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• Valuemap+denv

Valuemap+denv-equiv

Basic equivalence relation for valuemap+denv structures.

Definitions and Theorems

Function: valuemap+denv-equiv$inline

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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