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• Denv

Denv-equiv

Basic equivalence relation for denv structures.

Definitions and Theorems

Function: denv-equiv$inline

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

Theorem: denv-equiv-is-an-equivalence

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

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

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

Theorem: denv-fix-under-denv-equiv

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

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

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

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

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

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

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

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

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