		Search-engine friendly clone of the ACL2 documentation.

Congruence-rulelist

Congruence-rulelist-fix

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

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

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

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

Theorem: congruence-rulelist-p-of-congruence-rulelist-fix

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

Theorem: congruence-rulelist-fix-when-congruence-rulelist-p

(defthm congruence-rulelist-fix-when-congruence-rulelist-p
  (implies (congruence-rulelist-p x)
           (equal (congruence-rulelist-fix x) x)))

Function: congruence-rulelist-equiv$inline

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

Theorem: congruence-rulelist-equiv-is-an-equivalence

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

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

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

Theorem: congruence-rulelist-fix-under-congruence-rulelist-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(defthm
   cdr-of-congruence-rulelist-fix-x-under-congruence-rulelist-equiv
  (congruence-rulelist-equiv (cdr (congruence-rulelist-fix x))
                             (cdr x)))

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

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

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

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

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

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

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

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

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

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

Theorem: consp-of-congruence-rulelist-fix

(defthm consp-of-congruence-rulelist-fix
  (equal (consp (congruence-rulelist-fix x))
         (consp x)))

Theorem: congruence-rulelist-fix-under-iff

(defthm congruence-rulelist-fix-under-iff
  (iff (congruence-rulelist-fix x)
       (consp x)))

Theorem: congruence-rulelist-fix-of-cons

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

Theorem: len-of-congruence-rulelist-fix

(defthm len-of-congruence-rulelist-fix
  (equal (len (congruence-rulelist-fix x))
         (len x)))

Theorem: congruence-rulelist-fix-of-append

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

Theorem: congruence-rulelist-fix-of-repeat

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

Theorem: list-equiv-refines-congruence-rulelist-equiv

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

Theorem: nth-of-congruence-rulelist-fix

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

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

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

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

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

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

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

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

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