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• Rulelist

Rulelist-fix

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

Signature

(rulelist-fix x) → fty::newx

Arguments

x — Guard (rulelistp x).

Returns

fty::newx — Type (rulelistp fty::newx).

In the logic, we apply rule-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: rulelist-fix$inline

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

Theorem: rulelistp-of-rulelist-fix

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

Theorem: rulelist-fix-when-rulelistp

(defthm rulelist-fix-when-rulelistp
  (implies (rulelistp x)
           (equal (rulelist-fix x) x)))

Function: rulelist-equiv$inline

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

Theorem: rulelist-equiv-is-an-equivalence

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

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

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

Theorem: rulelist-fix-under-rulelist-equiv

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

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

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

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

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

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

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

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

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

Theorem: car-of-rulelist-fix-x-under-rule-equiv

(defthm car-of-rulelist-fix-x-under-rule-equiv
  (rule-equiv (car (rulelist-fix acl2::x))
              (car acl2::x)))

Theorem: car-rulelist-equiv-congruence-on-x-under-rule-equiv

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

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

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

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

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

Theorem: cons-of-rule-fix-x-under-rulelist-equiv

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

Theorem: cons-rule-equiv-congruence-on-x-under-rulelist-equiv

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

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

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

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

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

Theorem: consp-of-rulelist-fix

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

Theorem: rulelist-fix-under-iff

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

Theorem: rulelist-fix-of-cons

(defthm rulelist-fix-of-cons
  (equal (rulelist-fix (cons a x))
         (cons (rule-fix a) (rulelist-fix x))))

Theorem: len-of-rulelist-fix

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

Theorem: rulelist-fix-of-append

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

Theorem: rulelist-fix-of-repeat

(defthm rulelist-fix-of-repeat
  (equal (rulelist-fix (repeat acl2::n acl2::x))
         (repeat acl2::n (rule-fix acl2::x))))

Theorem: list-equiv-refines-rulelist-equiv

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

Theorem: nth-of-rulelist-fix

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

Theorem: rulelist-equiv-implies-rulelist-equiv-append-1

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

Theorem: rulelist-equiv-implies-rulelist-equiv-append-2

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

Theorem: rulelist-equiv-implies-rulelist-equiv-nthcdr-2

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

Theorem: rulelist-equiv-implies-rulelist-equiv-take-2

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