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• Modalist

Modalist-fix

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

Signature

(modalist-fix x) → fty::newx

Arguments

x — Guard (modalist-p x).

Returns

fty::newx — Type (modalist-p fty::newx).

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

Definitions and Theorems

Function: modalist-fix$inline

(defun modalist-fix$inline (x)
  (declare (xargs :guard (modalist-p x)))
  (let ((__function__ 'modalist-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             x
           (let ((rest (modalist-fix (cdr x))))
             (if (and (consp (car x))
                      (modname-p (caar x)))
                 (let ((fty::first-key (caar x))
                       (fty::first-val (module-fix (cdar x))))
                   (cons (cons fty::first-key fty::first-val)
                         rest))
               rest)))
         :exec x)))

Theorem: modalist-p-of-modalist-fix

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

Theorem: modalist-fix-when-modalist-p

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

Function: modalist-equiv$inline

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

Theorem: modalist-equiv-is-an-equivalence

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

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

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

Theorem: modalist-fix-under-modalist-equiv

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

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

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

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

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

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

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

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

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

Theorem: cons-of-module-fix-v-under-modalist-equiv

(defthm cons-of-module-fix-v-under-modalist-equiv
  (modalist-equiv (cons (cons acl2::k (module-fix acl2::v))
                        x)
                  (cons (cons acl2::k acl2::v) x)))

Theorem: cons-module-equiv-congruence-on-v-under-modalist-equiv

(defthm cons-module-equiv-congruence-on-v-under-modalist-equiv
  (implies (module-equiv acl2::v v-equiv)
           (modalist-equiv (cons (cons acl2::k acl2::v) x)
                           (cons (cons acl2::k v-equiv) x)))
  :rule-classes :congruence)

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

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

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

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

Theorem: modalist-fix-of-acons

(defthm modalist-fix-of-acons
  (equal (modalist-fix (cons (cons acl2::a acl2::b) x))
         (let ((rest (modalist-fix x)))
           (if (and (modname-p acl2::a))
               (let ((fty::first-key acl2::a)
                     (fty::first-val (module-fix acl2::b)))
                 (cons (cons fty::first-key fty::first-val)
                       rest))
             rest))))

Theorem: hons-assoc-equal-of-modalist-fix

(defthm hons-assoc-equal-of-modalist-fix
  (equal (hons-assoc-equal acl2::k (modalist-fix x))
         (let ((fty::pair (hons-assoc-equal acl2::k x)))
           (and (modname-p acl2::k)
                fty::pair
                (cons acl2::k
                      (module-fix (cdr fty::pair)))))))

Theorem: modalist-fix-of-append

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

Theorem: consp-car-of-modalist-fix

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