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• Env

Env-fix

(env-fix x) is a usual ACL2::fty omap fixing function.

Signature

(env-fix x) → *

Arguments

x — Guard (envp x).

Definitions and Theorems

Function: env-fix

(defun env-fix (x)
  (declare (xargs :guard (envp x)))
  (mbe :logic (if (envp x) x nil)
       :exec x))

Theorem: envp-of-env-fix

(defthm envp-of-env-fix
  (envp (env-fix x)))

Theorem: env-fix-when-envp

(defthm env-fix-when-envp
  (implies (envp x)
           (equal (env-fix x) x)))

Theorem: emptyp-env-fix

(defthm emptyp-env-fix
  (implies (or (omap::emptyp x) (not (envp x)))
           (omap::emptyp (env-fix x))))

Theorem: emptyp-of-env-fix-to-not-env-or-emptyp

(defthm emptyp-of-env-fix-to-not-env-or-emptyp
  (equal (omap::emptyp (env-fix x))
         (or (not (envp x)) (omap::emptyp x))))

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)