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• Api+senv

Api+senv-equiv

Basic equivalence relation for api+senv structures.

Definitions and Theorems

Function: api+senv-equiv$inline

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

Theorem: api+senv-equiv-is-an-equivalence

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

Theorem: api+senv-equiv-implies-equal-api+senv-fix-1

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

Theorem: api+senv-fix-under-api+senv-equiv

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

Theorem: equal-of-api+senv-fix-1-forward-to-api+senv-equiv

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

Theorem: equal-of-api+senv-fix-2-forward-to-api+senv-equiv

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

Theorem: api+senv-equiv-of-api+senv-fix-1-forward

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

Theorem: api+senv-equiv-of-api+senv-fix-2-forward

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