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Envmap

Envmap-fix

(envmap-fix x) is an fty alist fixing function that follows the fix-keys strategy.

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

Note that in the execution this is just an inline identity function.

Definitions and Theorems

Function: envmap-fix$inline

(defun envmap-fix$inline (x)
  (declare (xargs :guard (envmap-p x)))
  (let ((__function__ 'envmap-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             x
           (if (consp (car x))
               (cons (cons (pseudo-term-fix (caar x))
                           (svex-alist-fix (cdar x)))
                     (envmap-fix (cdr x)))
             (envmap-fix (cdr x))))
         :exec x)))

Theorem: envmap-p-of-envmap-fix

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

Theorem: envmap-fix-when-envmap-p

(defthm envmap-fix-when-envmap-p
  (implies (envmap-p x)
           (equal (envmap-fix x) x)))

Function: envmap-equiv$inline

(defun envmap-equiv$inline (x y)
  (declare (xargs :guard (and (envmap-p x) (envmap-p y))))
  (equal (envmap-fix x) (envmap-fix y)))

Theorem: envmap-equiv-is-an-equivalence

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

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

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

Theorem: envmap-fix-under-envmap-equiv

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

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

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

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

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

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

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

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

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

Theorem: cons-of-pseudo-term-fix-k-under-envmap-equiv

(defthm cons-of-pseudo-term-fix-k-under-envmap-equiv
  (envmap-equiv (cons (cons (pseudo-term-fix acl2::k) acl2::v)
                      x)
                (cons (cons acl2::k acl2::v) x)))

Theorem: cons-pseudo-term-equiv-congruence-on-k-under-envmap-equiv

(defthm cons-pseudo-term-equiv-congruence-on-k-under-envmap-equiv
  (implies (pseudo-term-equiv acl2::k k-equiv)
           (envmap-equiv (cons (cons acl2::k acl2::v) x)
                         (cons (cons k-equiv acl2::v) x)))
  :rule-classes :congruence)

Theorem: cons-of-svex-alist-fix-v-under-envmap-equiv

(defthm cons-of-svex-alist-fix-v-under-envmap-equiv
  (envmap-equiv (cons (cons acl2::k (svex-alist-fix acl2::v))
                      x)
                (cons (cons acl2::k acl2::v) x)))

Theorem: cons-svex-alist-equiv-congruence-on-v-under-envmap-equiv

(defthm cons-svex-alist-equiv-congruence-on-v-under-envmap-equiv
  (implies (svex-alist-equiv acl2::v v-equiv)
           (envmap-equiv (cons (cons acl2::k acl2::v) x)
                         (cons (cons acl2::k v-equiv) x)))
  :rule-classes :congruence)

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

(defthm cons-of-envmap-fix-y-under-envmap-equiv
  (envmap-equiv (cons x (envmap-fix y))
                (cons x y)))

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

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

Theorem: envmap-fix-of-acons

(defthm envmap-fix-of-acons
  (equal (envmap-fix (cons (cons acl2::a acl2::b) x))
         (cons (cons (pseudo-term-fix acl2::a)
                     (svex-alist-fix acl2::b))
               (envmap-fix x))))

Theorem: envmap-fix-of-append

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

Theorem: consp-car-of-envmap-fix

(defthm consp-car-of-envmap-fix
  (equal (consp (car (envmap-fix x)))
         (consp (envmap-fix x))))