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Lhprobe-constraintlist

Lhprobe-constraintlist-fix

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

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

In the logic, we apply lhprobe-constraint-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: lhprobe-constraintlist-fix$inline

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

Theorem: lhprobe-constraintlist-p-of-lhprobe-constraintlist-fix

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

Theorem: lhprobe-constraintlist-fix-when-lhprobe-constraintlist-p

(defthm lhprobe-constraintlist-fix-when-lhprobe-constraintlist-p
  (implies (lhprobe-constraintlist-p x)
           (equal (lhprobe-constraintlist-fix x)
                  x)))

Function: lhprobe-constraintlist-equiv$inline

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

Theorem: lhprobe-constraintlist-equiv-is-an-equivalence

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

Theorem: lhprobe-constraintlist-equiv-implies-equal-lhprobe-constraintlist-fix-1

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

Theorem: lhprobe-constraintlist-fix-under-lhprobe-constraintlist-equiv

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

Theorem: equal-of-lhprobe-constraintlist-fix-1-forward-to-lhprobe-constraintlist-equiv

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

Theorem: equal-of-lhprobe-constraintlist-fix-2-forward-to-lhprobe-constraintlist-equiv

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

Theorem: lhprobe-constraintlist-equiv-of-lhprobe-constraintlist-fix-1-forward

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

Theorem: lhprobe-constraintlist-equiv-of-lhprobe-constraintlist-fix-2-forward

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

Theorem: car-of-lhprobe-constraintlist-fix-x-under-lhprobe-constraint-equiv

(defthm
 car-of-lhprobe-constraintlist-fix-x-under-lhprobe-constraint-equiv
 (lhprobe-constraint-equiv (car (lhprobe-constraintlist-fix x))
                           (car x)))

Theorem: car-lhprobe-constraintlist-equiv-congruence-on-x-under-lhprobe-constraint-equiv

(defthm
 car-lhprobe-constraintlist-equiv-congruence-on-x-under-lhprobe-constraint-equiv
 (implies (lhprobe-constraintlist-equiv x x-equiv)
          (lhprobe-constraint-equiv (car x)
                                    (car x-equiv)))
 :rule-classes :congruence)

Theorem: cdr-of-lhprobe-constraintlist-fix-x-under-lhprobe-constraintlist-equiv

(defthm
 cdr-of-lhprobe-constraintlist-fix-x-under-lhprobe-constraintlist-equiv
 (lhprobe-constraintlist-equiv (cdr (lhprobe-constraintlist-fix x))
                               (cdr x)))

Theorem: cdr-lhprobe-constraintlist-equiv-congruence-on-x-under-lhprobe-constraintlist-equiv

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

Theorem: cons-of-lhprobe-constraint-fix-x-under-lhprobe-constraintlist-equiv

(defthm
 cons-of-lhprobe-constraint-fix-x-under-lhprobe-constraintlist-equiv
 (lhprobe-constraintlist-equiv (cons (lhprobe-constraint-fix x) y)
                               (cons x y)))

Theorem: cons-lhprobe-constraint-equiv-congruence-on-x-under-lhprobe-constraintlist-equiv

(defthm
 cons-lhprobe-constraint-equiv-congruence-on-x-under-lhprobe-constraintlist-equiv
 (implies (lhprobe-constraint-equiv x x-equiv)
          (lhprobe-constraintlist-equiv (cons x y)
                                        (cons x-equiv y)))
 :rule-classes :congruence)

Theorem: cons-of-lhprobe-constraintlist-fix-y-under-lhprobe-constraintlist-equiv

(defthm
 cons-of-lhprobe-constraintlist-fix-y-under-lhprobe-constraintlist-equiv
 (lhprobe-constraintlist-equiv
      (cons x (lhprobe-constraintlist-fix y))
      (cons x y)))

Theorem: cons-lhprobe-constraintlist-equiv-congruence-on-y-under-lhprobe-constraintlist-equiv

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

Theorem: consp-of-lhprobe-constraintlist-fix

(defthm consp-of-lhprobe-constraintlist-fix
  (equal (consp (lhprobe-constraintlist-fix x))
         (consp x)))

Theorem: lhprobe-constraintlist-fix-under-iff

(defthm lhprobe-constraintlist-fix-under-iff
  (iff (lhprobe-constraintlist-fix x)
       (consp x)))

Theorem: lhprobe-constraintlist-fix-of-cons

(defthm lhprobe-constraintlist-fix-of-cons
  (equal (lhprobe-constraintlist-fix (cons a x))
         (cons (lhprobe-constraint-fix a)
               (lhprobe-constraintlist-fix x))))

Theorem: len-of-lhprobe-constraintlist-fix

(defthm len-of-lhprobe-constraintlist-fix
  (equal (len (lhprobe-constraintlist-fix x))
         (len x)))

Theorem: lhprobe-constraintlist-fix-of-append

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

Theorem: lhprobe-constraintlist-fix-of-repeat

(defthm lhprobe-constraintlist-fix-of-repeat
  (equal (lhprobe-constraintlist-fix (repeat acl2::n x))
         (repeat acl2::n (lhprobe-constraint-fix x))))

Theorem: list-equiv-refines-lhprobe-constraintlist-equiv

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

Theorem: nth-of-lhprobe-constraintlist-fix

(defthm nth-of-lhprobe-constraintlist-fix
  (equal (nth acl2::n (lhprobe-constraintlist-fix x))
         (if (< (nfix acl2::n) (len x))
             (lhprobe-constraint-fix (nth acl2::n x))
           nil)))

Theorem: lhprobe-constraintlist-equiv-implies-lhprobe-constraintlist-equiv-append-1

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

Theorem: lhprobe-constraintlist-equiv-implies-lhprobe-constraintlist-equiv-append-2

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

Theorem: lhprobe-constraintlist-equiv-implies-lhprobe-constraintlist-equiv-nthcdr-2

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

Theorem: lhprobe-constraintlist-equiv-implies-lhprobe-constraintlist-equiv-take-2

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