for vars+modes structures.

Definitions and Theorems

Function: vars+modes-equiv$inline

(defun vars+modes-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (vars+modes-p acl2::x)
                              (vars+modes-p acl2::y))))
  (equal (vars+modes-fix acl2::x)
         (vars+modes-fix acl2::y)))

Theorem: vars+modes-equiv-is-an-equivalence

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

Theorem: vars+modes-equiv-implies-equal-vars+modes-fix-1

(defthm vars+modes-equiv-implies-equal-vars+modes-fix-1
  (implies (vars+modes-equiv acl2::x x-equiv)
           (equal (vars+modes-fix acl2::x)
                  (vars+modes-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: vars+modes-fix-under-vars+modes-equiv

(defthm vars+modes-fix-under-vars+modes-equiv
  (vars+modes-equiv (vars+modes-fix acl2::x)
                    acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-vars+modes-fix-1-forward-to-vars+modes-equiv

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

Theorem: equal-of-vars+modes-fix-2-forward-to-vars+modes-equiv

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

Theorem: vars+modes-equiv-of-vars+modes-fix-1-forward

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

Theorem: vars+modes-equiv-of-vars+modes-fix-2-forward

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