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Funenv

Funenv-fix

(funenv-fix x) is a usual ACL2::fty list fixing function.

Signature
(funenv-fix x) → fty::newx
Arguments: x — Guard (funenvp x).
Returns: fty::newx — Type (funenvp fty::newx).

In the logic, we apply funscope-fix to each member of the x. In the execution, none of that is actually necessary and this is just an inlined identity function.

Definitions and Theorems

Function: funenv-fix$inline

(defun funenv-fix$inline (x)
  (declare (xargs :guard (funenvp x)))
  (let ((__function__ 'funenv-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (funscope-fix (car x))
                 (funenv-fix (cdr x))))
         :exec x)))

Theorem: funenvp-of-funenv-fix

(defthm funenvp-of-funenv-fix
  (b* ((fty::newx (funenv-fix$inline x)))
    (funenvp fty::newx))
  :rule-classes :rewrite)

Theorem: funenv-fix-when-funenvp

(defthm funenv-fix-when-funenvp
  (implies (funenvp x)
           (equal (funenv-fix x) x)))

Function: funenv-equiv$inline

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

Theorem: funenv-equiv-is-an-equivalence

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

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

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

Theorem: funenv-fix-under-funenv-equiv

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

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

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

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

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

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

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

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

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

Theorem: car-of-funenv-fix-x-under-funscope-equiv

(defthm car-of-funenv-fix-x-under-funscope-equiv
  (funscope-equiv (car (funenv-fix acl2::x))
                  (car acl2::x)))

Theorem: car-funenv-equiv-congruence-on-x-under-funscope-equiv

(defthm car-funenv-equiv-congruence-on-x-under-funscope-equiv
  (implies (funenv-equiv acl2::x x-equiv)
           (funscope-equiv (car acl2::x)
                           (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-funenv-fix-x-under-funenv-equiv

(defthm cdr-of-funenv-fix-x-under-funenv-equiv
  (funenv-equiv (cdr (funenv-fix acl2::x))
                (cdr acl2::x)))

Theorem: cdr-funenv-equiv-congruence-on-x-under-funenv-equiv

(defthm cdr-funenv-equiv-congruence-on-x-under-funenv-equiv
  (implies (funenv-equiv acl2::x x-equiv)
           (funenv-equiv (cdr acl2::x)
                         (cdr x-equiv)))
  :rule-classes :congruence)

Theorem: cons-of-funscope-fix-x-under-funenv-equiv

(defthm cons-of-funscope-fix-x-under-funenv-equiv
  (funenv-equiv (cons (funscope-fix acl2::x) acl2::y)
                (cons acl2::x acl2::y)))

Theorem: cons-funscope-equiv-congruence-on-x-under-funenv-equiv

(defthm cons-funscope-equiv-congruence-on-x-under-funenv-equiv
  (implies (funscope-equiv acl2::x x-equiv)
           (funenv-equiv (cons acl2::x acl2::y)
                         (cons x-equiv acl2::y)))
  :rule-classes :congruence)

Theorem: cons-of-funenv-fix-y-under-funenv-equiv

(defthm cons-of-funenv-fix-y-under-funenv-equiv
  (funenv-equiv (cons acl2::x (funenv-fix acl2::y))
                (cons acl2::x acl2::y)))

Theorem: cons-funenv-equiv-congruence-on-y-under-funenv-equiv

(defthm cons-funenv-equiv-congruence-on-y-under-funenv-equiv
  (implies (funenv-equiv acl2::y y-equiv)
           (funenv-equiv (cons acl2::x acl2::y)
                         (cons acl2::x y-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-funenv-fix

(defthm consp-of-funenv-fix
  (equal (consp (funenv-fix acl2::x))
         (consp acl2::x)))

Theorem: funenv-fix-under-iff

(defthm funenv-fix-under-iff
  (iff (funenv-fix acl2::x)
       (consp acl2::x)))

Theorem: funenv-fix-of-cons

(defthm funenv-fix-of-cons
  (equal (funenv-fix (cons a x))
         (cons (funscope-fix a) (funenv-fix x))))

Theorem: len-of-funenv-fix

(defthm len-of-funenv-fix
  (equal (len (funenv-fix acl2::x))
         (len acl2::x)))

Theorem: funenv-fix-of-append

(defthm funenv-fix-of-append
  (equal (funenv-fix (append std::a std::b))
         (append (funenv-fix std::a)
                 (funenv-fix std::b))))

Theorem: funenv-fix-of-repeat

(defthm funenv-fix-of-repeat
  (equal (funenv-fix (repeat acl2::n acl2::x))
         (repeat acl2::n (funscope-fix acl2::x))))

Theorem: list-equiv-refines-funenv-equiv

(defthm list-equiv-refines-funenv-equiv
  (implies (list-equiv acl2::x acl2::y)
           (funenv-equiv acl2::x acl2::y))
  :rule-classes :refinement)

Theorem: nth-of-funenv-fix

(defthm nth-of-funenv-fix
  (equal (nth acl2::n (funenv-fix acl2::x))
         (if (< (nfix acl2::n) (len acl2::x))
             (funscope-fix (nth acl2::n acl2::x))
           nil)))

Theorem: funenv-equiv-implies-funenv-equiv-append-1

(defthm funenv-equiv-implies-funenv-equiv-append-1
  (implies (funenv-equiv acl2::x fty::x-equiv)
           (funenv-equiv (append acl2::x acl2::y)
                         (append fty::x-equiv acl2::y)))
  :rule-classes (:congruence))

Theorem: funenv-equiv-implies-funenv-equiv-append-2

(defthm funenv-equiv-implies-funenv-equiv-append-2
  (implies (funenv-equiv acl2::y fty::y-equiv)
           (funenv-equiv (append acl2::x acl2::y)
                         (append acl2::x fty::y-equiv)))
  :rule-classes (:congruence))

Theorem: funenv-equiv-implies-funenv-equiv-nthcdr-2

(defthm funenv-equiv-implies-funenv-equiv-nthcdr-2
  (implies (funenv-equiv acl2::l l-equiv)
           (funenv-equiv (nthcdr acl2::n acl2::l)
                         (nthcdr acl2::n l-equiv)))
  :rule-classes (:congruence))

Theorem: funenv-equiv-implies-funenv-equiv-take-2

(defthm funenv-equiv-implies-funenv-equiv-take-2
  (implies (funenv-equiv acl2::l l-equiv)
           (funenv-equiv (take acl2::n acl2::l)
                         (take acl2::n l-equiv)))
  :rule-classes (:congruence))