(BOOT-STRAP NQTHM)

[ 0.1 0.1 0.0 ]

GROUND-ZERO


(PROVE-LEMMA PLUS-RIGHT-ID2
             (REWRITE)
             (IMPLIES (NOT (NUMBERP Y))
                      (EQUAL (PLUS X Y) (FIX X)))
             ((ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This simplifies, unfolding the definition of FIX, to the following two new
goals:

Case 2. (IMPLIES (AND (NOT (NUMBERP Y))
                      (NOT (NUMBERP X)))
                 (EQUAL (PLUS X Y) 0)).

  This again simplifies, opening up the definitions of PLUS and EQUAL, to:

        T.

Case 1. (IMPLIES (AND (NOT (NUMBERP Y)) (NUMBERP X))
                 (EQUAL (PLUS X Y) X)),

  which we will name *1.


     We will appeal to induction.  There is only one plausible induction.  We
will induct according to the following scheme:
      (AND (IMPLIES (ZEROP X) (p X Y))
           (IMPLIES (AND (NOT (ZEROP X)) (p (SUB1 X) Y))
                    (p X Y))).
Linear arithmetic, the lemmas SUB1-LESSEQP and SUB1-LESSP, and the definition
of ZEROP establish that the measure (COUNT X) decreases according to the
well-founded relation LESSP in each induction step of the scheme.  The above
induction scheme generates the following two new formulas:

Case 2. (IMPLIES (AND (ZEROP X)
                      (NOT (NUMBERP Y))
                      (NUMBERP X))
                 (EQUAL (PLUS X Y) X)).

  This simplifies, expanding the functions ZEROP, NUMBERP, EQUAL, and PLUS, to:

        T.

Case 1. (IMPLIES (AND (NOT (ZEROP X))
                      (EQUAL (PLUS (SUB1 X) Y) (SUB1 X))
                      (NOT (NUMBERP Y))
                      (NUMBERP X))
                 (EQUAL (PLUS X Y) X)).

  This simplifies, using linear arithmetic, to the new conjecture:

        (IMPLIES (AND (EQUAL X 0)
                      (NOT (ZEROP X))
                      (EQUAL (PLUS (SUB1 X) Y) (SUB1 X))
                      (NOT (NUMBERP Y))
                      (NUMBERP X))
                 (EQUAL (PLUS X Y) X)),

  which again simplifies, expanding the function ZEROP, to:

        T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.0 0.0 ]

PLUS-RIGHT-ID2


(DISABLE PLUS-RIGHT-ID2)

[ 0.0 0.0 0.0 ]

PLUS-RIGHT-ID2-OFF


(PROVE-LEMMA PLUS-ADD1
             (REWRITE)
             (EQUAL (PLUS X (ADD1 Y))
                    (IF (NUMBERP Y)
                        (ADD1 (PLUS X Y))
                        (ADD1 X)))
             ((ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This simplifies, obviously, to two new formulas:

Case 2. (IMPLIES (NOT (NUMBERP Y))
                 (EQUAL (PLUS X (ADD1 Y)) (ADD1 X))),

  which again simplifies, rewriting with the lemma SUB1-TYPE-RESTRICTION, to
  the conjecture:

        (IMPLIES (NOT (NUMBERP Y))
                 (EQUAL (PLUS X 1) (ADD1 X))).

  But this again simplifies, using linear arithmetic, to:

        T.

Case 1. (IMPLIES (NUMBERP Y)
                 (EQUAL (PLUS X (ADD1 Y))
                        (ADD1 (PLUS X Y)))),

  which again simplifies, using linear arithmetic, to:

        T.

Q.E.D.


[ 0.0 0.0 0.0 ]

PLUS-ADD1


(DISABLE PLUS-ADD1)

[ 0.0 0.0 0.0 ]

PLUS-ADD1-OFF


(PROVE-LEMMA COMMUTATIVITY2-OF-PLUS
             (REWRITE)
             (EQUAL (PLUS X (PLUS Y Z))
                    (PLUS Y (PLUS X Z)))
             ((ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This simplifies, using linear arithmetic, to:

      T.

Q.E.D.


[ 0.0 0.0 0.0 ]

COMMUTATIVITY2-OF-PLUS


(DISABLE COMMUTATIVITY2-OF-PLUS)

[ 0.0 0.0 0.0 ]

COMMUTATIVITY2-OF-PLUS-OFF


(PROVE-LEMMA COMMUTATIVITY-OF-PLUS
             (REWRITE)
             (EQUAL (PLUS X Y) (PLUS Y X))
             ((ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This formula simplifies, using linear arithmetic, to:

      T.

Q.E.D.


[ 0.0 0.0 0.0 ]

COMMUTATIVITY-OF-PLUS


(DISABLE COMMUTATIVITY-OF-PLUS)

[ 0.0 0.0 0.0 ]

COMMUTATIVITY-OF-PLUS-OFF


(PROVE-LEMMA ASSOCIATIVITY-OF-PLUS
             (REWRITE)
             (EQUAL (PLUS (PLUS X Y) Z)
                    (PLUS X (PLUS Y Z)))
             ((ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This simplifies, using linear arithmetic, to:

      T.

Q.E.D.


[ 0.0 0.0 0.0 ]

ASSOCIATIVITY-OF-PLUS


(DISABLE ASSOCIATIVITY-OF-PLUS)

[ 0.0 0.0 0.0 ]

ASSOCIATIVITY-OF-PLUS-OFF


(PROVE-LEMMA PLUS-EQUAL-0
             (REWRITE)
             (EQUAL (EQUAL (PLUS A B) '0)
                    (AND (ZEROP A) (ZEROP B)))
             ((ENABLE PLUS-RIGHT-ID2 COMMUTATIVITY-OF-PLUS)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This simplifies, opening up the functions ZEROP and AND, to six new
conjectures:

Case 6. (IMPLIES (AND (NOT (EQUAL (PLUS A B) 0))
                      (NOT (NUMBERP A)))
                 (NOT (EQUAL B 0))),

  which again simplifies, applying PLUS-RIGHT-ID2 and COMMUTATIVITY-OF-PLUS,
  and unfolding the functions NUMBERP and EQUAL, to:

        T.

Case 5. (IMPLIES (AND (NOT (EQUAL (PLUS A B) 0))
                      (NOT (NUMBERP A)))
                 (NUMBERP B)).

  However this again simplifies, applying PLUS-RIGHT-ID2, and unfolding the
  function EQUAL, to:

        T.

Case 4. (IMPLIES (AND (NOT (EQUAL (PLUS A B) 0))
                      (EQUAL A 0))
                 (NOT (EQUAL B 0))).

  This again simplifies, using linear arithmetic, to:

        T.

Case 3. (IMPLIES (AND (NOT (EQUAL (PLUS A B) 0))
                      (EQUAL A 0))
                 (NUMBERP B)),

  which again simplifies, applying the lemma PLUS-RIGHT-ID2, and opening up
  the definitions of NUMBERP and EQUAL, to:

        T.

Case 2. (IMPLIES (AND (EQUAL (PLUS A B) 0)
                      (NOT (EQUAL A 0)))
                 (NOT (NUMBERP A))),

  which again simplifies, using linear arithmetic, to:

        T.

Case 1. (IMPLIES (AND (EQUAL (PLUS A B) 0)
                      (NOT (EQUAL B 0)))
                 (NOT (NUMBERP B))),

  which again simplifies, using linear arithmetic, to:

        T.

Q.E.D.


[ 0.0 0.0 0.0 ]

PLUS-EQUAL-0


(DISABLE PLUS-EQUAL-0)

[ 0.0 0.0 0.0 ]

PLUS-EQUAL-0-OFF


(PROVE-LEMMA DIFFERENCE-X-X
             (REWRITE)
             (EQUAL (DIFFERENCE X X) '0)
             ((ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This conjecture simplifies, using linear arithmetic, to:

      (IMPLIES (LESSP X X)
               (EQUAL (DIFFERENCE X X) 0)).

But this again simplifies, using linear arithmetic, to:

      T.

Q.E.D.


[ 0.0 0.0 0.0 ]

DIFFERENCE-X-X


(DISABLE DIFFERENCE-X-X)

[ 0.0 0.0 0.0 ]

DIFFERENCE-X-X-OFF


(PROVE-LEMMA DIFFERENCE-PLUS
             (REWRITE)
             (AND (EQUAL (DIFFERENCE (PLUS X Y) X)
                         (FIX Y))
                  (EQUAL (DIFFERENCE (PLUS Y X) X)
                         (FIX Y)))
             ((ENABLE PLUS-RIGHT-ID2 COMMUTATIVITY-OF-PLUS)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

WARNING:  Note that the proposed lemma DIFFERENCE-PLUS is to be stored as zero
type prescription rules, zero compound recognizer rules, zero linear rules,
and two replacement rules.


This formula can be simplified, using the abbreviation AND, to the following
two new conjectures:

Case 2. (EQUAL (DIFFERENCE (PLUS X Y) X)
               (FIX Y)).

  This simplifies, unfolding the function FIX, to the following two new
  formulas:

  Case 2.2.
          (IMPLIES (NOT (NUMBERP Y))
                   (EQUAL (DIFFERENCE (PLUS X Y) X) 0)).

    However this again simplifies, applying PLUS-RIGHT-ID2, to the following
    two new goals:

    Case 2.2.2.
            (IMPLIES (AND (NOT (NUMBERP Y))
                          (NOT (NUMBERP X)))
                     (EQUAL (DIFFERENCE 0 X) 0)).

      But this again simplifies, expanding EQUAL and DIFFERENCE, to:

            T.

    Case 2.2.1.
            (IMPLIES (AND (NOT (NUMBERP Y)) (NUMBERP X))
                     (EQUAL (DIFFERENCE X X) 0)),

      which again simplifies, using linear arithmetic, to:

            (IMPLIES (AND (LESSP X X)
                          (NOT (NUMBERP Y))
                          (NUMBERP X))
                     (EQUAL (DIFFERENCE X X) 0)).

      However this again simplifies, using linear arithmetic, to:

            T.

  Case 2.1.
          (IMPLIES (NUMBERP Y)
                   (EQUAL (DIFFERENCE (PLUS X Y) X) Y)),

    which again simplifies, using linear arithmetic, to the conjecture:

          (IMPLIES (AND (LESSP (PLUS X Y) X) (NUMBERP Y))
                   (EQUAL (DIFFERENCE (PLUS X Y) X) Y)).

    This again simplifies, using linear arithmetic, to:

          T.

Case 1. (EQUAL (DIFFERENCE (PLUS Y X) X)
               (FIX Y)),

  which simplifies, applying the lemma COMMUTATIVITY-OF-PLUS, and unfolding
  FIX, to two new conjectures:

  Case 1.2.
          (IMPLIES (NOT (NUMBERP Y))
                   (EQUAL (DIFFERENCE (PLUS X Y) X) 0)),

    which again simplifies, rewriting with the lemma PLUS-RIGHT-ID2, to two
    new formulas:

    Case 1.2.2.
            (IMPLIES (AND (NOT (NUMBERP Y))
                          (NOT (NUMBERP X)))
                     (EQUAL (DIFFERENCE 0 X) 0)),

      which again simplifies, expanding the definitions of EQUAL and
      DIFFERENCE, to:

            T.

    Case 1.2.1.
            (IMPLIES (AND (NOT (NUMBERP Y)) (NUMBERP X))
                     (EQUAL (DIFFERENCE X X) 0)),

      which again simplifies, using linear arithmetic, to:

            (IMPLIES (AND (LESSP X X)
                          (NOT (NUMBERP Y))
                          (NUMBERP X))
                     (EQUAL (DIFFERENCE X X) 0)).

      However this again simplifies, using linear arithmetic, to:

            T.

  Case 1.1.
          (IMPLIES (NUMBERP Y)
                   (EQUAL (DIFFERENCE (PLUS X Y) X) Y)),

    which again simplifies, using linear arithmetic, to the formula:

          (IMPLIES (AND (LESSP (PLUS X Y) X) (NUMBERP Y))
                   (EQUAL (DIFFERENCE (PLUS X Y) X) Y)).

    This again simplifies, using linear arithmetic, to:

          T.

Q.E.D.


[ 0.0 0.0 0.0 ]

DIFFERENCE-PLUS


(DISABLE DIFFERENCE-PLUS)

[ 0.0 0.0 0.0 ]

DIFFERENCE-PLUS-OFF


(PROVE-LEMMA PLUS-CANCELLATION
             (REWRITE)
             (EQUAL (EQUAL (PLUS A B) (PLUS A C))
                    (EQUAL (FIX B) (FIX C)))
             ((ENABLE PLUS-RIGHT-ID2 COMMUTATIVITY-OF-PLUS)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This conjecture simplifies, expanding the definition of FIX, to the following
seven new formulas:

Case 7. (IMPLIES (AND (NUMBERP C)
                      (NUMBERP B)
                      (NOT (EQUAL B C)))
                 (NOT (EQUAL (PLUS A B) (PLUS A C)))).

  This again simplifies, using linear arithmetic, to:

        T.

Case 6. (IMPLIES (AND (NUMBERP C)
                      (NOT (NUMBERP B))
                      (NOT (EQUAL 0 C)))
                 (NOT (EQUAL (PLUS A B) (PLUS A C)))),

  which again simplifies, rewriting with PLUS-RIGHT-ID2, to the following two
  new conjectures:

  Case 6.2.
          (IMPLIES (AND (NUMBERP C)
                        (NOT (NUMBERP B))
                        (NOT (EQUAL 0 C))
                        (NOT (NUMBERP A)))
                   (NOT (EQUAL 0 (PLUS A C)))).

    This again simplifies, using linear arithmetic, to:

          T.

  Case 6.1.
          (IMPLIES (AND (NUMBERP C)
                        (NOT (NUMBERP B))
                        (NOT (EQUAL 0 C))
                        (NUMBERP A))
                   (NOT (EQUAL A (PLUS A C)))),

    which again simplifies, using linear arithmetic, to:

          T.

Case 5. (IMPLIES (AND (NOT (NUMBERP C))
                      (NUMBERP B)
                      (NOT (EQUAL B 0)))
                 (NOT (EQUAL (PLUS A B) (PLUS A C)))),

  which again simplifies, rewriting with PLUS-RIGHT-ID2, to the following two
  new goals:

  Case 5.2.
          (IMPLIES (AND (NOT (NUMBERP C))
                        (NUMBERP B)
                        (NOT (EQUAL B 0))
                        (NOT (NUMBERP A)))
                   (NOT (EQUAL (PLUS A B) 0))).

    This again simplifies, using linear arithmetic, to:

          T.

  Case 5.1.
          (IMPLIES (AND (NOT (NUMBERP C))
                        (NUMBERP B)
                        (NOT (EQUAL B 0))
                        (NUMBERP A))
                   (NOT (EQUAL (PLUS A B) A))),

    which again simplifies, using linear arithmetic, to:

          T.

Case 4. (IMPLIES (AND (NOT (NUMBERP C))
                      (NOT (NUMBERP B)))
                 (EQUAL (EQUAL (PLUS A B) (PLUS A C))
                        T)),

  which again simplifies, rewriting with the lemma PLUS-RIGHT-ID2, and opening
  up EQUAL, to:

        T.

Case 3. (IMPLIES (AND (NOT (NUMBERP C)) (EQUAL B 0))
                 (EQUAL (EQUAL (PLUS A B) (PLUS A C))
                        T)),

  which again simplifies, rewriting with COMMUTATIVITY-OF-PLUS and
  PLUS-RIGHT-ID2, and expanding the functions EQUAL and PLUS, to:

        T.

Case 2. (IMPLIES (AND (NOT (NUMBERP B)) (EQUAL 0 C))
                 (EQUAL (EQUAL (PLUS A B) (PLUS A C))
                        T)).

  This again simplifies, applying PLUS-RIGHT-ID2 and COMMUTATIVITY-OF-PLUS,
  and opening up EQUAL and PLUS, to:

        T.

Case 1. (IMPLIES (AND (NUMBERP C)
                      (NUMBERP B)
                      (EQUAL B C))
                 (EQUAL (EQUAL (PLUS A B) (PLUS A C))
                        T)).

  But this again simplifies, expanding the function EQUAL, to:

        T.

Q.E.D.


[ 0.0 0.0 0.0 ]

PLUS-CANCELLATION


(DISABLE PLUS-CANCELLATION)

[ 0.0 0.0 0.0 ]

PLUS-CANCELLATION-OFF


(PROVE-LEMMA DIFFERENCE-0
             (REWRITE)
             (IMPLIES (NOT (LESSP Y X))
                      (EQUAL (DIFFERENCE X Y) '0))
             ((ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))


     Name the conjecture *1.


     Let us appeal to the induction principle.  The recursive terms in the
conjecture suggest four inductions.  However, they merge into one likely
candidate induction.  We will induct according to the following scheme:
      (AND (IMPLIES (OR (EQUAL X 0) (NOT (NUMBERP X)))
                    (p X Y))
           (IMPLIES (AND (NOT (OR (EQUAL X 0) (NOT (NUMBERP X))))
                         (OR (EQUAL Y 0) (NOT (NUMBERP Y))))
                    (p X Y))
           (IMPLIES (AND (NOT (OR (EQUAL X 0) (NOT (NUMBERP X))))
                         (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y))))
                         (p (SUB1 X) (SUB1 Y)))
                    (p X Y))).
Linear arithmetic, the lemmas SUB1-LESSEQP and SUB1-LESSP, and the definitions
of OR and NOT can be used to show that the measure (COUNT Y) decreases
according to the well-founded relation LESSP in each induction step of the
scheme.  Note, however, the inductive instance chosen for X.  The above
induction scheme leads to the following four new goals:

Case 4. (IMPLIES (AND (OR (EQUAL X 0) (NOT (NUMBERP X)))
                      (NOT (LESSP Y X)))
                 (EQUAL (DIFFERENCE X Y) 0)).

  This simplifies, expanding the definitions of NOT, OR, EQUAL, LESSP, and
  DIFFERENCE, to:

        T.

Case 3. (IMPLIES (AND (NOT (OR (EQUAL X 0) (NOT (NUMBERP X))))
                      (OR (EQUAL Y 0) (NOT (NUMBERP Y)))
                      (NOT (LESSP Y X)))
                 (EQUAL (DIFFERENCE X Y) 0)).

  This simplifies, unfolding NOT, OR, EQUAL, and LESSP, to:

        T.

Case 2. (IMPLIES (AND (NOT (OR (EQUAL X 0) (NOT (NUMBERP X))))
                      (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y))))
                      (LESSP (SUB1 Y) (SUB1 X))
                      (NOT (LESSP Y X)))
                 (EQUAL (DIFFERENCE X Y) 0)).

  This simplifies, using linear arithmetic, to:

        (IMPLIES (AND (LESSP X 1)
                      (NOT (OR (EQUAL X 0) (NOT (NUMBERP X))))
                      (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y))))
                      (LESSP (SUB1 Y) (SUB1 X))
                      (NOT (LESSP Y X)))
                 (EQUAL (DIFFERENCE X Y) 0)),

  which again simplifies, opening up the definitions of SUB1, NUMBERP, EQUAL,
  LESSP, NOT, and OR, to:

        T.

Case 1. (IMPLIES (AND (NOT (OR (EQUAL X 0) (NOT (NUMBERP X))))
                      (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y))))
                      (EQUAL (DIFFERENCE (SUB1 X) (SUB1 Y))
                             0)
                      (NOT (LESSP Y X)))
                 (EQUAL (DIFFERENCE X Y) 0)),

  which simplifies, using linear arithmetic, to three new conjectures:

  Case 1.3.
          (IMPLIES (AND (LESSP X Y)
                        (NOT (OR (EQUAL X 0) (NOT (NUMBERP X))))
                        (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y))))
                        (EQUAL (DIFFERENCE (SUB1 X) (SUB1 Y))
                               0)
                        (NOT (LESSP Y X)))
                   (EQUAL (DIFFERENCE X Y) 0)),

    which again simplifies, using linear arithmetic, to two new conjectures:

    Case 1.3.2.
            (IMPLIES (AND (LESSP (SUB1 X) (SUB1 Y))
                          (LESSP X Y)
                          (NOT (OR (EQUAL X 0) (NOT (NUMBERP X))))
                          (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y))))
                          (EQUAL (DIFFERENCE (SUB1 X) (SUB1 Y))
                                 0)
                          (NOT (LESSP Y X)))
                     (EQUAL (DIFFERENCE X Y) 0)),

      which again simplifies, unfolding the functions LESSP, NOT, OR,
      DIFFERENCE, and EQUAL, to:

            T.

    Case 1.3.1.
            (IMPLIES (AND (LESSP X 1)
                          (LESSP X Y)
                          (NOT (OR (EQUAL X 0) (NOT (NUMBERP X))))
                          (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y))))
                          (EQUAL (DIFFERENCE (SUB1 X) (SUB1 Y))
                                 0)
                          (NOT (LESSP Y X)))
                     (EQUAL (DIFFERENCE X Y) 0)),

      which again simplifies, unfolding SUB1, NUMBERP, EQUAL, LESSP, NOT, and
      OR, to:

            T.

  Case 1.2.
          (IMPLIES (AND (LESSP (SUB1 X) (SUB1 Y))
                        (NOT (OR (EQUAL X 0) (NOT (NUMBERP X))))
                        (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y))))
                        (EQUAL (DIFFERENCE (SUB1 X) (SUB1 Y))
                               0)
                        (NOT (LESSP Y X)))
                   (EQUAL (DIFFERENCE X Y) 0)),

    which again simplifies, expanding the definitions of NOT, OR, LESSP,
    DIFFERENCE, and EQUAL, to:

          T.

  Case 1.1.
          (IMPLIES (AND (LESSP X 1)
                        (NOT (OR (EQUAL X 0) (NOT (NUMBERP X))))
                        (NOT (OR (EQUAL Y 0) (NOT (NUMBERP Y))))
                        (EQUAL (DIFFERENCE (SUB1 X) (SUB1 Y))
                               0)
                        (NOT (LESSP Y X)))
                   (EQUAL (DIFFERENCE X Y) 0)),

    which again simplifies, opening up the definitions of SUB1, NUMBERP, EQUAL,
    LESSP, NOT, and OR, to:

          T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.0 0.0 ]

DIFFERENCE-0


(DISABLE DIFFERENCE-0)

[ 0.0 0.0 0.0 ]

DIFFERENCE-0-OFF


(PROVE-LEMMA EQUAL-DIFFERENCE-0
             (REWRITE)
             (EQUAL (EQUAL '0 (DIFFERENCE X Y))
                    (NOT (LESSP Y X)))
             ((ENABLE DIFFERENCE-0)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This conjecture simplifies, unfolding NOT, to two new conjectures:

Case 2. (IMPLIES (NOT (EQUAL 0 (DIFFERENCE X Y)))
                 (LESSP Y X)),

  which again simplifies, rewriting with the lemma DIFFERENCE-0, and opening
  up the definition of EQUAL, to:

        T.

Case 1. (IMPLIES (EQUAL 0 (DIFFERENCE X Y))
                 (NOT (LESSP Y X))),

  which again simplifies, using linear arithmetic, to:

        T.

Q.E.D.


[ 0.0 0.0 0.0 ]

EQUAL-DIFFERENCE-0


(DISABLE EQUAL-DIFFERENCE-0)

[ 0.0 0.0 0.0 ]

EQUAL-DIFFERENCE-0-OFF


(PROVE-LEMMA DIFFERENCE-CANCELLATION-0
             (REWRITE)
             (EQUAL (EQUAL X (DIFFERENCE X Y))
                    (AND (NUMBERP X)
                         (OR (EQUAL X '0) (ZEROP Y))))
             ((ENABLE DIFFERENCE-0)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This simplifies, opening up the functions ZEROP, OR, and AND, to five new
goals:

Case 5. (IMPLIES (EQUAL X (DIFFERENCE X Y))
                 (NUMBERP X)),

  which again simplifies, obviously, to:

        T.

Case 4. (IMPLIES (AND (EQUAL X (DIFFERENCE X Y))
                      (NOT (EQUAL X 0))
                      (NOT (EQUAL Y 0)))
                 (NOT (NUMBERP Y))).

  But this again simplifies, using linear arithmetic, to the conjecture:

        (IMPLIES (AND (LESSP X Y)
                      (EQUAL X (DIFFERENCE X Y))
                      (NOT (EQUAL X 0))
                      (NOT (EQUAL Y 0)))
                 (NOT (NUMBERP Y))).

  This again simplifies, using linear arithmetic and applying DIFFERENCE-0, to:

        T.

Case 3. (IMPLIES (AND (NOT (EQUAL X (DIFFERENCE X Y)))
                      (NUMBERP X))
                 (NUMBERP Y)).

  But this again simplifies, expanding the definition of DIFFERENCE, to:

        T.

Case 2. (IMPLIES (AND (NOT (EQUAL X (DIFFERENCE X Y)))
                      (NUMBERP X))
                 (NOT (EQUAL Y 0))),

  which again simplifies, using linear arithmetic, to:

        (IMPLIES (AND (LESSP X 0)
                      (NOT (EQUAL X (DIFFERENCE X 0))))
                 (NOT (NUMBERP X))).

  But this again simplifies, using linear arithmetic, to:

        T.

Case 1. (IMPLIES (AND (NOT (EQUAL X (DIFFERENCE X Y)))
                      (NUMBERP X))
                 (NOT (EQUAL X 0))),

  which again simplifies, using linear arithmetic, applying the lemma
  DIFFERENCE-0, and expanding EQUAL, to:

        T.

Q.E.D.


[ 0.0 0.0 0.0 ]

DIFFERENCE-CANCELLATION-0


(DISABLE DIFFERENCE-CANCELLATION-0)

[ 0.0 0.0 0.0 ]

DIFFERENCE-CANCELLATION-0-OFF


(PROVE-LEMMA DIFFERENCE-CANCELLATION-1
             (REWRITE)
             (EQUAL (EQUAL (DIFFERENCE X Y)
                           (DIFFERENCE Z Y))
                    (IF (LESSP X Y)
                        (NOT (LESSP Y Z))
                        (IF (LESSP Z Y)
                            (NOT (LESSP Y X))
                            (EQUAL (FIX X) (FIX Z)))))
             ((ENABLE DIFFERENCE-0 EQUAL-DIFFERENCE-0)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This simplifies, opening up NOT and FIX, to the following 11 new goals:

Case 11.(IMPLIES (AND (EQUAL (DIFFERENCE X Y)
                             (DIFFERENCE Z Y))
                      (NOT (LESSP X Y))
                      (NOT (LESSP Z Y))
                      (NOT (NUMBERP Z))
                      (NUMBERP X))
                 (EQUAL (EQUAL X 0) T)).

  This again simplifies, using linear arithmetic, applying DIFFERENCE-0 and
  EQUAL-DIFFERENCE-0, and expanding LESSP, to the following two new
  conjectures:

  Case 11.2.
          (IMPLIES (AND (NOT (LESSP Y X))
                        (NOT (LESSP X Y))
                        (EQUAL Y 0)
                        (NOT (NUMBERP Z))
                        (NUMBERP X))
                   (EQUAL X 0)).

    But this again simplifies, using linear arithmetic, to:

          T.

  Case 11.1.
          (IMPLIES (AND (NOT (LESSP Y X))
                        (NOT (LESSP X Y))
                        (NOT (NUMBERP Y))
                        (NOT (NUMBERP Z))
                        (NUMBERP X))
                   (EQUAL X 0)),

    which again simplifies, unfolding the definition of LESSP, to:

          T.

Case 10.(IMPLIES (AND (EQUAL (DIFFERENCE X Y)
                             (DIFFERENCE Z Y))
                      (NOT (LESSP X Y))
                      (NOT (LESSP Z Y))
                      (NUMBERP Z)
                      (NOT (NUMBERP X)))
                 (EQUAL (EQUAL 0 Z) T)),

  which again simplifies, using linear arithmetic, rewriting with DIFFERENCE-0
  and EQUAL-DIFFERENCE-0, and opening up the functions LESSP and EQUAL, to the
  following two new formulas:

  Case 10.2.
          (IMPLIES (AND (NOT (LESSP Y Z))
                        (EQUAL Y 0)
                        (NUMBERP Z)
                        (NOT (NUMBERP X)))
                   (EQUAL 0 Z)).

    But this again simplifies, using linear arithmetic, to:

          T.

  Case 10.1.
          (IMPLIES (AND (NOT (LESSP Y Z))
                        (NOT (NUMBERP Y))
                        (NUMBERP Z)
                        (NOT (NUMBERP X)))
                   (EQUAL 0 Z)),

    which again simplifies, expanding the function LESSP, to:

          T.

Case 9. (IMPLIES (AND (EQUAL (DIFFERENCE X Y)
                             (DIFFERENCE Z Y))
                      (NOT (LESSP X Y))
                      (NOT (LESSP Z Y))
                      (NUMBERP Z)
                      (NUMBERP X))
                 (EQUAL (EQUAL X Z) T)),

  which again simplifies, obviously, to the new formula:

        (IMPLIES (AND (EQUAL (DIFFERENCE X Y)
                             (DIFFERENCE Z Y))
                      (NOT (LESSP X Y))
                      (NOT (LESSP Z Y))
                      (NUMBERP Z)
                      (NUMBERP X))
                 (EQUAL X Z)),

  which again simplifies, using linear arithmetic, to:

        T.

Case 8. (IMPLIES (AND (NOT (EQUAL (DIFFERENCE X Y)
                                  (DIFFERENCE Z Y)))
                      (NOT (LESSP X Y))
                      (NOT (LESSP Z Y))
                      (NOT (NUMBERP Z)))
                 (NOT (EQUAL X 0))),

  which again simplifies, using linear arithmetic, appealing to the lemma
  DIFFERENCE-0, and unfolding the function EQUAL, to:

        T.

Case 7. (IMPLIES (AND (NOT (EQUAL (DIFFERENCE X Y)
                                  (DIFFERENCE Z Y)))
                      (NOT (LESSP X Y))
                      (NOT (LESSP Z Y))
                      (NOT (NUMBERP X)))
                 (NOT (EQUAL 0 Z))),

  which again simplifies, using linear arithmetic, applying DIFFERENCE-0, and
  unfolding the function EQUAL, to:

        T.

Case 6. (IMPLIES (AND (NOT (EQUAL (DIFFERENCE X Y)
                                  (DIFFERENCE Z Y)))
                      (NOT (LESSP X Y))
                      (NOT (LESSP Z Y))
                      (NUMBERP Z)
                      (NUMBERP X))
                 (NOT (EQUAL X Z))).

  This again simplifies, clearly, to:

        T.

Case 5. (IMPLIES (AND (NOT (EQUAL (DIFFERENCE X Y)
                                  (DIFFERENCE Z Y)))
                      (LESSP X Y))
                 (LESSP Y Z)).

  However this again simplifies, using linear arithmetic, applying
  DIFFERENCE-0, and expanding the function EQUAL, to:

        T.

Case 4. (IMPLIES (AND (NOT (EQUAL (DIFFERENCE X Y)
                                  (DIFFERENCE Z Y)))
                      (NOT (LESSP X Y))
                      (NOT (LESSP Z Y))
                      (NOT (NUMBERP Z)))
                 (NUMBERP X)).

  However this again simplifies, using linear arithmetic, applying
  DIFFERENCE-0, and unfolding the definition of EQUAL, to:

        T.

Case 3. (IMPLIES (AND (NOT (EQUAL (DIFFERENCE X Y)
                                  (DIFFERENCE Z Y)))
                      (NOT (LESSP X Y))
                      (LESSP Z Y))
                 (LESSP Y X)).

  But this again simplifies, using linear arithmetic, to three new goals:

  Case 3.3.
          (IMPLIES (AND (NOT (NUMBERP Y))
                        (NOT (EQUAL (DIFFERENCE X Y)
                                    (DIFFERENCE Z Y)))
                        (NOT (LESSP X Y))
                        (LESSP Z Y))
                   (LESSP Y X)),

    which again simplifies, appealing to the lemma DIFFERENCE-0, and expanding
    the definitions of DIFFERENCE and LESSP, to:

          T.

  Case 3.2.
          (IMPLIES (AND (NOT (NUMBERP X))
                        (NOT (EQUAL (DIFFERENCE X Y)
                                    (DIFFERENCE Z Y)))
                        (NOT (LESSP X Y))
                        (LESSP Z Y))
                   (LESSP Y X)),

    which again simplifies, using linear arithmetic, applying DIFFERENCE-0,
    and unfolding the function EQUAL, to:

          T.

  Case 3.1.
          (IMPLIES (AND (NUMBERP X)
                        (NUMBERP Y)
                        (NOT (EQUAL (DIFFERENCE X X)
                                    (DIFFERENCE Z X)))
                        (NOT (LESSP X X))
                        (LESSP Z X))
                   (LESSP X X)).

    However this again simplifies, using linear arithmetic, applying
    DIFFERENCE-0, and expanding the function EQUAL, to:

          T.

Case 2. (IMPLIES (AND (EQUAL (DIFFERENCE X Y)
                             (DIFFERENCE Z Y))
                      (LESSP X Y))
                 (NOT (LESSP Y Z))).

  However this again simplifies, using linear arithmetic and rewriting with
  the lemmas DIFFERENCE-0 and EQUAL-DIFFERENCE-0, to:

        T.

Case 1. (IMPLIES (AND (EQUAL (DIFFERENCE X Y)
                             (DIFFERENCE Z Y))
                      (NOT (LESSP X Y))
                      (LESSP Z Y))
                 (NOT (LESSP Y X))),

  which again simplifies, using linear arithmetic and rewriting with
  DIFFERENCE-0 and EQUAL-DIFFERENCE-0, to:

        T.

Q.E.D.


[ 0.0 0.0 0.0 ]

DIFFERENCE-CANCELLATION-1


(DISABLE DIFFERENCE-CANCELLATION-1)

[ 0.0 0.0 0.0 ]

DIFFERENCE-CANCELLATION-1-OFF


(PROVE-LEMMA TIMES-ZERO2
             (REWRITE)
             (IMPLIES (NOT (NUMBERP Y))
                      (EQUAL (TIMES X Y) '0))
             ((ENABLE PLUS-RIGHT-ID2 COMMUTATIVITY-OF-PLUS)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))


     Call the conjecture *1.


     Perhaps we can prove it by induction.  There is only one plausible
induction.  We will induct according to the following scheme:
      (AND (IMPLIES (ZEROP X) (p X Y))
           (IMPLIES (AND (NOT (ZEROP X)) (p (SUB1 X) Y))
                    (p X Y))).
Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP can be
used to prove that the measure (COUNT X) decreases according to the
well-founded relation LESSP in each induction step of the scheme.  The above
induction scheme leads to two new goals:

Case 2. (IMPLIES (AND (ZEROP X) (NOT (NUMBERP Y)))
                 (EQUAL (TIMES X Y) 0)),

  which simplifies, opening up the definitions of ZEROP, EQUAL, and TIMES, to:

        T.

Case 1. (IMPLIES (AND (NOT (ZEROP X))
                      (EQUAL (TIMES (SUB1 X) Y) 0)
                      (NOT (NUMBERP Y)))
                 (EQUAL (TIMES X Y) 0)),

  which simplifies, applying PLUS-RIGHT-ID2 and COMMUTATIVITY-OF-PLUS, and
  unfolding ZEROP, TIMES, NUMBERP, and EQUAL, to:

        T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.0 0.0 ]

TIMES-ZERO2


(DISABLE TIMES-ZERO2)

[ 0.0 0.0 0.0 ]

TIMES-ZERO2-OFF


(PROVE-LEMMA DISTRIBUTIVITY-OF-TIMES-OVER-PLUS
             (REWRITE)
             (EQUAL (TIMES X (PLUS Y Z))
                    (PLUS (TIMES X Y) (TIMES X Z)))
             ((ENABLE COMMUTATIVITY2-OF-PLUS ASSOCIATIVITY-OF-PLUS)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))


     Call the conjecture *1.


     We will try to prove it by induction.  There are four plausible
inductions.  They merge into two likely candidate inductions.  However, only
one is unflawed.  We will induct according to the following scheme:
      (AND (IMPLIES (ZEROP X) (p X Y Z))
           (IMPLIES (AND (NOT (ZEROP X)) (p (SUB1 X) Y Z))
                    (p X Y Z))).
Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP inform
us that the measure (COUNT X) decreases according to the well-founded relation
LESSP in each induction step of the scheme.  The above induction scheme
generates two new goals:

Case 2. (IMPLIES (ZEROP X)
                 (EQUAL (TIMES X (PLUS Y Z))
                        (PLUS (TIMES X Y) (TIMES X Z)))),

  which simplifies, opening up the functions ZEROP, EQUAL, TIMES, and PLUS, to:

        T.

Case 1. (IMPLIES (AND (NOT (ZEROP X))
                      (EQUAL (TIMES (SUB1 X) (PLUS Y Z))
                             (PLUS (TIMES (SUB1 X) Y)
                                   (TIMES (SUB1 X) Z))))
                 (EQUAL (TIMES X (PLUS Y Z))
                        (PLUS (TIMES X Y) (TIMES X Z)))),

  which simplifies, applying ASSOCIATIVITY-OF-PLUS and COMMUTATIVITY2-OF-PLUS,
  and unfolding the functions ZEROP and TIMES, to:

        T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.0 0.0 ]

DISTRIBUTIVITY-OF-TIMES-OVER-PLUS


(DISABLE DISTRIBUTIVITY-OF-TIMES-OVER-PLUS)

[ 0.0 0.0 0.0 ]

DISTRIBUTIVITY-OF-TIMES-OVER-PLUS-OFF


(PROVE-LEMMA TIMES-ADD1
             (REWRITE)
             (EQUAL (TIMES X (ADD1 Y))
                    (IF (NUMBERP Y)
                        (PLUS X (TIMES X Y))
                        (FIX X)))
             ((ENABLE PLUS-RIGHT-ID2 COMMUTATIVITY-OF-PLUS)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This conjecture simplifies, unfolding the definition of FIX, to three new
conjectures:

Case 3. (IMPLIES (AND (NOT (NUMBERP Y)) (NUMBERP X))
                 (EQUAL (TIMES X (ADD1 Y)) X)),

  which again simplifies, rewriting with SUB1-TYPE-RESTRICTION, to the new
  formula:

        (IMPLIES (AND (NOT (NUMBERP Y)) (NUMBERP X))
                 (EQUAL (TIMES X 1) X)),

  which we will name *1.

Case 2. (IMPLIES (AND (NOT (NUMBERP Y))
                      (NOT (NUMBERP X)))
                 (EQUAL (TIMES X (ADD1 Y)) 0)).

  However this again simplifies, applying SUB1-TYPE-RESTRICTION, and opening
  up the definitions of TIMES and EQUAL, to:

        T.

Case 1. (IMPLIES (NUMBERP Y)
                 (EQUAL (TIMES X (ADD1 Y))
                        (PLUS X (TIMES X Y)))),

  which we would usually push and work on later by induction.  But if we must
  use induction to prove the input conjecture, we prefer to induct on the
  original formulation of the problem.  Thus we will disregard all that we
  have previously done, give the name *1 to the original input, and work on it.


     So now let us consider:

(EQUAL (TIMES X (ADD1 Y))
       (IF (NUMBERP Y)
           (PLUS X (TIMES X Y))
           (FIX X))),

which we named *1 above.  We will appeal to induction.  The recursive terms in
the conjecture suggest three inductions.  However, they merge into one likely
candidate induction.  We will induct according to the following scheme:
      (AND (IMPLIES (ZEROP X) (p X Y))
           (IMPLIES (AND (NOT (ZEROP X)) (p (SUB1 X) Y))
                    (p X Y))).
Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP
establish that the measure (COUNT X) decreases according to the well-founded
relation LESSP in each induction step of the scheme.  The above induction
scheme leads to the following two new formulas:

Case 2. (IMPLIES (ZEROP X)
                 (EQUAL (TIMES X (ADD1 Y))
                        (IF (NUMBERP Y)
                            (PLUS X (TIMES X Y))
                            (FIX X)))).

  This simplifies, rewriting with PLUS-RIGHT-ID2 and COMMUTATIVITY-OF-PLUS,
  and expanding the definitions of ZEROP, EQUAL, TIMES, PLUS, FIX, and NUMBERP,
  to:

        T.

Case 1. (IMPLIES (AND (NOT (ZEROP X))
                      (EQUAL (TIMES (SUB1 X) (ADD1 Y))
                             (IF (NUMBERP Y)
                                 (PLUS (SUB1 X) (TIMES (SUB1 X) Y))
                                 (FIX (SUB1 X)))))
                 (EQUAL (TIMES X (ADD1 Y))
                        (IF (NUMBERP Y)
                            (PLUS X (TIMES X Y))
                            (FIX X)))),

  which simplifies, applying SUB1-TYPE-RESTRICTION and SUB1-ADD1, and
  unfolding the functions ZEROP, FIX, TIMES, and PLUS, to the following two
  new formulas:

  Case 1.2.
          (IMPLIES (AND (NOT (EQUAL X 0))
                        (NUMBERP X)
                        (NOT (NUMBERP Y))
                        (EQUAL (TIMES (SUB1 X) (ADD1 Y))
                               (SUB1 X)))
                   (EQUAL (TIMES X 1) X)).

    But this further simplifies, rewriting with SUB1-TYPE-RESTRICTION, and
    unfolding TIMES, to the new formula:

          (IMPLIES (AND (NOT (EQUAL X 0))
                        (NUMBERP X)
                        (NOT (NUMBERP Y))
                        (EQUAL (TIMES (SUB1 X) 1) (SUB1 X)))
                   (EQUAL (PLUS 1 (SUB1 X)) X)),

    which again simplifies, using linear arithmetic, to:

          T.

  Case 1.1.
          (IMPLIES (AND (NOT (EQUAL X 0))
                        (NUMBERP X)
                        (NUMBERP Y)
                        (EQUAL (TIMES (SUB1 X) (ADD1 Y))
                               (PLUS (SUB1 X) (TIMES (SUB1 X) Y))))
                   (EQUAL (ADD1 (PLUS Y (TIMES (SUB1 X) (ADD1 Y))))
                          (PLUS X Y (TIMES (SUB1 X) Y)))),

    which again simplifies, using linear arithmetic, to:

          T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.0 0.0 ]

TIMES-ADD1


(DISABLE TIMES-ADD1)

[ 0.0 0.0 0.0 ]

TIMES-ADD1-OFF


(PROVE-LEMMA COMMUTATIVITY-OF-TIMES
             (REWRITE)
             (EQUAL (TIMES X Y) (TIMES Y X))
             ((ENABLE TIMES-ZERO2 TIMES-ADD1)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))


     Give the conjecture the name *1.


     We will appeal to induction.  Two inductions are suggested by terms in
the conjecture, both of which are flawed.  We limit our consideration to the
two suggested by the largest number of nonprimitive recursive functions in the
conjecture.  Since both of these are equally likely, we will choose
arbitrarily.  We will induct according to the following scheme:
      (AND (IMPLIES (ZEROP X) (p X Y))
           (IMPLIES (AND (NOT (ZEROP X)) (p (SUB1 X) Y))
                    (p X Y))).
Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP inform
us that the measure (COUNT X) decreases according to the well-founded relation
LESSP in each induction step of the scheme.  The above induction scheme
produces the following two new conjectures:

Case 2. (IMPLIES (ZEROP X)
                 (EQUAL (TIMES X Y) (TIMES Y X))).

  This simplifies, applying TIMES-ZERO2, and opening up the functions ZEROP,
  EQUAL, and TIMES, to:

        (IMPLIES (EQUAL X 0)
                 (EQUAL 0 (TIMES Y 0))).

  This again simplifies, obviously, to:

        (EQUAL 0 (TIMES Y 0)),

  which we will name *1.1.

Case 1. (IMPLIES (AND (NOT (ZEROP X))
                      (EQUAL (TIMES (SUB1 X) Y)
                             (TIMES Y (SUB1 X))))
                 (EQUAL (TIMES X Y) (TIMES Y X))).

  This simplifies, unfolding the definitions of ZEROP and TIMES, to the new
  goal:

        (IMPLIES (AND (NOT (EQUAL X 0))
                      (NUMBERP X)
                      (EQUAL (TIMES (SUB1 X) Y)
                             (TIMES Y (SUB1 X))))
                 (EQUAL (PLUS Y (TIMES Y (SUB1 X)))
                        (TIMES Y X))).

  Applying the lemma SUB1-ELIM, replace X by (ADD1 Z) to eliminate (SUB1 X).
  We employ the type restriction lemma noted when SUB1 was introduced to
  restrict the new variable.  This produces the new goal:

        (IMPLIES (AND (NUMBERP Z)
                      (NOT (EQUAL (ADD1 Z) 0))
                      (EQUAL (TIMES Z Y) (TIMES Y Z)))
                 (EQUAL (PLUS Y (TIMES Y Z))
                        (TIMES Y (ADD1 Z)))),

  which further simplifies, applying the lemma TIMES-ADD1, to:

        T.


     So we now return to:

      (EQUAL 0 (TIMES Y 0)),

which is formula *1.1 above.  We will try to prove it by induction.  There is
only one suggested induction.  We will induct according to the following
scheme:
      (AND (IMPLIES (ZEROP Y) (p Y))
           (IMPLIES (AND (NOT (ZEROP Y)) (p (SUB1 Y)))
                    (p Y))).
Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP
establish that the measure (COUNT Y) decreases according to the well-founded
relation LESSP in each induction step of the scheme.  The above induction
scheme produces two new conjectures:

Case 2. (IMPLIES (ZEROP Y)
                 (EQUAL 0 (TIMES Y 0))),

  which simplifies, unfolding ZEROP, TIMES, and EQUAL, to:

        T.

Case 1. (IMPLIES (AND (NOT (ZEROP Y))
                      (EQUAL 0 (TIMES (SUB1 Y) 0)))
                 (EQUAL 0 (TIMES Y 0))),

  which simplifies, unfolding the definitions of ZEROP, TIMES, PLUS, and EQUAL,
  to:

        T.


     That finishes the proof of *1.1, which, in turn, finishes the proof of *1.
Q.E.D.


[ 0.0 0.0 0.0 ]

COMMUTATIVITY-OF-TIMES


(DISABLE COMMUTATIVITY-OF-TIMES)

[ 0.0 0.0 0.0 ]

COMMUTATIVITY-OF-TIMES-OFF


(PROVE-LEMMA COMMUTATIVITY2-OF-TIMES
             (REWRITE)
             (EQUAL (TIMES X (TIMES Y Z))
                    (TIMES Y (TIMES X Z)))
             ((ENABLE DISTRIBUTIVITY-OF-TIMES-OVER-PLUS COMMUTATIVITY-OF-TIMES)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))


     Call the conjecture *1.


     Perhaps we can prove it by induction.  Four inductions are suggested by
terms in the conjecture.  They merge into two likely candidate inductions,
both of which are unflawed.  Since both of these are equally likely, we will
choose arbitrarily.  We will induct according to the following scheme:
      (AND (IMPLIES (ZEROP Y) (p X Y Z))
           (IMPLIES (AND (NOT (ZEROP Y)) (p X (SUB1 Y) Z))
                    (p X Y Z))).
Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP can be
used to prove that the measure (COUNT Y) decreases according to the
well-founded relation LESSP in each induction step of the scheme.  The above
induction scheme leads to two new goals:

Case 2. (IMPLIES (ZEROP Y)
                 (EQUAL (TIMES X Y Z) (TIMES Y X Z))),

  which simplifies, applying the lemma COMMUTATIVITY-OF-TIMES, and opening up
  the definitions of ZEROP, EQUAL, and TIMES, to:

        T.

Case 1. (IMPLIES (AND (NOT (ZEROP Y))
                      (EQUAL (TIMES X (SUB1 Y) Z)
                             (TIMES (SUB1 Y) X Z)))
                 (EQUAL (TIMES X Y Z) (TIMES Y X Z))),

  which simplifies, rewriting with DISTRIBUTIVITY-OF-TIMES-OVER-PLUS, and
  opening up the functions ZEROP and TIMES, to:

        T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.0 0.0 ]

COMMUTATIVITY2-OF-TIMES


(DISABLE COMMUTATIVITY2-OF-TIMES)

[ 0.0 0.0 0.0 ]

COMMUTATIVITY2-OF-TIMES-OFF


(PROVE-LEMMA ASSOCIATIVITY-OF-TIMES
             (REWRITE)
             (EQUAL (TIMES (TIMES X Y) Z)
                    (TIMES X (TIMES Y Z)))
             ((ENABLE COMMUTATIVITY-OF-TIMES COMMUTATIVITY2-OF-TIMES)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This simplifies, rewriting with COMMUTATIVITY-OF-TIMES and
COMMUTATIVITY2-OF-TIMES, to:

      T.

Q.E.D.


[ 0.0 0.0 0.0 ]

ASSOCIATIVITY-OF-TIMES


(DISABLE ASSOCIATIVITY-OF-TIMES)

[ 0.0 0.0 0.0 ]

ASSOCIATIVITY-OF-TIMES-OFF


(PROVE-LEMMA EQUAL-TIMES-0
             (REWRITE)
             (EQUAL (EQUAL (TIMES X Y) '0)
                    (OR (ZEROP X) (ZEROP Y)))
             ((ENABLE PLUS-EQUAL-0 TIMES-ZERO2 COMMUTATIVITY-OF-TIMES)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This simplifies, opening up the functions ZEROP and OR, to five new
conjectures:

Case 5. (IMPLIES (NOT (EQUAL (TIMES X Y) 0))
                 (NUMBERP Y)),

  which again simplifies, applying TIMES-ZERO2, and unfolding the function
  EQUAL, to:

        T.

Case 4. (IMPLIES (NOT (EQUAL (TIMES X Y) 0))
                 (NOT (EQUAL Y 0))).

  However this again simplifies, applying COMMUTATIVITY-OF-TIMES, and
  unfolding the functions EQUAL and TIMES, to:

        T.

Case 3. (IMPLIES (NOT (EQUAL (TIMES X Y) 0))
                 (NUMBERP X)).

  This again simplifies, opening up the definitions of TIMES and EQUAL, to:

        T.

Case 2. (IMPLIES (NOT (EQUAL (TIMES X Y) 0))
                 (NOT (EQUAL X 0))),

  which again simplifies, using linear arithmetic, to:

        T.

Case 1. (IMPLIES (AND (EQUAL (TIMES X Y) 0)
                      (NOT (EQUAL X 0))
                      (NUMBERP X)
                      (NOT (EQUAL Y 0)))
                 (NOT (NUMBERP Y))),

  which we will name *1.


     Perhaps we can prove it by induction.  There is only one plausible
induction.  We will induct according to the following scheme:
      (AND (IMPLIES (ZEROP X) (p Y X))
           (IMPLIES (AND (NOT (ZEROP X)) (p Y (SUB1 X)))
                    (p Y X))).
Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP can be
used to establish that the measure (COUNT X) decreases according to the
well-founded relation LESSP in each induction step of the scheme.  The above
induction scheme produces three new conjectures:

Case 3. (IMPLIES (AND (ZEROP X)
                      (EQUAL (TIMES X Y) 0)
                      (NOT (EQUAL X 0))
                      (NUMBERP X)
                      (NOT (EQUAL Y 0)))
                 (NOT (NUMBERP Y))),

  which simplifies, unfolding the function ZEROP, to:

        T.

Case 2. (IMPLIES (AND (NOT (ZEROP X))
                      (NOT (EQUAL (TIMES (SUB1 X) Y) 0))
                      (EQUAL (TIMES X Y) 0)
                      (NOT (EQUAL X 0))
                      (NUMBERP X)
                      (NOT (EQUAL Y 0)))
                 (NOT (NUMBERP Y))),

  which simplifies, rewriting with the lemma PLUS-EQUAL-0, and expanding the
  functions ZEROP and TIMES, to:

        T.

Case 1. (IMPLIES (AND (NOT (ZEROP X))
                      (EQUAL (SUB1 X) 0)
                      (EQUAL (TIMES X Y) 0)
                      (NOT (EQUAL X 0))
                      (NUMBERP X)
                      (NOT (EQUAL Y 0)))
                 (NOT (NUMBERP Y))),

  which simplifies, rewriting with the lemma PLUS-EQUAL-0, and expanding the
  definitions of ZEROP and TIMES, to:

        T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.0 0.0 ]

EQUAL-TIMES-0


(DISABLE EQUAL-TIMES-0)

[ 0.0 0.0 0.0 ]

EQUAL-TIMES-0-OFF


(DEFN EXP
      (I J)
      (IF (ZEROP J)
          '1
          (TIMES I (EXP I (SUB1 J))))
      NIL)

     Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP
inform us that the measure (COUNT J) decreases according to the well-founded
relation LESSP in each recursive call.  Hence, EXP is accepted under the
definitional principle.  From the definition we can conclude that
(NUMBERP (EXP I J)) is a theorem.




[ 0.0 0.0 0.0 ]

EXP


(PROVE-LEMMA EXP-PLUS
             (REWRITE)
             (EQUAL (EXP I (PLUS J K))
                    (TIMES (EXP I J) (EXP I K)))
             ((ENABLE COMMUTATIVITY-OF-TIMES ASSOCIATIVITY-OF-TIMES EXP)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))


     Call the conjecture *1.


     We will try to prove it by induction.  There are three plausible
inductions.  They merge into two likely candidate inductions.  However, only
one is unflawed.  We will induct according to the following scheme:
      (AND (IMPLIES (ZEROP J) (p I J K))
           (IMPLIES (AND (NOT (ZEROP J)) (p I (SUB1 J) K))
                    (p I J K))).
Linear arithmetic, the lemmas SUB1-LESSEQP and SUB1-LESSP, and the definition
of ZEROP inform us that the measure (COUNT J) decreases according to the
well-founded relation LESSP in each induction step of the scheme.  The above
induction scheme generates two new goals:

Case 2. (IMPLIES (ZEROP J)
                 (EQUAL (EXP I (PLUS J K))
                        (TIMES (EXP I J) (EXP I K)))),

  which simplifies, opening up the functions ZEROP, EQUAL, PLUS, and EXP, to
  four new formulas:

  Case 2.4.
          (IMPLIES (AND (EQUAL J 0) (NOT (NUMBERP K)))
                   (EQUAL (EXP I 0)
                          (TIMES 1 (EXP I K)))),

    which again simplifies, opening up EQUAL, EXP, and TIMES, to:

          T.

  Case 2.3.
          (IMPLIES (AND (EQUAL J 0) (NUMBERP K))
                   (EQUAL (EXP I K)
                          (TIMES 1 (EXP I K)))),

    which again simplifies, using linear arithmetic, to:

          T.

  Case 2.2.
          (IMPLIES (AND (NOT (NUMBERP J))
                        (NOT (NUMBERP K)))
                   (EQUAL (EXP I 0)
                          (TIMES 1 (EXP I K)))),

    which again simplifies, opening up the definitions of EQUAL, EXP, and
    TIMES, to:

          T.

  Case 2.1.
          (IMPLIES (AND (NOT (NUMBERP J)) (NUMBERP K))
                   (EQUAL (EXP I K)
                          (TIMES 1 (EXP I K)))),

    which again simplifies, using linear arithmetic, to:

          T.

Case 1. (IMPLIES (AND (NOT (ZEROP J))
                      (EQUAL (EXP I (PLUS (SUB1 J) K))
                             (TIMES (EXP I (SUB1 J)) (EXP I K))))
                 (EQUAL (EXP I (PLUS J K))
                        (TIMES (EXP I J) (EXP I K)))),

  which simplifies, rewriting with COMMUTATIVITY-OF-TIMES, SUB1-ADD1, and
  ASSOCIATIVITY-OF-TIMES, and opening up the functions ZEROP, PLUS, and EXP,
  to:

        T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.0 0.1 ]

EXP-PLUS


(DISABLE EXP-PLUS)

[ 0.0 0.0 0.0 ]

EXP-PLUS-OFF


(PROVE-LEMMA EQUAL-LESSP
             (REWRITE)
             (EQUAL (EQUAL (LESSP X Y) Z)
                    (IF (LESSP X Y)
                        (EQUAL '*1*TRUE Z)
                        (EQUAL '*1*FALSE Z)))
             ((ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This simplifies, clearly, to the following four new formulas:

Case 4. (IMPLIES (AND (EQUAL (LESSP X Y) Z)
                      (NOT (LESSP X Y)))
                 (NOT Z)).

  This again simplifies, trivially, to:

        T.

Case 3. (IMPLIES (AND (NOT (EQUAL (LESSP X Y) Z))
                      (NOT (LESSP X Y)))
                 Z).

  This again simplifies, clearly, to:

        T.

Case 2. (IMPLIES (AND (NOT (EQUAL (LESSP X Y) Z))
                      (LESSP X Y))
                 (NOT (EQUAL T Z))).

  This again simplifies, obviously, to:

        T.

Case 1. (IMPLIES (AND (EQUAL (LESSP X Y) Z)
                      (LESSP X Y))
                 (EQUAL (EQUAL T Z) T)).

  This again simplifies, obviously, to:

        T.

Q.E.D.


[ 0.0 0.0 0.0 ]

EQUAL-LESSP


(DISABLE EQUAL-LESSP)

[ 0.0 0.0 0.0 ]

EQUAL-LESSP-OFF


(PROVE-LEMMA DIFFERENCE-ELIM
             (ELIM)
             (IMPLIES (AND (NUMBERP Y) (NOT (LESSP Y X)))
                      (EQUAL (PLUS X (DIFFERENCE Y X)) Y))
             ((ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This conjecture simplifies, using linear arithmetic, to:

      T.

Q.E.D.


[ 0.0 0.0 0.0 ]

DIFFERENCE-ELIM


(PROVE-LEMMA REMAINDER-QUOTIENT
             (REWRITE)
             (EQUAL (PLUS (REMAINDER X Y)
                          (TIMES Y (QUOTIENT X Y)))
                    (FIX X))
             ((ENABLE COMMUTATIVITY2-OF-PLUS COMMUTATIVITY-OF-PLUS TIMES-ZERO2
                      TIMES-ADD1 COMMUTATIVITY-OF-TIMES)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This formula simplifies, opening up the function FIX, to the following two new
goals:

Case 2. (IMPLIES (NOT (NUMBERP X))
                 (EQUAL (PLUS (REMAINDER X Y)
                              (TIMES Y (QUOTIENT X Y)))
                        0)).

  However this again simplifies, rewriting with COMMUTATIVITY-OF-TIMES, and
  opening up LESSP, REMAINDER, QUOTIENT, EQUAL, TIMES, and PLUS, to:

        T.

Case 1. (IMPLIES (NUMBERP X)
                 (EQUAL (PLUS (REMAINDER X Y)
                              (TIMES Y (QUOTIENT X Y)))
                        X)).

  Call the above conjecture *1.


     We will appeal to induction.  Three inductions are suggested by terms in
the conjecture.  They merge into two likely candidate inductions.  However,
only one is unflawed.  We will induct according to the following scheme:
      (AND (IMPLIES (ZEROP Y) (p X Y))
           (IMPLIES (AND (NOT (ZEROP Y)) (LESSP X Y))
                    (p X Y))
           (IMPLIES (AND (NOT (ZEROP Y))
                         (NOT (LESSP X Y))
                         (p (DIFFERENCE X Y) Y))
                    (p X Y))).
Linear arithmetic, the lemmas COUNT-NUMBERP and COUNT-NOT-LESSP, and the
definition of ZEROP can be used to show that the measure (COUNT X) decreases
according to the well-founded relation LESSP in each induction step of the
scheme.  The above induction scheme produces three new goals:

Case 3. (IMPLIES (AND (ZEROP Y) (NUMBERP X))
                 (EQUAL (PLUS (REMAINDER X Y)
                              (TIMES Y (QUOTIENT X Y)))
                        X)),

  which simplifies, applying COMMUTATIVITY-OF-PLUS, TIMES-ZERO2, and
  COMMUTATIVITY-OF-TIMES, and unfolding the functions ZEROP, EQUAL, REMAINDER,
  QUOTIENT, TIMES, and PLUS, to:

        T.

Case 2. (IMPLIES (AND (NOT (ZEROP Y))
                      (LESSP X Y)
                      (NUMBERP X))
                 (EQUAL (PLUS (REMAINDER X Y)
                              (TIMES Y (QUOTIENT X Y)))
                        X)).

  This simplifies, rewriting with the lemmas COMMUTATIVITY-OF-TIMES and
  COMMUTATIVITY-OF-PLUS, and opening up the functions ZEROP, REMAINDER,
  QUOTIENT, EQUAL, TIMES, and PLUS, to:

        T.

Case 1. (IMPLIES (AND (NOT (ZEROP Y))
                      (NOT (LESSP X Y))
                      (EQUAL (PLUS (REMAINDER (DIFFERENCE X Y) Y)
                                   (TIMES Y
                                          (QUOTIENT (DIFFERENCE X Y) Y)))
                             (DIFFERENCE X Y))
                      (NUMBERP X))
                 (EQUAL (PLUS (REMAINDER X Y)
                              (TIMES Y (QUOTIENT X Y)))
                        X)).

  This simplifies, rewriting with TIMES-ADD1 and COMMUTATIVITY2-OF-PLUS, and
  opening up the functions ZEROP, REMAINDER, and QUOTIENT, to the goal:

        (IMPLIES (AND (NOT (EQUAL Y 0))
                      (NUMBERP Y)
                      (NOT (LESSP X Y))
                      (EQUAL (PLUS (REMAINDER (DIFFERENCE X Y) Y)
                                   (TIMES Y
                                          (QUOTIENT (DIFFERENCE X Y) Y)))
                             (DIFFERENCE X Y))
                      (NUMBERP X))
                 (EQUAL (PLUS Y (DIFFERENCE X Y)) X)).

  But this again simplifies, using linear arithmetic, to:

        T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.0 0.0 ]

REMAINDER-QUOTIENT


(DISABLE REMAINDER-QUOTIENT)

[ 0.0 0.0 0.0 ]

REMAINDER-QUOTIENT-OFF


(PROVE-LEMMA REMAINDER-WRT-1
             (REWRITE)
             (EQUAL (REMAINDER Y '1) '0)
             ((ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))


     Call the conjecture *1.


     We will try to prove it by induction.  There is only one suggested
induction.  We will induct according to the following scheme:
      (AND (IMPLIES (ZEROP 1) (p Y))
           (IMPLIES (AND (NOT (ZEROP 1)) (LESSP Y 1))
                    (p Y))
           (IMPLIES (AND (NOT (ZEROP 1))
                         (NOT (LESSP Y 1))
                         (p (DIFFERENCE Y 1)))
                    (p Y))).
Linear arithmetic, the lemmas COUNT-NUMBERP and COUNT-NOT-LESSP, and the
definition of ZEROP can be used to establish that the measure (COUNT Y)
decreases according to the well-founded relation LESSP in each induction step
of the scheme.  The above induction scheme generates the following three new
formulas:

Case 3. (IMPLIES (ZEROP 1)
                 (EQUAL (REMAINDER Y 1) 0)).

  This simplifies, expanding the definition of ZEROP, to:

        T.

Case 2. (IMPLIES (AND (NOT (ZEROP 1)) (LESSP Y 1))
                 (EQUAL (REMAINDER Y 1) 0)).

  This simplifies, opening up the functions ZEROP, REMAINDER, EQUAL, and
  NUMBERP, to the new goal:

        (IMPLIES (AND (LESSP Y 1) (NUMBERP Y))
                 (EQUAL Y 0)),

  which again simplifies, using linear arithmetic, to:

        T.

Case 1. (IMPLIES (AND (NOT (ZEROP 1))
                      (NOT (LESSP Y 1))
                      (EQUAL (REMAINDER (DIFFERENCE Y 1) 1)
                             0))
                 (EQUAL (REMAINDER Y 1) 0)),

  which simplifies, expanding ZEROP, REMAINDER, EQUAL, and NUMBERP, to:

        T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.0 0.0 ]

REMAINDER-WRT-1


(DISABLE REMAINDER-WRT-1)

[ 0.0 0.0 0.0 ]

REMAINDER-WRT-1-OFF


(PROVE-LEMMA REMAINDER-WRT-12
             (REWRITE)
             (IMPLIES (NOT (NUMBERP X))
                      (EQUAL (REMAINDER Y X) (FIX Y)))
             ((ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This simplifies, unfolding the definitions of REMAINDER and FIX, to:

      T.

Q.E.D.


[ 0.0 0.0 0.0 ]

REMAINDER-WRT-12


(DISABLE REMAINDER-WRT-12)

[ 0.0 0.0 0.0 ]

REMAINDER-WRT-12-OFF


(PROVE-LEMMA LESSP-REMAINDER2
             (REWRITE GENERALIZE)
             (EQUAL (LESSP (REMAINDER X Y) Y)
                    (NOT (ZEROP Y)))
             ((ENABLE EQUAL-LESSP REMAINDER-WRT-12)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This conjecture simplifies, applying EQUAL-LESSP, and unfolding the functions
ZEROP and NOT, to three new goals:

Case 3. (IMPLIES (AND (NOT (LESSP (REMAINDER X Y) Y))
                      (NOT (EQUAL Y 0)))
                 (NOT (NUMBERP Y))),

  which we will name *1.

Case 2. (IMPLIES (LESSP (REMAINDER X Y) Y)
                 (NUMBERP Y)).

  This again simplifies, applying REMAINDER-WRT-12, and unfolding the
  definition of LESSP, to:

        T.

Case 1. (IMPLIES (LESSP (REMAINDER X Y) Y)
                 (NOT (EQUAL Y 0))).

  This again simplifies, using linear arithmetic, to:

        T.


     So next consider:

      (IMPLIES (AND (NOT (LESSP (REMAINDER X Y) Y))
                    (NOT (EQUAL Y 0)))
               (NOT (NUMBERP Y))),

named *1 above.  Let us appeal to the induction principle.  The recursive
terms in the conjecture suggest two inductions.  However, only one is unflawed.
We will induct according to the following scheme:
      (AND (IMPLIES (ZEROP Y) (p Y X))
           (IMPLIES (AND (NOT (ZEROP Y)) (LESSP X Y))
                    (p Y X))
           (IMPLIES (AND (NOT (ZEROP Y))
                         (NOT (LESSP X Y))
                         (p Y (DIFFERENCE X Y)))
                    (p Y X))).
Linear arithmetic, the lemmas COUNT-NUMBERP and COUNT-NOT-LESSP, and the
definition of ZEROP inform us that the measure (COUNT X) decreases according
to the well-founded relation LESSP in each induction step of the scheme.  The
above induction scheme produces three new conjectures:

Case 3. (IMPLIES (AND (ZEROP Y)
                      (NOT (LESSP (REMAINDER X Y) Y))
                      (NOT (EQUAL Y 0)))
                 (NOT (NUMBERP Y))),

  which simplifies, unfolding the function ZEROP, to:

        T.

Case 2. (IMPLIES (AND (NOT (ZEROP Y))
                      (LESSP X Y)
                      (NOT (LESSP (REMAINDER X Y) Y))
                      (NOT (EQUAL Y 0)))
                 (NOT (NUMBERP Y))),

  which simplifies, unfolding ZEROP and REMAINDER, to:

        (IMPLIES (AND (LESSP X Y)
                      (NOT (NUMBERP X))
                      (NOT (LESSP 0 Y))
                      (NOT (EQUAL Y 0)))
                 (NOT (NUMBERP Y))).

  This again simplifies, using linear arithmetic, to:

        T.

Case 1. (IMPLIES (AND (NOT (ZEROP Y))
                      (NOT (LESSP X Y))
                      (LESSP (REMAINDER (DIFFERENCE X Y) Y)
                             Y)
                      (NOT (LESSP (REMAINDER X Y) Y))
                      (NOT (EQUAL Y 0)))
                 (NOT (NUMBERP Y))),

  which simplifies, expanding the functions ZEROP and REMAINDER, to:

        T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.0 0.0 ]

LESSP-REMAINDER2


(DISABLE LESSP-REMAINDER2)

[ 0.0 0.0 0.0 ]

LESSP-REMAINDER2-OFF


(PROVE-LEMMA REMAINDER-X-X
             (REWRITE)
             (EQUAL (REMAINDER X X) '0)
             ((ENABLE DIFFERENCE-0)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This conjecture simplifies, rewriting with DIFFERENCE-0, and expanding the
functions NUMBERP, REMAINDER, LESSP, and EQUAL, to:

      (IMPLIES (AND (NOT (EQUAL X 0)) (NUMBERP X))
               (NOT (LESSP X X))),

which again simplifies, using linear arithmetic, to:

      T.

Q.E.D.


[ 0.0 0.0 0.0 ]

REMAINDER-X-X


(DISABLE REMAINDER-X-X)

[ 0.0 0.0 0.0 ]

REMAINDER-X-X-OFF


(PROVE-LEMMA REMAINDER-QUOTIENT-ELIM
             (ELIM)
             (IMPLIES (AND (NOT (ZEROP Y)) (NUMBERP X))
                      (EQUAL (PLUS (REMAINDER X Y)
                                   (TIMES Y (QUOTIENT X Y)))
                             X))
             ((ENABLE REMAINDER-QUOTIENT)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This formula can be simplified, using the abbreviations ZEROP, NOT, AND, and
IMPLIES, to:

      (IMPLIES (AND (NOT (EQUAL Y 0))
                    (NUMBERP Y)
                    (NUMBERP X))
               (EQUAL (PLUS (REMAINDER X Y)
                            (TIMES Y (QUOTIENT X Y)))
                      X)),

which simplifies, rewriting with REMAINDER-QUOTIENT, to:

      T.

Q.E.D.


[ 0.0 0.0 0.0 ]

REMAINDER-QUOTIENT-ELIM


(PROVE-LEMMA LESSP-TIMES-1
             (REWRITE)
             (IMPLIES (NOT (ZEROP I))
                      (NOT (LESSP (TIMES I J) J)))
             ((ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

WARNING:  Note that the proposed lemma LESSP-TIMES-1 is to be stored as zero
type prescription rules, zero compound recognizer rules, one linear rule, and
zero replacement rules.


This formula can be simplified, using the abbreviations ZEROP, NOT, and
IMPLIES, to:

      (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I))
               (NOT (LESSP (TIMES I J) J))),

which we will name *1.


     We will appeal to induction.  There are two plausible inductions.
However, only one is unflawed.  We will induct according to the following
scheme:
      (AND (IMPLIES (ZEROP I) (p I J))
           (IMPLIES (AND (NOT (ZEROP I)) (p (SUB1 I) J))
                    (p I J))).
Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP can be
used to show that the measure (COUNT I) decreases according to the
well-founded relation LESSP in each induction step of the scheme.  The above
induction scheme produces three new conjectures:

Case 3. (IMPLIES (AND (ZEROP I)
                      (NOT (EQUAL I 0))
                      (NUMBERP I))
                 (NOT (LESSP (TIMES I J) J))),

  which simplifies, opening up the definition of ZEROP, to:

        T.

Case 2. (IMPLIES (AND (NOT (ZEROP I))
                      (EQUAL (SUB1 I) 0)
                      (NOT (EQUAL I 0))
                      (NUMBERP I))
                 (NOT (LESSP (TIMES I J) J))),

  which simplifies, unfolding the functions ZEROP and TIMES, to the goal:

        (IMPLIES (AND (EQUAL (SUB1 I) 0)
                      (NOT (EQUAL I 0))
                      (NUMBERP I))
                 (NOT (LESSP (PLUS J (TIMES (SUB1 I) J))
                             J))).

  However this again simplifies, using linear arithmetic, to:

        T.

Case 1. (IMPLIES (AND (NOT (ZEROP I))
                      (NOT (LESSP (TIMES (SUB1 I) J) J))
                      (NOT (EQUAL I 0))
                      (NUMBERP I))
                 (NOT (LESSP (TIMES I J) J))),

  which simplifies, opening up ZEROP and TIMES, to:

        (IMPLIES (AND (NOT (LESSP (TIMES (SUB1 I) J) J))
                      (NOT (EQUAL I 0))
                      (NUMBERP I))
                 (NOT (LESSP (PLUS J (TIMES (SUB1 I) J))
                             J))).

  But this again simplifies, using linear arithmetic, to:

        T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.0 0.0 ]

LESSP-TIMES-1


(DISABLE LESSP-TIMES-1)

[ 0.0 0.0 0.0 ]

LESSP-TIMES-1-OFF


(PROVE-LEMMA LESSP-TIMES-2
             (REWRITE)
             (IMPLIES (NOT (ZEROP I))
                      (NOT (LESSP (TIMES J I) J)))
             ((ENABLE COMMUTATIVITY-OF-TIMES LESSP-TIMES-1)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

WARNING:  Note that the proposed lemma LESSP-TIMES-2 is to be stored as zero
type prescription rules, zero compound recognizer rules, one linear rule, and
zero replacement rules.


This formula can be simplified, using the abbreviations ZEROP, NOT, and
IMPLIES, to:

      (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I))
               (NOT (LESSP (TIMES J I) J))),

which simplifies, rewriting with the lemma COMMUTATIVITY-OF-TIMES, to:

      (IMPLIES (AND (NOT (EQUAL I 0)) (NUMBERP I))
               (NOT (LESSP (TIMES I J) J))).

But this again simplifies, using linear arithmetic and applying the lemma
LESSP-TIMES-1, to:

      T.

Q.E.D.


[ 0.0 0.0 0.0 ]

LESSP-TIMES-2


(DISABLE LESSP-TIMES-2)

[ 0.0 0.0 0.0 ]

LESSP-TIMES-2-OFF


(PROVE-LEMMA LESSP-QUOTIENT1
             (REWRITE)
             (EQUAL (LESSP (QUOTIENT I J) I)
                    (AND (NOT (ZEROP I))
                         (OR (ZEROP J) (NOT (EQUAL J '1)))))
             ((ENABLE PLUS-EQUAL-0 DIFFERENCE-0 EQUAL-DIFFERENCE-0
                      EQUAL-TIMES-0 EQUAL-LESSP LESSP-REMAINDER2
                      REMAINDER-QUOTIENT-ELIM LESSP-TIMES-1)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This formula simplifies, rewriting with EQUAL-LESSP, and opening up the
definitions of ZEROP, NOT, OR, and AND, to the following six new formulas:

Case 6. (IMPLIES (AND (NOT (LESSP (QUOTIENT I J) I))
                      (NOT (EQUAL I 0))
                      (NUMBERP I))
                 (NOT (EQUAL J 0))).

  But this again simplifies, opening up EQUAL, QUOTIENT, and LESSP, to:

        T.

Case 5. (IMPLIES (AND (NOT (LESSP (QUOTIENT I J) I))
                      (NOT (EQUAL I 0))
                      (NUMBERP I))
                 (NUMBERP J)),

  which again simplifies, unfolding the definitions of QUOTIENT, EQUAL, and
  LESSP, to:

        T.

Case 4. (IMPLIES (AND (NOT (LESSP (QUOTIENT I J) I))
                      (NOT (EQUAL I 0))
                      (NUMBERP I))
                 (EQUAL J 1)).

  Applying the lemma REMAINDER-QUOTIENT-ELIM, replace I by
  (PLUS Z (TIMES J X)) to eliminate (QUOTIENT I J) and (REMAINDER I J).  We
  employ LESSP-REMAINDER2, the type restriction lemma noted when QUOTIENT was
  introduced, and the type restriction lemma noted when REMAINDER was
  introduced to restrict the new variables.  We thus obtain the following
  three new formulas:

  Case 4.3.
          (IMPLIES (AND (EQUAL J 0)
                        (NOT (LESSP (QUOTIENT I J) I))
                        (NOT (EQUAL I 0))
                        (NUMBERP I))
                   (EQUAL J 1)).

    However this further simplifies, opening up the definitions of EQUAL,
    QUOTIENT, and LESSP, to:

          T.

  Case 4.2.
          (IMPLIES (AND (NOT (NUMBERP J))
                        (NOT (LESSP (QUOTIENT I J) I))
                        (NOT (EQUAL I 0))
                        (NUMBERP I))
                   (EQUAL J 1)),

    which further simplifies, opening up QUOTIENT, EQUAL, and LESSP, to:

          T.

  Case 4.1.
          (IMPLIES (AND (NUMBERP X)
                        (NUMBERP Z)
                        (EQUAL (LESSP Z J) (NOT (ZEROP J)))
                        (NUMBERP J)
                        (NOT (EQUAL J 0))
                        (NOT (LESSP X (PLUS Z (TIMES J X))))
                        (NOT (EQUAL (PLUS Z (TIMES J X)) 0)))
                   (EQUAL J 1)),

    which further simplifies, rewriting with the lemmas EQUAL-TIMES-0 and
    PLUS-EQUAL-0, and expanding the functions ZEROP and NOT, to two new goals:

    Case 4.1.2.
            (IMPLIES (AND (NUMBERP X)
                          (NUMBERP Z)
                          (LESSP Z J)
                          (NUMBERP J)
                          (NOT (EQUAL J 0))
                          (NOT (LESSP X (PLUS Z (TIMES J X))))
                          (NOT (EQUAL Z 0)))
                     (EQUAL J 1)),

      which again simplifies, using linear arithmetic and applying
      LESSP-TIMES-1, to:

            T.

    Case 4.1.1.
            (IMPLIES (AND (NUMBERP X)
                          (NUMBERP Z)
                          (LESSP Z J)
                          (NUMBERP J)
                          (NOT (EQUAL J 0))
                          (NOT (LESSP X (PLUS Z (TIMES J X))))
                          (NOT (EQUAL X 0)))
                     (EQUAL J 1)),

      which we would usually push and work on later by induction.  But if we
      must use induction to prove the input conjecture, we prefer to induct on
      the original formulation of the problem.  Thus we will disregard all
      that we have previously done, give the name *1 to the original input,
      and work on it.


     So now let us consider:

(EQUAL (LESSP (QUOTIENT I J) I)
       (AND (NOT (ZEROP I))
            (OR (ZEROP J) (NOT (EQUAL J 1))))).

We gave this the name *1 above.  Perhaps we can prove it by induction.  Two
inductions are suggested by terms in the conjecture, both of which are
unflawed.  So we will choose the one suggested by the largest number of
nonprimitive recursive functions.  We will induct according to the following
scheme:
      (AND (IMPLIES (ZEROP J) (p I J))
           (IMPLIES (AND (NOT (ZEROP J)) (LESSP I J))
                    (p I J))
           (IMPLIES (AND (NOT (ZEROP J))
                         (NOT (LESSP I J))
                         (p (DIFFERENCE I J) J))
                    (p I J))).
Linear arithmetic, the lemmas COUNT-NUMBERP and COUNT-NOT-LESSP, and the
definition of ZEROP inform us that the measure (COUNT I) decreases according
to the well-founded relation LESSP in each induction step of the scheme.  The
above induction scheme leads to the following three new formulas:

Case 3. (IMPLIES (ZEROP J)
                 (EQUAL (LESSP (QUOTIENT I J) I)
                        (AND (NOT (ZEROP I))
                             (OR (ZEROP J) (NOT (EQUAL J 1)))))).

  This simplifies, expanding the definitions of ZEROP, EQUAL, QUOTIENT, LESSP,
  NOT, OR, and AND, to the following four new goals:

  Case 3.4.
          (IMPLIES (AND (EQUAL J 0)
                        (NOT (EQUAL I 0))
                        (NOT (NUMBERP I)))
                   (EQUAL (NUMBERP I) F)).

    This again simplifies, clearly, to:

          T.

  Case 3.3.
          (IMPLIES (AND (EQUAL J 0)
                        (NOT (EQUAL I 0))
                        (NUMBERP I))
                   (EQUAL (NUMBERP I) T)).

    This again simplifies, clearly, to:

          T.

  Case 3.2.
          (IMPLIES (AND (NOT (NUMBERP J))
                        (NOT (EQUAL I 0))
                        (NOT (NUMBERP I)))
                   (EQUAL (NUMBERP I) F)).

    This again simplifies, clearly, to:

          T.

  Case 3.1.
          (IMPLIES (AND (NOT (NUMBERP J))
                        (NOT (EQUAL I 0))
                        (NUMBERP I))
                   (EQUAL (NUMBERP I) T)).

    This again simplifies, trivially, to:

          T.

Case 2. (IMPLIES (AND (NOT (ZEROP J)) (LESSP I J))
                 (EQUAL (LESSP (QUOTIENT I J) I)
                        (AND (NOT (ZEROP I))
                             (OR (ZEROP J) (NOT (EQUAL J 1)))))).

  This simplifies, expanding the definitions of ZEROP, QUOTIENT, EQUAL, LESSP,
  NOT, OR, and AND, to the following three new goals:

  Case 2.3.
          (IMPLIES (AND (NOT (EQUAL J 0))
                        (NUMBERP J)
                        (LESSP I J)
                        (NOT (EQUAL I 0))
                        (EQUAL J 1))
                   (EQUAL (NUMBERP I) F)).

    This again simplifies, using linear arithmetic, to:

          (IMPLIES (AND (NOT (NUMBERP I))
                        (NOT (EQUAL 1 0))
                        (NUMBERP 1)
                        (LESSP I 1)
                        (NOT (EQUAL I 0)))
                   (EQUAL (NUMBERP I) F)).

    This again simplifies, clearly, to:

          T.

  Case 2.2.
          (IMPLIES (AND (NOT (EQUAL J 0))
                        (NUMBERP J)
                        (LESSP I J)
                        (NOT (EQUAL I 0))
                        (NOT (NUMBERP I)))
                   (EQUAL (NUMBERP I) F)).

    This again simplifies, obviously, to:

          T.

  Case 2.1.
          (IMPLIES (AND (NOT (EQUAL J 0))
                        (NUMBERP J)
                        (LESSP I J)
                        (NOT (EQUAL I 0))
                        (NUMBERP I)
                        (NOT (EQUAL J 1)))
                   (EQUAL (NUMBERP I) T)).

    This again simplifies, obviously, to:

          T.

Case 1. (IMPLIES (AND (NOT (ZEROP J))
                      (NOT (LESSP I J))
                      (EQUAL (LESSP (QUOTIENT (DIFFERENCE I J) J)
                                    (DIFFERENCE I J))
                             (AND (NOT (ZEROP (DIFFERENCE I J)))
                                  (OR (ZEROP J) (NOT (EQUAL J 1))))))
                 (EQUAL (LESSP (QUOTIENT I J) I)
                        (AND (NOT (ZEROP I))
                             (OR (ZEROP J) (NOT (EQUAL J 1)))))).

  This simplifies, applying EQUAL-DIFFERENCE-0, EQUAL-LESSP, and SUB1-ADD1,
  and unfolding the functions ZEROP, NOT, OR, AND, EQUAL, QUOTIENT, and LESSP,
  to four new goals:

  Case 1.4.
          (IMPLIES (AND (NOT (EQUAL J 0))
                        (NUMBERP J)
                        (NOT (LESSP I J))
                        (NOT (LESSP (QUOTIENT (DIFFERENCE I J) J)
                                    (DIFFERENCE I J)))
                        (EQUAL J 1))
                   (NOT (LESSP (QUOTIENT I 1) I))),

    which again simplifies, unfolding EQUAL and NUMBERP, to the goal:

          (IMPLIES (AND (NOT (LESSP I 1))
                        (NOT (LESSP (QUOTIENT (DIFFERENCE I 1) 1)
                                    (DIFFERENCE I 1))))
                   (NOT (LESSP (QUOTIENT I 1) I))).

    However this again simplifies, applying SUB1-ADD1, and opening up the
    functions NUMBERP, EQUAL, QUOTIENT, and LESSP, to the new formula:

          (IMPLIES (AND (NOT (LESSP I 1))
                        (NOT (LESSP (QUOTIENT (DIFFERENCE I 1) 1)
                                    (DIFFERENCE I 1)))
                        (NOT (EQUAL I 0))
                        (NUMBERP I))
                   (NOT (LESSP (QUOTIENT (DIFFERENCE I 1) 1)
                               (SUB1 I)))),

    which again simplifies, using linear arithmetic, to:

          T.

  Case 1.3.
          (IMPLIES (AND (NOT (EQUAL J 0))
                        (NUMBERP J)
                        (NOT (LESSP I J))
                        (NOT (LESSP (QUOTIENT (DIFFERENCE I J) J)
                                    (DIFFERENCE I J)))
                        (NOT (LESSP J I))
                        (NOT (EQUAL I 0))
                        (NUMBERP I)
                        (EQUAL J 1))
                   (EQUAL (LESSP (QUOTIENT (DIFFERENCE I J) J)
                                 (SUB1 I))
                          F)),

    which again simplifies, using linear arithmetic, to:

          (IMPLIES (AND (NOT (EQUAL 1 0))
                        (NUMBERP 1)
                        (NOT (LESSP 1 1))
                        (NOT (LESSP (QUOTIENT (DIFFERENCE 1 1) 1)
                                    (DIFFERENCE 1 1)))
                        (NOT (LESSP 1 1))
                        (NOT (EQUAL 1 0))
                        (NUMBERP 1))
                   (EQUAL (LESSP (QUOTIENT (DIFFERENCE 1 1) 1)
                                 (SUB1 1))
                          F)).

    However this again simplifies, unfolding EQUAL, NUMBERP, LESSP, DIFFERENCE,
    QUOTIENT, and SUB1, to:

          T.

  Case 1.2.
          (IMPLIES (AND (NOT (EQUAL J 0))
                        (NUMBERP J)
                        (NOT (LESSP I J))
                        (NOT (LESSP (QUOTIENT (DIFFERENCE I J) J)
                                    (DIFFERENCE I J)))
                        (NOT (LESSP J I))
                        (NOT (EQUAL I 0))
                        (NUMBERP I)
                        (NOT (EQUAL J 1)))
                   (EQUAL (LESSP (QUOTIENT (DIFFERENCE I J) J)
                                 (SUB1 I))
                          T)),

    which again simplifies, using linear arithmetic, to the conjecture:

          (IMPLIES (AND (NOT (EQUAL I 0))
                        (NUMBERP I)
                        (NOT (LESSP I I))
                        (NOT (LESSP (QUOTIENT (DIFFERENCE I I) I)
                                    (DIFFERENCE I I)))
                        (NOT (LESSP I I))
                        (NOT (EQUAL I 0))
                        (NUMBERP I)
                        (NOT (EQUAL I 1)))
                   (EQUAL (LESSP (QUOTIENT (DIFFERENCE I I) I)
                                 (SUB1 I))
                          T)).

    This again simplifies, applying DIFFERENCE-0, and expanding the
    definitions of LESSP, EQUAL, and QUOTIENT, to the new formula:

          (IMPLIES (AND (NOT (LESSP I I))
                        (NOT (EQUAL I 0))
                        (NUMBERP I)
                        (NOT (EQUAL I 1)))
                   (NOT (EQUAL (SUB1 I) 0))),

    which again simplifies, using linear arithmetic, to:

          T.

  Case 1.1.
          (IMPLIES (AND (NOT (EQUAL J 0))
                        (NUMBERP J)
                        (NOT (LESSP I J))
                        (LESSP (QUOTIENT (DIFFERENCE I J) J)
                               (DIFFERENCE I J))
                        (LESSP J I)
                        (NOT (EQUAL J 1))
                        (NOT (EQUAL I 0))
                        (NUMBERP I))
                   (EQUAL (LESSP (QUOTIENT (DIFFERENCE I J) J)
                                 (SUB1 I))
                          T)),

    which again simplifies, obviously, to:

          (IMPLIES (AND (NOT (EQUAL J 0))
                        (NUMBERP J)
                        (NOT (LESSP I J))
                        (LESSP (QUOTIENT (DIFFERENCE I J) J)
                               (DIFFERENCE I J))
                        (LESSP J I)
                        (NOT (EQUAL J 1))
                        (NOT (EQUAL I 0))
                        (NUMBERP I))
                   (LESSP (QUOTIENT (DIFFERENCE I J) J)
                          (SUB1 I))),

    which again simplifies, using linear arithmetic, to:

          T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.0 0.0 ]

LESSP-QUOTIENT1


(DISABLE LESSP-QUOTIENT1)

[ 0.0 0.0 0.0 ]

LESSP-QUOTIENT1-OFF


(PROVE-LEMMA LESSP-REMAINDER1
             (REWRITE)
             (EQUAL (LESSP (REMAINDER X Y) X)
                    (AND (NOT (ZEROP Y))
                         (AND (NOT (ZEROP X))
                              (NOT (LESSP X Y)))))
             ((ENABLE PLUS-EQUAL-0 COMMUTATIVITY-OF-TIMES EQUAL-TIMES-0
                      EQUAL-LESSP REMAINDER-WRT-12 LESSP-REMAINDER2
                      REMAINDER-QUOTIENT-ELIM)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This formula simplifies, rewriting with EQUAL-LESSP, and opening up the
definitions of ZEROP, NOT, and AND, to the following six new formulas:

Case 6. (IMPLIES (AND (NOT (LESSP (REMAINDER X Y) X))
                      (NOT (EQUAL Y 0))
                      (NUMBERP Y)
                      (NOT (EQUAL X 0))
                      (NUMBERP X))
                 (LESSP X Y)).

  Appealing to the lemma REMAINDER-QUOTIENT-ELIM, we now replace X by
  (PLUS Z (TIMES Y V)) to eliminate (REMAINDER X Y) and (QUOTIENT X Y).  We
  rely upon LESSP-REMAINDER2, the type restriction lemma noted when REMAINDER
  was introduced, and the type restriction lemma noted when QUOTIENT was
  introduced to constrain the new variables.  We must thus prove:

        (IMPLIES (AND (NUMBERP Z)
                      (EQUAL (LESSP Z Y) (NOT (ZEROP Y)))
                      (NUMBERP V)
                      (NOT (LESSP Z (PLUS Z (TIMES Y V))))
                      (NOT (EQUAL Y 0))
                      (NUMBERP Y)
                      (NOT (EQUAL (PLUS Z (TIMES Y V)) 0)))
                 (LESSP (PLUS Z (TIMES Y V)) Y)).

  However this further simplifies, appealing to the lemmas
  COMMUTATIVITY-OF-TIMES, EQUAL-TIMES-0, and PLUS-EQUAL-0, and unfolding the
  definitions of ZEROP and NOT, to two new conjectures:

  Case 6.2.
          (IMPLIES (AND (NUMBERP Z)
                        (LESSP Z Y)
                        (NUMBERP V)
                        (NOT (LESSP Z (PLUS Z (TIMES V Y))))
                        (NOT (EQUAL Y 0))
                        (NUMBERP Y)
                        (NOT (EQUAL Z 0)))
                   (LESSP (PLUS Z (TIMES V Y)) Y)),

    which again simplifies, using linear arithmetic, to:

          T.

  Case 6.1.
          (IMPLIES (AND (NUMBERP Z)
                        (LESSP Z Y)
                        (NUMBERP V)
                        (NOT (LESSP Z (PLUS Z (TIMES V Y))))
                        (NOT (EQUAL Y 0))
                        (NUMBERP Y)
                        (NOT (EQUAL V 0)))
                   (LESSP (PLUS Z (TIMES V Y)) Y)),

    which again simplifies, using linear arithmetic, to:

          T.

Case 5. (IMPLIES (LESSP (REMAINDER X Y) X)
                 (NOT (LESSP X Y))),

  which again simplifies, unfolding the functions REMAINDER and LESSP, to two
  new formulas:

  Case 5.2.
          (IMPLIES (AND (NOT (NUMBERP X))
                        (LESSP 0 X)
                        (NOT (EQUAL Y 0)))
                   (NOT (NUMBERP Y))),

    which again simplifies, opening up LESSP, to:

          T.

  Case 5.1.
          (IMPLIES (AND (NUMBERP X) (LESSP X X))
                   (NOT (LESSP X Y))),

    which again simplifies, using linear arithmetic, to:

          T.

Case 4. (IMPLIES (LESSP (REMAINDER X Y) X)
                 (NUMBERP X)),

  which again simplifies, expanding the functions LESSP and REMAINDER, to:

        T.

Case 3. (IMPLIES (LESSP (REMAINDER X Y) X)
                 (NOT (EQUAL X 0))),

  which again simplifies, using linear arithmetic, to:

        T.

Case 2. (IMPLIES (LESSP (REMAINDER X Y) X)
                 (NUMBERP Y)),

  which again simplifies, rewriting with REMAINDER-WRT-12, to the following
  two new conjectures:

  Case 2.2.
          (IMPLIES (AND (NOT (NUMBERP X)) (LESSP 0 X))
                   (NUMBERP Y)).

    However this again simplifies, unfolding the function LESSP, to:

          T.

  Case 2.1.
          (IMPLIES (AND (NUMBERP X) (LESSP X X))
                   (NUMBERP Y)),

    which again simplifies, using linear arithmetic, to:

          T.

Case 1. (IMPLIES (LESSP (REMAINDER X Y) X)
                 (NOT (EQUAL Y 0))),

  which again simplifies, opening up the definitions of EQUAL and REMAINDER,
  to two new formulas:

  Case 1.2.
          (IMPLIES (NOT (NUMBERP X))
                   (NOT (LESSP 0 X))),

    which again simplifies, unfolding the function LESSP, to:

          T.

  Case 1.1.
          (IMPLIES (NUMBERP X)
                   (NOT (LESSP X X))),

    which again simplifies, using linear arithmetic, to:

          T.

Q.E.D.


[ 0.0 0.0 0.0 ]

LESSP-REMAINDER1


(DISABLE LESSP-REMAINDER1)

[ 0.0 0.0 0.0 ]

LESSP-REMAINDER1-OFF


(PROVE-LEMMA DIFFERENCE-PLUS1
             (REWRITE)
             (EQUAL (DIFFERENCE (PLUS X Y) X)
                    (FIX Y))
             ((ENABLE DIFFERENCE-PLUS)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This conjecture simplifies, appealing to the lemma DIFFERENCE-PLUS, and
expanding the function FIX, to:

      T.

Q.E.D.


[ 0.0 0.0 0.0 ]

DIFFERENCE-PLUS1


(DISABLE DIFFERENCE-PLUS1)

[ 0.0 0.0 0.0 ]

DIFFERENCE-PLUS1-OFF


(PROVE-LEMMA DIFFERENCE-PLUS2
             (REWRITE)
             (EQUAL (DIFFERENCE (PLUS Y X) X)
                    (FIX Y))
             ((ENABLE COMMUTATIVITY-OF-PLUS DIFFERENCE-PLUS1)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This conjecture simplifies, appealing to the lemmas COMMUTATIVITY-OF-PLUS and
DIFFERENCE-PLUS1, and expanding the function FIX, to:

      T.

Q.E.D.


[ 0.0 0.0 0.0 ]

DIFFERENCE-PLUS2


(DISABLE DIFFERENCE-PLUS2)

[ 0.0 0.0 0.0 ]

DIFFERENCE-PLUS2-OFF


(PROVE-LEMMA DIFFERENCE-PLUS-CANCELATION
             (REWRITE)
             (EQUAL (DIFFERENCE (PLUS X Y) (PLUS X Z))
                    (DIFFERENCE Y Z))
             ((ENABLE DIFFERENCE-0)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This simplifies, using linear arithmetic, to the following two new conjectures:

Case 2. (IMPLIES (LESSP (PLUS X Y) (PLUS X Z))
                 (EQUAL (DIFFERENCE (PLUS X Y) (PLUS X Z))
                        (DIFFERENCE Y Z))).

  But this again simplifies, using linear arithmetic, applying DIFFERENCE-0,
  and opening up the function EQUAL, to:

        T.

Case 1. (IMPLIES (LESSP Y Z)
                 (EQUAL (DIFFERENCE (PLUS X Y) (PLUS X Z))
                        (DIFFERENCE Y Z))).

  But this again simplifies, using linear arithmetic, applying the lemma
  DIFFERENCE-0, and expanding EQUAL, to:

        T.

Q.E.D.


[ 0.0 0.0 0.0 ]

DIFFERENCE-PLUS-CANCELATION


(DISABLE DIFFERENCE-PLUS-CANCELATION)

[ 0.0 0.0 0.0 ]

DIFFERENCE-PLUS-CANCELATION-OFF


(PROVE-LEMMA TIMES-DIFFERENCE
             (REWRITE)
             (EQUAL (TIMES X (DIFFERENCE C W))
                    (DIFFERENCE (TIMES C X) (TIMES W X)))
             ((ENABLE PLUS-EQUAL-0 DIFFERENCE-0 EQUAL-DIFFERENCE-0
                      COMMUTATIVITY-OF-TIMES EQUAL-TIMES-0 DIFFERENCE-ELIM
                      DIFFERENCE-PLUS-CANCELATION)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

.

Appealing to the lemma DIFFERENCE-ELIM, we now replace C by (PLUS W Z) to
eliminate (DIFFERENCE C W).  We use the type restriction lemma noted when
DIFFERENCE was introduced to constrain the new variable.  The result is three
new goals:

Case 3. (IMPLIES (LESSP C W)
                 (EQUAL (TIMES X (DIFFERENCE C W))
                        (DIFFERENCE (TIMES C X)
                                    (TIMES W X)))),

  which simplifies, using linear arithmetic, appealing to the lemmas
  DIFFERENCE-0, COMMUTATIVITY-OF-TIMES, and EQUAL-DIFFERENCE-0, and unfolding
  EQUAL and TIMES, to:

        (IMPLIES (LESSP C W)
                 (NOT (LESSP (TIMES W X) (TIMES C X)))),

  which we would usually push and work on later by induction.  But if we must
  use induction to prove the input conjecture, we prefer to induct on the
  original formulation of the problem.  Thus we will disregard all that we
  have previously done, give the name *1 to the original input, and work on it.


     So now let us consider:

(EQUAL (TIMES X (DIFFERENCE C W))
       (DIFFERENCE (TIMES C X) (TIMES W X))),

which we named *1 above.  We will appeal to induction.  There are five
plausible inductions.  They merge into two likely candidate inductions.
However, only one is unflawed.  We will induct according to the following
scheme:
      (AND (IMPLIES (ZEROP C) (p X C W))
           (IMPLIES (AND (NOT (ZEROP C)) (ZEROP W))
                    (p X C W))
           (IMPLIES (AND (NOT (ZEROP C))
                         (NOT (ZEROP W))
                         (p X (SUB1 C) (SUB1 W)))
                    (p X C W))).
Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP
establish that the measure (COUNT C) decreases according to the well-founded
relation LESSP in each induction step of the scheme.  Note, however, the
inductive instance chosen for W.  The above induction scheme produces the
following three new goals:

Case 3. (IMPLIES (ZEROP C)
                 (EQUAL (TIMES X (DIFFERENCE C W))
                        (DIFFERENCE (TIMES C X)
                                    (TIMES W X)))).

  This simplifies, using linear arithmetic, rewriting with DIFFERENCE-0 and
  COMMUTATIVITY-OF-TIMES, and unfolding the functions ZEROP, EQUAL, and TIMES,
  to:

        T.

Case 2. (IMPLIES (AND (NOT (ZEROP C)) (ZEROP W))
                 (EQUAL (TIMES X (DIFFERENCE C W))
                        (DIFFERENCE (TIMES C X)
                                    (TIMES W X)))),

  which simplifies, appealing to the lemmas COMMUTATIVITY-OF-TIMES,
  PLUS-EQUAL-0, and EQUAL-TIMES-0, and unfolding the functions ZEROP, EQUAL,
  DIFFERENCE, and TIMES, to four new conjectures:

  Case 2.4.
          (IMPLIES (AND (NOT (EQUAL C 0))
                        (NUMBERP C)
                        (EQUAL W 0)
                        (NOT (NUMBERP X)))
                   (EQUAL (PLUS X (TIMES (SUB1 C) X))
                          0)),

    which again simplifies, applying EQUAL-TIMES-0, and opening up the
    function PLUS, to:

          T.

  Case 2.3.
          (IMPLIES (AND (NOT (EQUAL C 0))
                        (NUMBERP C)
                        (EQUAL W 0)
                        (EQUAL X 0))
                   (EQUAL (PLUS X (TIMES (SUB1 C) X))
                          0)).

    But this again simplifies, applying COMMUTATIVITY-OF-TIMES, and expanding
    the functions EQUAL, TIMES, and PLUS, to:

          T.

  Case 2.2.
          (IMPLIES (AND (NOT (EQUAL C 0))
                        (NUMBERP C)
                        (NOT (NUMBERP W))
                        (NOT (NUMBERP X)))
                   (EQUAL (PLUS X (TIMES (SUB1 C) X))
                          0)).

    This again simplifies, applying EQUAL-TIMES-0, and unfolding the
    definition of PLUS, to:

          T.

  Case 2.1.
          (IMPLIES (AND (NOT (EQUAL C 0))
                        (NUMBERP C)
                        (NOT (NUMBERP W))
                        (EQUAL X 0))
                   (EQUAL (PLUS X (TIMES (SUB1 C) X))
                          0)).

    However this again simplifies, rewriting with COMMUTATIVITY-OF-TIMES, and
    expanding the definitions of EQUAL, TIMES, and PLUS, to:

          T.

Case 1. (IMPLIES (AND (NOT (ZEROP C))
                      (NOT (ZEROP W))
                      (EQUAL (TIMES X
                                    (DIFFERENCE (SUB1 C) (SUB1 W)))
                             (DIFFERENCE (TIMES (SUB1 C) X)
                                         (TIMES (SUB1 W) X))))
                 (EQUAL (TIMES X (DIFFERENCE C W))
                        (DIFFERENCE (TIMES C X)
                                    (TIMES W X)))).

  This simplifies, rewriting with DIFFERENCE-PLUS-CANCELATION, and unfolding
  the definitions of ZEROP, DIFFERENCE, and TIMES, to:

        T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.0 0.0 ]

TIMES-DIFFERENCE


(DISABLE TIMES-DIFFERENCE)

[ 0.0 0.0 0.0 ]

TIMES-DIFFERENCE-OFF


(DEFN DIVIDES
      (X Y)
      (ZEROP (REMAINDER Y X))
      NIL)

     Note that (OR (FALSEP (DIVIDES X Y)) (TRUEP (DIVIDES X Y))) is a theorem.




[ 0.0 0.0 0.0 ]

DIVIDES


(PROVE-LEMMA DIVIDES-TIMES
             (REWRITE)
             (EQUAL (REMAINDER (TIMES X Z) Z) '0)
             ((ENABLE COMMUTATIVITY-OF-TIMES EQUAL-TIMES-0 REMAINDER-WRT-12
                      DIFFERENCE-PLUS1)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))


     Name the conjecture *1.


     Perhaps we can prove it by induction.  There is only one plausible
induction.  We will induct according to the following scheme:
      (AND (IMPLIES (ZEROP X) (p X Z))
           (IMPLIES (AND (NOT (ZEROP X)) (p (SUB1 X) Z))
                    (p X Z))).
Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP
establish that the measure (COUNT X) decreases according to the well-founded
relation LESSP in each induction step of the scheme.  The above induction
scheme produces the following two new goals:

Case 2. (IMPLIES (ZEROP X)
                 (EQUAL (REMAINDER (TIMES X Z) Z) 0)).

  This simplifies, expanding ZEROP, EQUAL, TIMES, LESSP, NUMBERP, and
  REMAINDER, to:

        T.

Case 1. (IMPLIES (AND (NOT (ZEROP X))
                      (EQUAL (REMAINDER (TIMES (SUB1 X) Z) Z)
                             0))
                 (EQUAL (REMAINDER (TIMES X Z) Z) 0)).

  This simplifies, rewriting with DIFFERENCE-PLUS1, and unfolding the
  definitions of ZEROP, TIMES, and REMAINDER, to three new goals:

  Case 1.3.
          (IMPLIES (AND (NOT (EQUAL X 0))
                        (NUMBERP X)
                        (EQUAL (REMAINDER (TIMES (SUB1 X) Z) Z)
                               0)
                        (NOT (NUMBERP Z)))
                   (EQUAL (PLUS Z (TIMES (SUB1 X) Z))
                          0)),

    which again simplifies, applying REMAINDER-WRT-12 and EQUAL-TIMES-0, and
    unfolding the function PLUS, to:

          T.

  Case 1.2.
          (IMPLIES (AND (NOT (EQUAL X 0))
                        (NUMBERP X)
                        (EQUAL (REMAINDER (TIMES (SUB1 X) Z) Z)
                               0)
                        (EQUAL Z 0))
                   (EQUAL (PLUS Z (TIMES (SUB1 X) Z))
                          0)).

    However this again simplifies, applying COMMUTATIVITY-OF-TIMES, and
    unfolding the functions EQUAL, TIMES, REMAINDER, and PLUS, to:

          T.

  Case 1.1.
          (IMPLIES (AND (NOT (EQUAL X 0))
                        (NUMBERP X)
                        (EQUAL (REMAINDER (TIMES (SUB1 X) Z) Z)
                               0)
                        (LESSP (PLUS Z (TIMES (SUB1 X) Z)) Z))
                   (EQUAL (PLUS Z (TIMES (SUB1 X) Z))
                          0)).

    However this again simplifies, using linear arithmetic, to:

          T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.0 0.0 ]

DIVIDES-TIMES


(DISABLE DIVIDES-TIMES)

[ 0.0 0.0 0.0 ]

DIVIDES-TIMES-OFF


(PROVE-LEMMA DIFFERENCE-PLUS3
             (REWRITE)
             (EQUAL (DIFFERENCE (PLUS B (PLUS A C)) A)
                    (PLUS B C))
             ((ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This simplifies, using linear arithmetic, to:

      (IMPLIES (LESSP (PLUS B A C) A)
               (EQUAL (DIFFERENCE (PLUS B A C) A)
                      (PLUS B C))),

which again simplifies, using linear arithmetic, to:

      T.

Q.E.D.


[ 0.0 0.0 0.0 ]

DIFFERENCE-PLUS3


(DISABLE DIFFERENCE-PLUS3)

[ 0.0 0.0 0.0 ]

DIFFERENCE-PLUS3-OFF


(PROVE-LEMMA DIFFERENCE-ADD1-CANCELLATION
             (REWRITE)
             (EQUAL (DIFFERENCE (ADD1 (PLUS Y Z)) Z)
                    (ADD1 Y))
             ((ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This conjecture simplifies, using linear arithmetic, to:

      (IMPLIES (LESSP (ADD1 (PLUS Y Z)) Z)
               (EQUAL (DIFFERENCE (ADD1 (PLUS Y Z)) Z)
                      (ADD1 Y))),

which again simplifies, using linear arithmetic, to:

      T.

Q.E.D.


[ 0.0 0.0 0.0 ]

DIFFERENCE-ADD1-CANCELLATION


(DISABLE DIFFERENCE-ADD1-CANCELLATION)

[ 0.0 0.0 0.0 ]

DIFFERENCE-ADD1-CANCELLATION-OFF


(PROVE-LEMMA REMAINDER-ADD1
             (REWRITE)
             (IMPLIES (AND (NOT (ZEROP Y))
                           (NOT (EQUAL Y '1)))
                      (NOT (EQUAL (REMAINDER (ADD1 (TIMES X Y)) Y)
                                  '0)))
             ((ENABLE COMMUTATIVITY-OF-PLUS DIFFERENCE-0
                      COMMUTATIVITY-OF-TIMES DIFFERENCE-ADD1-CANCELLATION)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This conjecture can be simplified, using the abbreviations ZEROP, NOT, AND,
and IMPLIES, to:

      (IMPLIES (AND (NOT (EQUAL Y 0))
                    (NUMBERP Y)
                    (NOT (EQUAL Y 1)))
               (NOT (EQUAL (REMAINDER (ADD1 (TIMES X Y)) Y)
                           0))).

Name the above subgoal *1.


     We will appeal to induction.  There is only one plausible induction.  We
will induct according to the following scheme:
      (AND (IMPLIES (ZEROP X) (p X Y))
           (IMPLIES (AND (NOT (ZEROP X)) (p (SUB1 X) Y))
                    (p X Y))).
Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP inform
us that the measure (COUNT X) decreases according to the well-founded relation
LESSP in each induction step of the scheme.  The above induction scheme leads
to the following two new formulas:

Case 2. (IMPLIES (AND (ZEROP X)
                      (NOT (EQUAL Y 0))
                      (NUMBERP Y)
                      (NOT (EQUAL Y 1)))
                 (NOT (EQUAL (REMAINDER (ADD1 (TIMES X Y)) Y)
                             0))).

  This simplifies, using linear arithmetic, applying the lemma DIFFERENCE-0,
  and opening up the definitions of ZEROP, EQUAL, TIMES, ADD1, NUMBERP,
  REMAINDER, and LESSP, to the following two new goals:

  Case 2.2.
          (IMPLIES (AND (EQUAL X 0)
                        (NOT (EQUAL Y 0))
                        (NUMBERP Y)
                        (NOT (EQUAL Y 1)))
                   (LESSP 1 Y)).

    This again simplifies, using linear arithmetic, to:

          T.

  Case 2.1.
          (IMPLIES (AND (NOT (NUMBERP X))
                        (NOT (EQUAL Y 0))
                        (NUMBERP Y)
                        (NOT (EQUAL Y 1)))
                   (LESSP 1 Y)),

    which again simplifies, using linear arithmetic, to:

          T.

Case 1. (IMPLIES (AND (NOT (ZEROP X))
                      (NOT (EQUAL (REMAINDER (ADD1 (TIMES (SUB1 X) Y))
                                             Y)
                                  0))
                      (NOT (EQUAL Y 0))
                      (NUMBERP Y)
                      (NOT (EQUAL Y 1)))
                 (NOT (EQUAL (REMAINDER (ADD1 (TIMES X Y)) Y)
                             0))),

  which simplifies, expanding the functions ZEROP and TIMES, to:

        (IMPLIES (AND (NOT (EQUAL X 0))
                      (NUMBERP X)
                      (NOT (EQUAL (REMAINDER (ADD1 (TIMES (SUB1 X) Y))
                                             Y)
                                  0))
                      (NOT (EQUAL Y 0))
                      (NUMBERP Y)
                      (NOT (EQUAL Y 1)))
                 (NOT (EQUAL (REMAINDER (ADD1 (PLUS Y (TIMES (SUB1 X) Y)))
                                        Y)
                             0))).

  But this further simplifies, rewriting with the lemma COMMUTATIVITY-OF-TIMES,
  to:

        (IMPLIES (AND (NOT (EQUAL X 0))
                      (NUMBERP X)
                      (NOT (EQUAL (REMAINDER (ADD1 (TIMES Y (SUB1 X)))
                                             Y)
                                  0))
                      (NOT (EQUAL Y 0))
                      (NUMBERP Y)
                      (NOT (EQUAL Y 1)))
                 (NOT (EQUAL (REMAINDER (ADD1 (PLUS Y (TIMES Y (SUB1 X))))
                                        Y)
                             0))).

  Appealing to the lemma SUB1-ELIM, we now replace X by (ADD1 Z) to eliminate
  (SUB1 X).  We employ the type restriction lemma noted when SUB1 was
  introduced to constrain the new variable.  We must thus prove the conjecture:

        (IMPLIES (AND (NUMBERP Z)
                      (NOT (EQUAL (ADD1 Z) 0))
                      (NOT (EQUAL (REMAINDER (ADD1 (TIMES Y Z)) Y)
                                  0))
                      (NOT (EQUAL Y 0))
                      (NUMBERP Y)
                      (NOT (EQUAL Y 1)))
                 (NOT (EQUAL (REMAINDER (ADD1 (PLUS Y (TIMES Y Z)))
                                        Y)
                             0))).

  This further simplifies, trivially, to the new goal:

        (IMPLIES (AND (NUMBERP Z)
                      (NOT (EQUAL (REMAINDER (ADD1 (TIMES Y Z)) Y)
                                  0))
                      (NOT (EQUAL Y 0))
                      (NUMBERP Y)
                      (NOT (EQUAL Y 1)))
                 (NOT (EQUAL (REMAINDER (ADD1 (PLUS Y (TIMES Y Z)))
                                        Y)
                             0))),

  which we generalize by replacing (TIMES Y Z) by A.  We restrict the new
  variable by recalling the type restriction lemma noted when TIMES was
  introduced.  We would thus like to prove:

        (IMPLIES (AND (NUMBERP A)
                      (NUMBERP Z)
                      (NOT (EQUAL (REMAINDER (ADD1 A) Y) 0))
                      (NOT (EQUAL Y 0))
                      (NUMBERP Y)
                      (NOT (EQUAL Y 1)))
                 (NOT (EQUAL (REMAINDER (ADD1 (PLUS Y A)) Y)
                             0))),

  which further simplifies, appealing to the lemmas COMMUTATIVITY-OF-PLUS,
  DIFFERENCE-ADD1-CANCELLATION, and SUB1-ADD1, and unfolding LESSP and
  REMAINDER, to:

        (IMPLIES (AND (NUMBERP A)
                      (NUMBERP Z)
                      (NOT (EQUAL (REMAINDER (ADD1 A) Y) 0))
                      (NOT (EQUAL Y 0))
                      (NUMBERP Y)
                      (NOT (EQUAL Y 1))
                      (LESSP (PLUS A Y) (SUB1 Y)))
                 (NOT (EQUAL (ADD1 (PLUS A Y)) 0))).

  This finally simplifies, using linear arithmetic, to:

        T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.1 0.0 ]

REMAINDER-ADD1


(DISABLE REMAINDER-ADD1)

[ 0.0 0.0 0.0 ]

REMAINDER-ADD1-OFF


(PROVE-LEMMA DIVIDES-PLUS-REWRITE1
             (REWRITE)
             (IMPLIES (AND (EQUAL (REMAINDER X Z) '0)
                           (EQUAL (REMAINDER Y Z) '0))
                      (EQUAL (REMAINDER (PLUS X Y) Z) '0))
             ((ENABLE PLUS-RIGHT-ID2 COMMUTATIVITY2-OF-PLUS
                      COMMUTATIVITY-OF-PLUS ASSOCIATIVITY-OF-PLUS
                      COMMUTATIVITY-OF-TIMES DIFFERENCE-ELIM REMAINDER-WRT-12
                      LESSP-REMAINDER2 REMAINDER-QUOTIENT-ELIM DIVIDES-TIMES
                      DIFFERENCE-PLUS3)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

.

Applying the lemma REMAINDER-QUOTIENT-ELIM, replace X by (PLUS V (TIMES Z W))
to eliminate (REMAINDER X Z) and (QUOTIENT X Z).  We employ LESSP-REMAINDER2,
the type restriction lemma noted when REMAINDER was introduced, and the type
restriction lemma noted when QUOTIENT was introduced to restrict the new
variables.  We would thus like to prove the following four new conjectures:

Case 4. (IMPLIES (AND (NOT (NUMBERP X))
                      (EQUAL (REMAINDER X Z) 0)
                      (EQUAL (REMAINDER Y Z) 0))
                 (EQUAL (REMAINDER (PLUS X Y) Z) 0)).

  However this simplifies, expanding LESSP, REMAINDER, EQUAL, and PLUS, to the
  conjecture:

        (IMPLIES (AND (NOT (NUMBERP X))
                      (EQUAL (REMAINDER Y Z) 0)
                      (NOT (NUMBERP Y)))
                 (EQUAL (REMAINDER 0 Z) 0)).

  But this again simplifies, expanding the definitions of LESSP, REMAINDER,
  EQUAL, and NUMBERP, to:

        T.

Case 3. (IMPLIES (AND (EQUAL Z 0)
                      (EQUAL (REMAINDER X Z) 0)
                      (EQUAL (REMAINDER Y Z) 0))
                 (EQUAL (REMAINDER (PLUS X Y) Z) 0)),

  which simplifies, applying PLUS-RIGHT-ID2 and COMMUTATIVITY-OF-PLUS, and
  expanding the definitions of EQUAL, REMAINDER, PLUS, and NUMBERP, to:

        T.

Case 2. (IMPLIES (AND (NOT (NUMBERP Z))
                      (EQUAL (REMAINDER X Z) 0)
                      (EQUAL (REMAINDER Y Z) 0))
                 (EQUAL (REMAINDER (PLUS X Y) Z) 0)).

  But this simplifies, applying REMAINDER-WRT-12, PLUS-RIGHT-ID2, and
  COMMUTATIVITY-OF-PLUS, and opening up the definitions of PLUS, NUMBERP, and
  EQUAL, to:

        T.

Case 1. (IMPLIES (AND (NUMBERP V)
                      (EQUAL (LESSP V Z) (NOT (ZEROP Z)))
                      (NUMBERP W)
                      (NUMBERP Z)
                      (NOT (EQUAL Z 0))
                      (EQUAL V 0)
                      (EQUAL (REMAINDER Y Z) 0))
                 (EQUAL (REMAINDER (PLUS (PLUS V (TIMES Z W)) Y)
                                   Z)
                        0)).

  However this simplifies, applying COMMUTATIVITY-OF-TIMES and
  COMMUTATIVITY-OF-PLUS, and unfolding the functions NUMBERP, EQUAL, LESSP,
  ZEROP, NOT, and PLUS, to the new formula:

        (IMPLIES (AND (NUMBERP W)
                      (NUMBERP Z)
                      (NOT (EQUAL Z 0))
                      (EQUAL (REMAINDER Y Z) 0))
                 (EQUAL (REMAINDER (PLUS Y (TIMES W Z)) Z)
                        0)).

  Applying the lemma REMAINDER-QUOTIENT-ELIM, replace Y by
  (PLUS V (TIMES Z D)) to eliminate (REMAINDER Y Z) and (QUOTIENT Y Z).  We
  rely upon LESSP-REMAINDER2, the type restriction lemma noted when REMAINDER
  was introduced, and the type restriction lemma noted when QUOTIENT was
  introduced to restrict the new variables.  This produces the following two
  new formulas:

  Case 1.2.
          (IMPLIES (AND (NOT (NUMBERP Y))
                        (NUMBERP W)
                        (NUMBERP Z)
                        (NOT (EQUAL Z 0))
                        (EQUAL (REMAINDER Y Z) 0))
                   (EQUAL (REMAINDER (PLUS Y (TIMES W Z)) Z)
                          0)).

    However this further simplifies, rewriting with DIVIDES-TIMES, and opening
    up LESSP, REMAINDER, EQUAL, and PLUS, to:

          T.

  Case 1.1.
          (IMPLIES (AND (NUMBERP V)
                        (EQUAL (LESSP V Z) (NOT (ZEROP Z)))
                        (NUMBERP D)
                        (NUMBERP W)
                        (NUMBERP Z)
                        (NOT (EQUAL Z 0))
                        (EQUAL V 0))
                   (EQUAL (REMAINDER (PLUS (PLUS V (TIMES Z D))
                                           (TIMES W Z))
                                     Z)
                          0)).

    However this further simplifies, applying COMMUTATIVITY-OF-TIMES, and
    opening up NUMBERP, EQUAL, LESSP, ZEROP, NOT, and PLUS, to:

          (IMPLIES (AND (NUMBERP D)
                        (NUMBERP W)
                        (NUMBERP Z)
                        (NOT (EQUAL Z 0)))
                   (EQUAL (REMAINDER (PLUS (TIMES D Z) (TIMES W Z))
                                     Z)
                          0)),

    which we would normally push and work on later by induction.  But if we
    must use induction to prove the input conjecture, we prefer to induct on
    the original formulation of the problem.  Thus we will disregard all that
    we have previously done, give the name *1 to the original input, and work
    on it.


     So now let us return to:

(IMPLIES (AND (EQUAL (REMAINDER X Z) 0)
              (EQUAL (REMAINDER Y Z) 0))
         (EQUAL (REMAINDER (PLUS X Y) Z) 0)).

We named this *1.  We will try to prove it by induction.  There are three
plausible inductions, two of which are unflawed.  So we will choose the one
suggested by the largest number of nonprimitive recursive functions.  We will
induct according to the following scheme:
      (AND (IMPLIES (ZEROP Z) (p X Y Z))
           (IMPLIES (AND (NOT (ZEROP Z)) (LESSP X Z))
                    (p X Y Z))
           (IMPLIES (AND (NOT (ZEROP Z))
                         (NOT (LESSP X Z))
                         (p (DIFFERENCE X Z) Y Z))
                    (p X Y Z))).
Linear arithmetic, the lemmas COUNT-NUMBERP and COUNT-NOT-LESSP, and the
definition of ZEROP inform us that the measure (COUNT X) decreases according
to the well-founded relation LESSP in each induction step of the scheme.  The
above induction scheme produces four new goals:

Case 4. (IMPLIES (AND (ZEROP Z)
                      (EQUAL (REMAINDER X Z) 0)
                      (EQUAL (REMAINDER Y Z) 0))
                 (EQUAL (REMAINDER (PLUS X Y) Z) 0)),

  which simplifies, rewriting with the lemmas PLUS-RIGHT-ID2,
  COMMUTATIVITY-OF-PLUS, and REMAINDER-WRT-12, and expanding ZEROP, EQUAL,
  REMAINDER, PLUS, and NUMBERP, to:

        T.

Case 3. (IMPLIES (AND (NOT (ZEROP Z))
                      (LESSP X Z)
                      (EQUAL (REMAINDER X Z) 0)
                      (EQUAL (REMAINDER Y Z) 0))
                 (EQUAL (REMAINDER (PLUS X Y) Z) 0)),

  which simplifies, unfolding ZEROP, REMAINDER, EQUAL, and PLUS, to two new
  conjectures:

  Case 3.2.
          (IMPLIES (AND (NOT (EQUAL Z 0))
                        (NUMBERP Z)
                        (LESSP X Z)
                        (EQUAL X 0)
                        (EQUAL (REMAINDER Y Z) 0)
                        (NOT (NUMBERP Y)))
                   (EQUAL (REMAINDER 0 Z) 0)),

    which again simplifies, unfolding EQUAL, LESSP, REMAINDER, and NUMBERP, to:

          T.

  Case 3.1.
          (IMPLIES (AND (NOT (EQUAL Z 0))
                        (NUMBERP Z)
                        (LESSP X Z)
                        (NOT (NUMBERP X))
                        (EQUAL (REMAINDER Y Z) 0)
                        (NOT (NUMBERP Y)))
                   (EQUAL (REMAINDER 0 Z) 0)),

    which again simplifies, unfolding the functions LESSP, REMAINDER, EQUAL,
    and NUMBERP, to:

          T.

Case 2. (IMPLIES (AND (NOT (ZEROP Z))
                      (NOT (LESSP X Z))
                      (NOT (EQUAL (REMAINDER (DIFFERENCE X Z) Z)
                                  0))
                      (EQUAL (REMAINDER X Z) 0)
                      (EQUAL (REMAINDER Y Z) 0))
                 (EQUAL (REMAINDER (PLUS X Y) Z) 0)),

  which simplifies, unfolding the definitions of ZEROP and REMAINDER, to:

        T.

Case 1. (IMPLIES (AND (NOT (ZEROP Z))
                      (NOT (LESSP X Z))
                      (EQUAL (REMAINDER (PLUS (DIFFERENCE X Z) Y)
                                        Z)
                             0)
                      (EQUAL (REMAINDER X Z) 0)
                      (EQUAL (REMAINDER Y Z) 0))
                 (EQUAL (REMAINDER (PLUS X Y) Z) 0)),

  which simplifies, rewriting with COMMUTATIVITY-OF-PLUS, and opening up ZEROP
  and REMAINDER, to:

        (IMPLIES (AND (NOT (EQUAL Z 0))
                      (NUMBERP Z)
                      (NOT (LESSP X Z))
                      (EQUAL (REMAINDER (PLUS Y (DIFFERENCE X Z))
                                        Z)
                             0)
                      (EQUAL (REMAINDER (DIFFERENCE X Z) Z)
                             0)
                      (EQUAL (REMAINDER Y Z) 0))
                 (EQUAL (REMAINDER (PLUS X Y) Z) 0)).

  Applying the lemmas DIFFERENCE-ELIM and REMAINDER-QUOTIENT-ELIM, replace X
  by (PLUS Z V) to eliminate (DIFFERENCE X Z) and V by (PLUS W (TIMES Z D)) to
  eliminate (REMAINDER V Z) and (QUOTIENT V Z).  We use the type restriction
  lemma noted when DIFFERENCE was introduced, LESSP-REMAINDER2, the type
  restriction lemma noted when REMAINDER was introduced, and the type
  restriction lemma noted when QUOTIENT was introduced to restrict the new
  variables.  This produces the following two new conjectures:

  Case 1.2.
          (IMPLIES (AND (NOT (NUMBERP X))
                        (NOT (EQUAL Z 0))
                        (NUMBERP Z)
                        (NOT (LESSP X Z))
                        (EQUAL (REMAINDER (PLUS Y (DIFFERENCE X Z))
                                          Z)
                               0)
                        (EQUAL (REMAINDER (DIFFERENCE X Z) Z)
                               0)
                        (EQUAL (REMAINDER Y Z) 0))
                   (EQUAL (REMAINDER (PLUS X Y) Z) 0)).

    This further simplifies, unfolding the function LESSP, to:

          T.

  Case 1.1.
          (IMPLIES (AND (NUMBERP W)
                        (EQUAL (LESSP W Z) (NOT (ZEROP Z)))
                        (NUMBERP D)
                        (NOT (EQUAL Z 0))
                        (NUMBERP Z)
                        (NOT (LESSP (PLUS Z W (TIMES Z D)) Z))
                        (EQUAL (REMAINDER (PLUS Y W (TIMES Z D)) Z)
                               0)
                        (EQUAL W 0)
                        (EQUAL (REMAINDER Y Z) 0))
                   (EQUAL (REMAINDER (PLUS (PLUS Z W (TIMES Z D)) Y)
                                     Z)
                          0)),

    which further simplifies, rewriting with the lemmas COMMUTATIVITY-OF-TIMES,
    COMMUTATIVITY-OF-PLUS, COMMUTATIVITY2-OF-PLUS, ASSOCIATIVITY-OF-PLUS, and
    DIFFERENCE-PLUS3, and unfolding the functions NUMBERP, EQUAL, LESSP, ZEROP,
    NOT, PLUS, and REMAINDER, to:

          (IMPLIES (AND (NUMBERP D)
                        (NOT (EQUAL Z 0))
                        (NUMBERP Z)
                        (NOT (LESSP (PLUS Z (TIMES D Z)) Z))
                        (EQUAL (REMAINDER (PLUS Y (TIMES D Z)) Z)
                               0)
                        (EQUAL (REMAINDER Y Z) 0)
                        (LESSP (PLUS Y Z (TIMES D Z)) Z))
                   (EQUAL (PLUS Y Z (TIMES D Z)) 0)).

    However this again simplifies, using linear arithmetic, to:

          T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.1 0.0 ]

DIVIDES-PLUS-REWRITE1


(DISABLE DIVIDES-PLUS-REWRITE1)

[ 0.0 0.0 0.0 ]

DIVIDES-PLUS-REWRITE1-OFF


(PROVE-LEMMA DIVIDES-PLUS-REWRITE2
             (REWRITE)
             (IMPLIES (AND (EQUAL (REMAINDER X Z) '0)
                           (NOT (EQUAL (REMAINDER Y Z) '0)))
                      (NOT (EQUAL (REMAINDER (PLUS X Y) Z) '0)))
             ((ENABLE COMMUTATIVITY2-OF-PLUS COMMUTATIVITY-OF-PLUS
                      ASSOCIATIVITY-OF-PLUS COMMUTATIVITY-OF-TIMES
                      DIFFERENCE-ELIM REMAINDER-WRT-12 LESSP-REMAINDER2
                      REMAINDER-QUOTIENT-ELIM DIFFERENCE-PLUS3)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

.

Applying the lemma REMAINDER-QUOTIENT-ELIM, replace X by (PLUS V (TIMES Z W))
to eliminate (REMAINDER X Z) and (QUOTIENT X Z).  We use LESSP-REMAINDER2, the
type restriction lemma noted when REMAINDER was introduced, and the type
restriction lemma noted when QUOTIENT was introduced to restrict the new
variables.  We would thus like to prove the following four new formulas:

Case 4. (IMPLIES (AND (NOT (NUMBERP X))
                      (EQUAL (REMAINDER X Z) 0)
                      (NOT (EQUAL (REMAINDER Y Z) 0)))
                 (NOT (EQUAL (REMAINDER (PLUS X Y) Z) 0))).

  But this simplifies, unfolding the definitions of LESSP, REMAINDER, EQUAL,
  and PLUS, to:

        (IMPLIES (AND (NOT (NUMBERP X))
                      (NOT (EQUAL (REMAINDER Y Z) 0))
                      (NOT (NUMBERP Y)))
                 (NOT (EQUAL (REMAINDER 0 Z) 0))).

  This again simplifies, expanding the functions LESSP, REMAINDER, and EQUAL,
  to:

        T.

Case 3. (IMPLIES (AND (EQUAL Z 0)
                      (EQUAL (REMAINDER X Z) 0)
                      (NOT (EQUAL (REMAINDER Y Z) 0)))
                 (NOT (EQUAL (REMAINDER (PLUS X Y) Z) 0))),

  which simplifies, expanding the functions EQUAL, REMAINDER, and PLUS, to:

        T.

Case 2. (IMPLIES (AND (NOT (NUMBERP Z))
                      (EQUAL (REMAINDER X Z) 0)
                      (NOT (EQUAL (REMAINDER Y Z) 0)))
                 (NOT (EQUAL (REMAINDER (PLUS X Y) Z) 0))),

  which simplifies, rewriting with REMAINDER-WRT-12, and expanding EQUAL and
  PLUS, to:

        T.

Case 1. (IMPLIES (AND (NUMBERP V)
                      (EQUAL (LESSP V Z) (NOT (ZEROP Z)))
                      (NUMBERP W)
                      (NUMBERP Z)
                      (NOT (EQUAL Z 0))
                      (EQUAL V 0)
                      (NOT (EQUAL (REMAINDER Y Z) 0)))
                 (NOT (EQUAL (REMAINDER (PLUS (PLUS V (TIMES Z W)) Y)
                                        Z)
                             0))).

  This simplifies, rewriting with COMMUTATIVITY-OF-TIMES and
  COMMUTATIVITY-OF-PLUS, and expanding the functions NUMBERP, EQUAL, LESSP,
  ZEROP, NOT, and PLUS, to:

        (IMPLIES (AND (NUMBERP W)
                      (NUMBERP Z)
                      (NOT (EQUAL Z 0))
                      (NOT (EQUAL (REMAINDER Y Z) 0)))
                 (NOT (EQUAL (REMAINDER (PLUS Y (TIMES W Z)) Z)
                             0))).

  Applying the lemma REMAINDER-QUOTIENT-ELIM, replace Y by
  (PLUS V (TIMES Z D)) to eliminate (REMAINDER Y Z) and (QUOTIENT Y Z).  We
  rely upon LESSP-REMAINDER2, the type restriction lemma noted when REMAINDER
  was introduced, and the type restriction lemma noted when QUOTIENT was
  introduced to restrict the new variables.  We thus obtain the following two
  new conjectures:

  Case 1.2.
          (IMPLIES (AND (NOT (NUMBERP Y))
                        (NUMBERP W)
                        (NUMBERP Z)
                        (NOT (EQUAL Z 0))
                        (NOT (EQUAL (REMAINDER Y Z) 0)))
                   (NOT (EQUAL (REMAINDER (PLUS Y (TIMES W Z)) Z)
                               0))).

    This further simplifies, unfolding the definitions of LESSP, REMAINDER,
    and EQUAL, to:

          T.

  Case 1.1.
          (IMPLIES (AND (NUMBERP V)
                        (EQUAL (LESSP V Z) (NOT (ZEROP Z)))
                        (NUMBERP D)
                        (NUMBERP W)
                        (NUMBERP Z)
                        (NOT (EQUAL Z 0))
                        (NOT (EQUAL V 0)))
                   (NOT (EQUAL (REMAINDER (PLUS (PLUS V (TIMES Z D))
                                                (TIMES W Z))
                                          Z)
                               0))),

    which further simplifies, rewriting with COMMUTATIVITY-OF-TIMES and
    ASSOCIATIVITY-OF-PLUS, and opening up ZEROP and NOT, to:

          (IMPLIES (AND (NUMBERP V)
                        (LESSP V Z)
                        (NUMBERP D)
                        (NUMBERP W)
                        (NUMBERP Z)
                        (NOT (EQUAL Z 0))
                        (NOT (EQUAL V 0)))
                   (NOT (EQUAL (REMAINDER (PLUS V (TIMES D Z) (TIMES W Z))
                                          Z)
                               0))),

    which we would normally push and work on later by induction.  But if we
    must use induction to prove the input conjecture, we prefer to induct on
    the original formulation of the problem.  Thus we will disregard all that
    we have previously done, give the name *1 to the original input, and work
    on it.


     So now let us return to:

(IMPLIES (AND (EQUAL (REMAINDER X Z) 0)
              (NOT (EQUAL (REMAINDER Y Z) 0)))
         (NOT (EQUAL (REMAINDER (PLUS X Y) Z) 0))).

We named this *1.  We will try to prove it by induction.  Three inductions are
suggested by terms in the conjecture, two of which are unflawed.  So we will
choose the one suggested by the largest number of nonprimitive recursive
functions.  We will induct according to the following scheme:
      (AND (IMPLIES (ZEROP Z) (p X Y Z))
           (IMPLIES (AND (NOT (ZEROP Z)) (LESSP X Z))
                    (p X Y Z))
           (IMPLIES (AND (NOT (ZEROP Z))
                         (NOT (LESSP X Z))
                         (p (DIFFERENCE X Z) Y Z))
                    (p X Y Z))).
Linear arithmetic, the lemmas COUNT-NUMBERP and COUNT-NOT-LESSP, and the
definition of ZEROP inform us that the measure (COUNT X) decreases according
to the well-founded relation LESSP in each induction step of the scheme.  The
above induction scheme generates four new formulas:

Case 4. (IMPLIES (AND (ZEROP Z)
                      (EQUAL (REMAINDER X Z) 0)
                      (NOT (EQUAL (REMAINDER Y Z) 0)))
                 (NOT (EQUAL (REMAINDER (PLUS X Y) Z) 0))),

  which simplifies, rewriting with the lemma REMAINDER-WRT-12, and unfolding
  the definitions of ZEROP, EQUAL, REMAINDER, and PLUS, to:

        T.

Case 3. (IMPLIES (AND (NOT (ZEROP Z))
                      (LESSP X Z)
                      (EQUAL (REMAINDER X Z) 0)
                      (NOT (EQUAL (REMAINDER Y Z) 0)))
                 (NOT (EQUAL (REMAINDER (PLUS X Y) Z) 0))),

  which simplifies, expanding the functions ZEROP, REMAINDER, EQUAL, and PLUS,
  to two new goals:

  Case 3.2.
          (IMPLIES (AND (NOT (EQUAL Z 0))
                        (NUMBERP Z)
                        (LESSP X Z)
                        (EQUAL X 0)
                        (NOT (EQUAL (REMAINDER Y Z) 0))
                        (NOT (NUMBERP Y)))
                   (NOT (EQUAL (REMAINDER 0 Z) 0))),

    which again simplifies, opening up EQUAL, LESSP, and REMAINDER, to:

          T.

  Case 3.1.
          (IMPLIES (AND (NOT (EQUAL Z 0))
                        (NUMBERP Z)
                        (LESSP X Z)
                        (NOT (NUMBERP X))
                        (NOT (EQUAL (REMAINDER Y Z) 0))
                        (NOT (NUMBERP Y)))
                   (NOT (EQUAL (REMAINDER 0 Z) 0))),

    which again simplifies, expanding the functions LESSP, REMAINDER, and
    EQUAL, to:

          T.

Case 2. (IMPLIES (AND (NOT (ZEROP Z))
                      (NOT (LESSP X Z))
                      (NOT (EQUAL (REMAINDER (DIFFERENCE X Z) Z)
                                  0))
                      (EQUAL (REMAINDER X Z) 0)
                      (NOT (EQUAL (REMAINDER Y Z) 0)))
                 (NOT (EQUAL (REMAINDER (PLUS X Y) Z) 0))),

  which simplifies, unfolding the functions ZEROP and REMAINDER, to:

        T.

Case 1. (IMPLIES (AND (NOT (ZEROP Z))
                      (NOT (LESSP X Z))
                      (NOT (EQUAL (REMAINDER (PLUS (DIFFERENCE X Z) Y)
                                             Z)
                                  0))
                      (EQUAL (REMAINDER X Z) 0)
                      (NOT (EQUAL (REMAINDER Y Z) 0)))
                 (NOT (EQUAL (REMAINDER (PLUS X Y) Z) 0))),

  which simplifies, rewriting with the lemma COMMUTATIVITY-OF-PLUS, and
  opening up the definitions of ZEROP and REMAINDER, to the goal:

        (IMPLIES (AND (NOT (EQUAL Z 0))
                      (NUMBERP Z)
                      (NOT (LESSP X Z))
                      (NOT (EQUAL (REMAINDER (PLUS Y (DIFFERENCE X Z))
                                             Z)
                                  0))
                      (EQUAL (REMAINDER (DIFFERENCE X Z) Z)
                             0)
                      (NOT (EQUAL (REMAINDER Y Z) 0)))
                 (NOT (EQUAL (REMAINDER (PLUS X Y) Z) 0))).

  Appealing to the lemmas DIFFERENCE-ELIM and REMAINDER-QUOTIENT-ELIM, we now
  replace X by (PLUS Z V) to eliminate (DIFFERENCE X Z) and V by
  (PLUS W (TIMES Z D)) to eliminate (REMAINDER V Z) and (QUOTIENT V Z).  We
  use the type restriction lemma noted when DIFFERENCE was introduced,
  LESSP-REMAINDER2, the type restriction lemma noted when REMAINDER was
  introduced, and the type restriction lemma noted when QUOTIENT was
  introduced to constrain the new variables.  We must thus prove two new
  formulas:

  Case 1.2.
          (IMPLIES (AND (NOT (NUMBERP X))
                        (NOT (EQUAL Z 0))
                        (NUMBERP Z)
                        (NOT (LESSP X Z))
                        (NOT (EQUAL (REMAINDER (PLUS Y (DIFFERENCE X Z))
                                               Z)
                                    0))
                        (EQUAL (REMAINDER (DIFFERENCE X Z) Z)
                               0)
                        (NOT (EQUAL (REMAINDER Y Z) 0)))
                   (NOT (EQUAL (REMAINDER (PLUS X Y) Z) 0))),

    which further simplifies, opening up LESSP, to:

          T.

  Case 1.1.
          (IMPLIES (AND (NUMBERP W)
                        (EQUAL (LESSP W Z) (NOT (ZEROP Z)))
                        (NUMBERP D)
                        (NOT (EQUAL Z 0))
                        (NUMBERP Z)
                        (NOT (LESSP (PLUS Z W (TIMES Z D)) Z))
                        (NOT (EQUAL (REMAINDER (PLUS Y W (TIMES Z D)) Z)
                                    0))
                        (EQUAL W 0)
                        (NOT (EQUAL (REMAINDER Y Z) 0)))
                   (NOT (EQUAL (REMAINDER (PLUS (PLUS Z W (TIMES Z D)) Y)
                                          Z)
                               0))),

    which further simplifies, applying COMMUTATIVITY-OF-TIMES,
    COMMUTATIVITY-OF-PLUS, COMMUTATIVITY2-OF-PLUS, ASSOCIATIVITY-OF-PLUS, and
    DIFFERENCE-PLUS3, and unfolding the definitions of NUMBERP, EQUAL, LESSP,
    ZEROP, NOT, PLUS, and REMAINDER, to:

          (IMPLIES (AND (NUMBERP D)
                        (NOT (EQUAL Z 0))
                        (NUMBERP Z)
                        (NOT (LESSP (PLUS Z (TIMES D Z)) Z))
                        (NOT (EQUAL (REMAINDER (PLUS Y (TIMES D Z)) Z)
                                    0))
                        (NOT (EQUAL (REMAINDER Y Z) 0))
                        (LESSP (PLUS Y Z (TIMES D Z)) Z))
                   (NOT (EQUAL (PLUS Y Z (TIMES D Z)) 0))),

    which again simplifies, using linear arithmetic, to:

          T.


     That finishes the proof of *1.  Q.E.D.


[ 0.0 0.1 0.0 ]

DIVIDES-PLUS-REWRITE2


(DISABLE DIVIDES-PLUS-REWRITE2)

[ 0.0 0.0 0.0 ]

DIVIDES-PLUS-REWRITE2-OFF


(PROVE-LEMMA DIVIDES-PLUS-REWRITE
             (REWRITE)
             (IMPLIES (EQUAL (REMAINDER X Z) '0)
                      (EQUAL (EQUAL (REMAINDER (PLUS X Y) Z) '0)
                             (EQUAL (REMAINDER Y Z) '0)))
             ((ENABLE DIVIDES-PLUS-REWRITE1 DIVIDES-PLUS-REWRITE2)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This conjecture simplifies, clearly, to two new conjectures:

Case 2. (IMPLIES (AND (EQUAL (REMAINDER X Z) 0)
                      (NOT (EQUAL (REMAINDER Y Z) 0)))
                 (NOT (EQUAL (REMAINDER (PLUS X Y) Z) 0))),

  which again simplifies, applying DIVIDES-PLUS-REWRITE2, to:

        T.

Case 1. (IMPLIES (AND (EQUAL (REMAINDER X Z) 0)
                      (EQUAL (REMAINDER Y Z) 0))
                 (EQUAL (EQUAL (REMAINDER (PLUS X Y) Z) 0)
                        T)).

  However this again simplifies, applying DIVIDES-PLUS-REWRITE1, and unfolding
  the function EQUAL, to:

        T.

Q.E.D.


[ 0.0 0.0 0.0 ]

DIVIDES-PLUS-REWRITE


(DISABLE DIVIDES-PLUS-REWRITE)

[ 0.0 0.0 0.0 ]

DIVIDES-PLUS-REWRITE-OFF


(PROVE-LEMMA LESSP-PLUS-CANCELATION
             (REWRITE)
             (EQUAL (LESSP (PLUS X Y) (PLUS X Z))
                    (LESSP Y Z))
             ((ENABLE EQUAL-LESSP)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This simplifies, rewriting with EQUAL-LESSP, to two new formulas:

Case 2. (IMPLIES (NOT (LESSP (PLUS X Y) (PLUS X Z)))
                 (NOT (LESSP Y Z))),

  which again simplifies, using linear arithmetic, to:

        T.

Case 1. (IMPLIES (LESSP (PLUS X Y) (PLUS X Z))
                 (LESSP Y Z)),

  which again simplifies, using linear arithmetic, to:

        T.

Q.E.D.


[ 0.0 0.0 0.0 ]

LESSP-PLUS-CANCELATION


(DISABLE LESSP-PLUS-CANCELATION)

[ 0.0 0.0 0.0 ]

LESSP-PLUS-CANCELATION-OFF


(PROVE-LEMMA DIVIDES-PLUS-REWRITE-COMMUTED
             (REWRITE)
             (IMPLIES (EQUAL (REMAINDER X Z) '0)
                      (EQUAL (EQUAL (REMAINDER (PLUS Y X) Z) '0)
                             (EQUAL (REMAINDER Y Z) '0)))
             ((ENABLE COMMUTATIVITY-OF-PLUS DIVIDES-PLUS-REWRITE)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This conjecture simplifies, applying COMMUTATIVITY-OF-PLUS and
DIVIDES-PLUS-REWRITE, to:

      T.

Q.E.D.


[ 0.0 0.0 0.0 ]

DIVIDES-PLUS-REWRITE-COMMUTED


(DISABLE DIVIDES-PLUS-REWRITE-COMMUTED)

[ 0.0 0.0 0.0 ]

DIVIDES-PLUS-REWRITE-COMMUTED-OFF


(PROVE-LEMMA EUCLID
             (REWRITE)
             (IMPLIES (EQUAL (REMAINDER X Z) '0)
                      (EQUAL (EQUAL (REMAINDER (DIFFERENCE Y X) Z)
                                    '0)
                             (IF (LESSP X Y)
                                 (EQUAL (REMAINDER Y Z) '0)
                                 '*1*TRUE)))
             ((ENABLE PLUS-RIGHT-ID2 COMMUTATIVITY2-OF-PLUS
                      COMMUTATIVITY-OF-PLUS DIFFERENCE-0
                      COMMUTATIVITY-OF-TIMES DIFFERENCE-ELIM REMAINDER-WRT-12
                      LESSP-REMAINDER2 REMAINDER-QUOTIENT-ELIM DIVIDES-TIMES
                      DIVIDES-PLUS-REWRITE1 DIVIDES-PLUS-REWRITE-COMMUTED)
              (ENABLE-THEORY GROUND-ZERO)
              (DISABLE-THEORY T)))

This formula simplifies, obviously, to the following three new conjectures:

Case 3. (IMPLIES (AND (EQUAL (REMAINDER X Z) 0)
                      (NOT (EQUAL (REMAINDER (DIFFERENCE Y X) Z)
                                  0)))
                 (LESSP X Y)).

  This again simplifies, applying DIFFERENCE-0, and expanding LESSP, EQUAL,
  NUMBERP, and REMAINDER, to:

        T.

Case 2. (IMPLIES (AND (EQUAL (REMAINDER X Z) 0)
                      (NOT (EQUAL (REMAINDER (DIFFERENCE Y X) Z)
                                  0)))
                 (NOT (EQUAL (REMAINDER Y Z) 0))).

  Appealing to the lemmas DIFFERENCE-ELIM and REMAINDER-QUOTIENT-ELIM, we now
  replace Y by (PLUS X V) to eliminate (DIFFERENCE Y X) and V by
  (PLUS W (TIMES Z D)) to eliminate (REMAINDER V Z) and (QUOTIENT V Z).  We
  rely upon the type restriction lemma noted when DIFFERENCE was introduced,
  LESSP-REMAINDER2, the type restriction lemma noted when REMAINDER was
  introduced, and the type restriction lemma noted when QUOTIENT was
  introduced to constrain the new variables.  We must thus prove five new
  goals:

  Case 2.5.
          (IMPLIES (AND (LESSP Y X)
                        (EQUAL (REMAINDER X Z) 0)
                        (NOT (EQUAL (REMAINDER (DIFFERENCE Y X) Z)
                                    0)))
                   (NOT (EQUAL (REMAINDER Y Z) 0))),

    which further simplifies, using linear arithmetic, rewriting with
    DIFFERENCE-0, and expanding the definitions of LESSP, EQUAL, NUMBERP, and
    REMAINDER, to:

          T.

  Case 2.4.
          (IMPLIES (AND (NOT (NUMBERP Y))
                        (EQUAL (REMAINDER X Z) 0)
                        (NOT (EQUAL (REMAINDER (DIFFERENCE Y X) Z)
                                    0)))
                   (NOT (EQUAL (REMAINDER