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• Fun-env

Fun-env-fix

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

Signature

(fun-env-fix x) → *

Arguments

x — Guard (fun-envp x).

Definitions and Theorems

Function: fun-env-fix

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

Theorem: fun-envp-of-fun-env-fix

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

Theorem: fun-env-fix-when-fun-envp

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

Theorem: emptyp-fun-env-fix

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

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

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

Function: fun-env-equiv$inline

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

Theorem: fun-env-equiv-is-an-equivalence

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

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

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

Theorem: fun-env-fix-under-fun-env-equiv

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

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

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

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

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

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

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

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

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