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Env-equiv

Basic equivalence relation for env structures.

Definitions and Theorems

Function: env-equiv$inline

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

Theorem: env-equiv-is-an-equivalence

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

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

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

Theorem: env-fix-under-env-equiv

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

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

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

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

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

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

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

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

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