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• Values+denv

Values+denv-equiv

Basic equivalence relation for values+denv structures.

Definitions and Theorems

Function: values+denv-equiv$inline

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

Theorem: values+denv-equiv-is-an-equivalence

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

Theorem: values+denv-equiv-implies-equal-values+denv-fix-1

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

Theorem: values+denv-fix-under-values+denv-equiv

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

Theorem: equal-of-values+denv-fix-1-forward-to-values+denv-equiv

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

Theorem: equal-of-values+denv-fix-2-forward-to-values+denv-equiv

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

Theorem: values+denv-equiv-of-values+denv-fix-1-forward

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

Theorem: values+denv-equiv-of-values+denv-fix-2-forward

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