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Lhslist

Lhslist-fix

(lhslist-fix x) is a usual fty list fixing function.

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

In the logic, we apply lhs-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: lhslist-fix$inline

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

Theorem: lhslist-p-of-lhslist-fix

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

Theorem: lhslist-fix-when-lhslist-p

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

Function: lhslist-equiv$inline

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

Theorem: lhslist-equiv-is-an-equivalence

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

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

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

Theorem: lhslist-fix-under-lhslist-equiv

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

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

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

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

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

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

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

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

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

Theorem: car-of-lhslist-fix-x-under-lhs-equiv

(defthm car-of-lhslist-fix-x-under-lhs-equiv
  (lhs-equiv (car (lhslist-fix x))
             (car x)))

Theorem: car-lhslist-equiv-congruence-on-x-under-lhs-equiv

(defthm car-lhslist-equiv-congruence-on-x-under-lhs-equiv
  (implies (lhslist-equiv x x-equiv)
           (lhs-equiv (car x) (car x-equiv)))
  :rule-classes :congruence)

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

(defthm cdr-of-lhslist-fix-x-under-lhslist-equiv
  (lhslist-equiv (cdr (lhslist-fix x))
                 (cdr x)))

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

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

Theorem: cons-of-lhs-fix-x-under-lhslist-equiv

(defthm cons-of-lhs-fix-x-under-lhslist-equiv
  (lhslist-equiv (cons (lhs-fix x) y)
                 (cons x y)))

Theorem: cons-lhs-equiv-congruence-on-x-under-lhslist-equiv

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

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

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

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

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

Theorem: consp-of-lhslist-fix

(defthm consp-of-lhslist-fix
  (equal (consp (lhslist-fix x))
         (consp x)))

Theorem: lhslist-fix-under-iff

(defthm lhslist-fix-under-iff
  (iff (lhslist-fix x) (consp x)))

Theorem: lhslist-fix-of-cons

(defthm lhslist-fix-of-cons
  (equal (lhslist-fix (cons a x))
         (cons (lhs-fix a) (lhslist-fix x))))

Theorem: len-of-lhslist-fix

(defthm len-of-lhslist-fix
  (equal (len (lhslist-fix x)) (len x)))

Theorem: lhslist-fix-of-append

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

Theorem: lhslist-fix-of-repeat

(defthm lhslist-fix-of-repeat
  (equal (lhslist-fix (repeat acl2::n x))
         (repeat acl2::n (lhs-fix x))))

Theorem: list-equiv-refines-lhslist-equiv

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

Theorem: nth-of-lhslist-fix

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

Theorem: lhslist-equiv-implies-lhslist-equiv-append-1

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

Theorem: lhslist-equiv-implies-lhslist-equiv-append-2

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

Theorem: lhslist-equiv-implies-lhslist-equiv-nthcdr-2

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

Theorem: lhslist-equiv-implies-lhslist-equiv-take-2

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