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Senv

Senv-equiv

Basic equivalence relation for senv structures.

Definitions and Theorems

Function: senv-equiv$inline

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

Theorem: senv-equiv-is-an-equivalence

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

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

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

Theorem: senv-fix-under-senv-equiv

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

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

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

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

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

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

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

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

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