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Env-fix

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

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

In the logic, we apply env-block-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: env-fix$inline

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

Theorem: envp-of-env-fix

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

Theorem: env-fix-when-envp

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

Theorem: car-of-env-fix-x-under-env-block-equiv

(defthm car-of-env-fix-x-under-env-block-equiv
  (env-block-equiv (car (env-fix acl2::x))
                   (car acl2::x)))

Theorem: car-env-equiv-congruence-on-x-under-env-block-equiv

(defthm car-env-equiv-congruence-on-x-under-env-block-equiv
  (implies (env-equiv acl2::x x-equiv)
           (env-block-equiv (car acl2::x)
                            (car x-equiv)))
  :rule-classes :congruence)

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

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

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

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

Theorem: cons-of-env-block-fix-x-under-env-equiv

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

Theorem: cons-env-block-equiv-congruence-on-x-under-env-equiv

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

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

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

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

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

Theorem: consp-of-env-fix

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

Theorem: env-fix-under-iff

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

Theorem: env-fix-of-cons

(defthm env-fix-of-cons
  (equal (env-fix (cons a x))
         (cons (env-block-fix a) (env-fix x))))

Theorem: len-of-env-fix

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

Theorem: env-fix-of-append

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

Theorem: env-fix-of-repeat

(defthm env-fix-of-repeat
  (equal (env-fix (repeat acl2::n acl2::x))
         (repeat acl2::n (env-block-fix acl2::x))))

Theorem: list-equiv-refines-env-equiv

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

Theorem: nth-of-env-fix

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

Theorem: env-equiv-implies-env-equiv-append-1

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

Theorem: env-equiv-implies-env-equiv-append-2

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

Theorem: env-equiv-implies-env-equiv-nthcdr-2

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

Theorem: env-equiv-implies-env-equiv-take-2

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