Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/24100)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

1.1 Rule Removal

1.1.1 Dependency Pair Transformation

There are 301 ruless (increase limit for explicit display).

1.1.1.1 Reduction Pair Processor

There are 176 ruless (increase limit for explicit display).

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

• The 1^st component contains the pair

  4#(3(3(1(2(3(4(0(0(2(x1))))))))))	→	4#(3(3(1(2(4(3(0(0(2(x1))))))))))	(275)
  4#(3(3(5(0(1(2(3(1(1(x1))))))))))	→	4#(3(3(5(0(1(3(2(1(1(x1))))))))))	(218)
  4#(3(3(4(1(4(1(0(4(2(x1))))))))))	→	4#(3(1(1(3(4(4(0(4(2(x1))))))))))	(314)
  4#(4(3(0(3(2(5(3(0(5(x1))))))))))	→	4#(4(0(3(3(2(5(3(0(5(x1))))))))))	(429)

  1.1.1.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [4#(x1)]	=	1 · x1
  [3(x1)]	=	1 · x1
  [1(x1)]	=	1
  [2(x1)]	=	0
  [4(x1)]	=	1 + 1 · x1
  [0(x1)]	=	0
  [5(x1)]	=	0

  the pair

  4#(3(3(4(1(4(1(0(4(2(x1))))))))))

  →

  4#(3(1(1(3(4(4(0(4(2(x1))))))))))

  (314)

  could be deleted.

  1.1.1.1.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [4#(x1)]	=	1 · x1
  [3(x1)]	=	1 + 1 · x1
  [1(x1)]	=	1
  [2(x1)]	=	0
  [4(x1)]	=	1 · x1
  [0(x1)]	=	1
  [5(x1)]	=	0

  the pair

  4#(4(3(0(3(2(5(3(0(5(x1))))))))))

  →

  4#(4(0(3(3(2(5(3(0(5(x1))))))))))

  (429)

  could be deleted.

  1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [5(x1)]	=	3 + 1 · x1
  [1(x1)]	=	1 · x1
  [0(x1)]	=	1 · x1
  [3(x1)]	=	3 + 1 · x1
  [2(x1)]	=	1 + 2 · x1
  [4(x1)]	=	1 · x1
  [4#(x1)]	=	1 · x1

  the pair

  4#(3(3(5(0(1(2(3(1(1(x1))))))))))

  →

  4#(3(3(5(0(1(3(2(1(1(x1))))))))))

  (218)

  and the rules

  4(3(3(5(0(1(2(3(1(1(x1))))))))))	→	4(3(3(5(0(1(3(2(1(1(x1))))))))))	(89)
  0(1(0(2(5(2(2(2(3(1(x1))))))))))	→	0(1(0(5(2(2(2(2(3(1(x1))))))))))	(92)
  5(3(0(0(4(2(2(5(4(1(x1))))))))))	→	5(2(4(0(3(1(0(5(2(4(x1))))))))))	(93)
  5(2(2(1(2(3(3(0(1(2(x1))))))))))	→	5(2(2(1(3(2(3(0(1(2(x1))))))))))	(98)
  1(4(2(5(3(4(4(2(1(2(x1))))))))))	→	5(2(4(4(1(3(4(2(1(2(x1))))))))))	(99)
  1(2(4(2(3(1(4(3(1(2(x1))))))))))	→	1(2(4(3(2(1(4(3(1(2(x1))))))))))	(100)
  0(2(5(0(4(2(3(4(0(3(x1))))))))))	→	3(2(4(0(5(4(2(0(0(3(x1))))))))))	(108)
  5(4(0(3(5(1(2(5(3(3(x1))))))))))	→	5(0(3(5(5(2(4(1(3(3(x1))))))))))	(110)
  3(2(4(2(0(2(3(1(5(3(x1))))))))))	→	2(3(2(0(5(4(1(3(2(3(x1))))))))))	(114)
  3(1(0(2(0(2(2(3(1(4(x1))))))))))	→	3(1(0(2(0(1(2(3(2(4(x1))))))))))	(116)
  0(2(0(1(2(3(1(0(1(5(x1))))))))))	→	2(0(0(1(3(2(1(0(1(5(x1))))))))))	(123)
  5(0(2(3(5(5(1(3(5(5(x1))))))))))	→	5(0(3(2(5(5(1(3(5(5(x1))))))))))	(126)

  could be deleted.

  1.1.1.1.1.1.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [4#(x1)]	=	1 · x1
  [3(x1)]	=	1 + 1 · x1
  [1(x1)]	=	1 · x1
  [2(x1)]	=	1 · x1
  [4(x1)]	=	1
  [0(x1)]	=	0
  [5(x1)]	=	0

  the pair

  4#(3(3(1(2(3(4(0(0(2(x1))))))))))

  →

  4#(3(3(1(2(4(3(0(0(2(x1))))))))))

  (275)

  could be deleted.

  1.1.1.1.1.1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  2#(3(1(5(2(0(2(1(5(1(x1))))))))))	→	2#(3(1(5(2(2(0(1(5(1(x1))))))))))	(263)
  2#(3(1(0(5(5(0(1(0(1(x1))))))))))	→	2#(3(1(0(5(0(5(1(0(1(x1))))))))))	(211)
  2#(5(5(4(3(1(0(5(3(2(x1))))))))))	→	2#(5(5(4(1(3(0(5(3(2(x1))))))))))	(308)
  2#(4(0(4(5(4(1(1(5(2(x1))))))))))	→	2#(4(0(4(4(5(1(1(5(2(x1))))))))))	(321)
  2#(3(5(3(3(4(0(0(4(4(x1))))))))))	→	2#(3(5(3(3(0(4(0(4(4(x1))))))))))	(413)
  2#(1(4(3(1(5(3(1(5(4(x1))))))))))	→	2#(4(1(3(1(5(3(1(5(4(x1))))))))))	(426)
  2#(3(2(4(4(3(0(0(3(5(x1))))))))))	→	2#(3(2(4(3(4(0(0(3(5(x1))))))))))	(444)

  1.1.1.1.1.2 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [2#(x1)]	=	1 · x1
  [3(x1)]	=	1
  [1(x1)]	=	1
  [5(x1)]	=	0
  [2(x1)]	=	0
  [0(x1)]	=	1
  [4(x1)]	=	0

  the pair

  2#(1(4(3(1(5(3(1(5(4(x1))))))))))

  →

  2#(4(1(3(1(5(3(1(5(4(x1))))))))))

  (426)

  could be deleted.

  1.1.1.1.1.2.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [2#(x1)]	=	1 · x1
  [3(x1)]	=	1 · x1
  [1(x1)]	=	0
  [5(x1)]	=	1 · x1
  [2(x1)]	=	0
  [0(x1)]	=	0
  [4(x1)]	=	1

  the pair

  2#(3(5(3(3(4(0(0(4(4(x1))))))))))

  →

  2#(3(5(3(3(0(4(0(4(4(x1))))))))))

  (413)

  could be deleted.

  1.1.1.1.1.2.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [2#(x1)]	=	1 · x1
  [3(x1)]	=	1 + 1 · x1
  [1(x1)]	=	1
  [5(x1)]	=	1 · x1
  [2(x1)]	=	0
  [0(x1)]	=	1
  [4(x1)]	=	1 · x1

  the pair

  2#(5(5(4(3(1(0(5(3(2(x1))))))))))

  →

  2#(5(5(4(1(3(0(5(3(2(x1))))))))))

  (308)

  could be deleted.

  1.1.1.1.1.2.1.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [5(x1)]	=	2 + 2 · x1
  [1(x1)]	=	1 · x1
  [0(x1)]	=	1 · x1
  [3(x1)]	=	1 + 2 · x1
  [2(x1)]	=	2 · x1
  [4(x1)]	=	1 · x1
  [2#(x1)]	=	1 · x1

  the rules

  5(5(1(0(3(2(5(1(0(0(x1))))))))))	→	5(5(0(5(2(1(3(1(0(0(x1))))))))))	(78)
  0(3(5(1(5(4(3(5(0(0(x1))))))))))	→	0(3(5(5(4(5(1(3(0(0(x1))))))))))	(80)
  4(3(3(5(0(1(2(3(1(1(x1))))))))))	→	4(3(3(5(0(1(3(2(1(1(x1))))))))))	(89)
  3(4(3(1(5(1(5(1(2(1(x1))))))))))	→	1(3(1(5(4(5(3(1(2(1(x1))))))))))	(90)
  0(1(0(2(5(2(2(2(3(1(x1))))))))))	→	0(1(0(5(2(2(2(2(3(1(x1))))))))))	(92)
  5(3(0(0(4(2(2(5(4(1(x1))))))))))	→	5(2(4(0(3(1(0(5(2(4(x1))))))))))	(93)
  5(2(2(1(2(3(3(0(1(2(x1))))))))))	→	5(2(2(1(3(2(3(0(1(2(x1))))))))))	(98)
  1(4(2(5(3(4(4(2(1(2(x1))))))))))	→	5(2(4(4(1(3(4(2(1(2(x1))))))))))	(99)
  1(2(4(2(3(1(4(3(1(2(x1))))))))))	→	1(2(4(3(2(1(4(3(1(2(x1))))))))))	(100)
  0(2(5(0(4(2(3(4(0(3(x1))))))))))	→	3(2(4(0(5(4(2(0(0(3(x1))))))))))	(108)
  3(5(0(0(4(2(1(0(2(3(x1))))))))))	→	0(5(0(3(2(4(1(0(2(3(x1))))))))))	(109)
  5(4(0(3(5(1(2(5(3(3(x1))))))))))	→	5(0(3(5(5(2(4(1(3(3(x1))))))))))	(110)
  3(2(4(2(0(2(3(1(5(3(x1))))))))))	→	2(3(2(0(5(4(1(3(2(3(x1))))))))))	(114)
  3(1(0(2(0(2(2(3(1(4(x1))))))))))	→	3(1(0(2(0(1(2(3(2(4(x1))))))))))	(116)
  0(2(0(1(2(3(1(0(1(5(x1))))))))))	→	2(0(0(1(3(2(1(0(1(5(x1))))))))))	(123)
  3(4(1(1(5(1(5(1(1(5(x1))))))))))	→	1(4(5(1(3(1(5(1(1(5(x1))))))))))	(124)
  5(0(2(3(5(5(1(3(5(5(x1))))))))))	→	5(0(3(2(5(5(1(3(5(5(x1))))))))))	(126)

  could be deleted.

  1.1.1.1.1.2.1.1.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [3(x1)]	=	2 · x1
  [0(x1)]	=	1 · x1
  [2(x1)]	=	2 + 1 · x1
  [1(x1)]	=	2 · x1
  [4(x1)]	=	1 · x1
  [5(x1)]	=	2 · x1
  [2#(x1)]	=	2 · x1

  the rule

  0(2(1(4(2(3(2(0(3(1(x1))))))))))

  →

  0(2(4(2(3(2(1(0(3(1(x1))))))))))

  (91)

  could be deleted.

  1.1.1.1.1.2.1.1.1.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [3(x1)]	=	1 · x1
  [0(x1)]	=	1 · x1
  [2(x1)]	=	1 + 2 · x1
  [1(x1)]	=	1 · x1
  [4(x1)]	=	2 · x1
  [5(x1)]	=	1 · x1
  [2#(x1)]	=	2 · x1

  the rule

  3(4(3(1(0(0(4(2(5(0(x1))))))))))

  →

  3(4(3(1(0(0(2(4(5(0(x1))))))))))

  (84)

  could be deleted.

  1.1.1.1.1.2.1.1.1.1.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [3(x1)]	=	2 · x1
  [0(x1)]	=	2 + 2 · x1
  [2(x1)]	=	3 + 2 · x1
  [1(x1)]	=	1 · x1
  [4(x1)]	=	1 · x1
  [5(x1)]	=	1 · x1
  [2#(x1)]	=	1 · x1

  the pair

  2#(3(1(5(2(0(2(1(5(1(x1))))))))))

  →

  2#(3(1(5(2(2(0(1(5(1(x1))))))))))

  (263)

  and the rules

  3(3(0(2(0(1(0(4(0(0(x1))))))))))	→	3(3(2(0(0(1(0(4(0(0(x1))))))))))	(79)
  0(0(4(3(0(1(0(2(2(0(x1))))))))))	→	0(4(0(0(3(1(0(2(2(0(x1))))))))))	(81)
  2(3(1(5(2(0(2(1(5(1(x1))))))))))	→	2(3(1(5(2(2(0(1(5(1(x1))))))))))	(95)
  0(5(3(4(3(0(3(1(5(2(x1))))))))))	→	0(5(3(0(4(3(3(1(5(2(x1))))))))))	(105)
  5(4(1(5(3(0(4(1(0(3(x1))))))))))	→	5(4(0(5(4(3(1(1(0(3(x1))))))))))	(106)
  0(3(0(0(2(4(3(0(5(3(x1))))))))))	→	0(3(0(2(4(0(3(0(5(3(x1))))))))))	(113)
  3(0(0(3(0(0(0(2(3(4(x1))))))))))	→	3(0(0(3(0(0(2(0(3(4(x1))))))))))	(118)
  4(4(3(0(3(2(5(3(0(5(x1))))))))))	→	4(4(0(3(3(2(5(3(0(5(x1))))))))))	(122)

  could be deleted.

  1.1.1.1.1.2.1.1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    2#(3(2(4(4(3(0(0(3(5(x1))))))))))	→	2#(3(2(4(3(4(0(0(3(5(x1))))))))))	(444)
    2#(3(1(0(5(5(0(1(0(1(x1))))))))))	→	2#(3(1(0(5(0(5(1(0(1(x1))))))))))	(211)

    1.1.1.1.1.2.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [3(x1)]	=	2 · x1
    [5(x1)]	=	1 + 2 · x1
    [4(x1)]	=	1 · x1
    [0(x1)]	=	1 · x1
    [2(x1)]	=	2 · x1
    [1(x1)]	=	2 · x1
    [2#(x1)]	=	1 · x1

    the rule

    3(3(4(2(1(5(3(3(4(0(x1))))))))))

    →

    3(3(4(2(5(1(3(3(4(0(x1))))))))))

    (83)

    could be deleted.

    1.1.1.1.1.2.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [3(x1)]	=	1 · x1
    [5(x1)]	=	1 · x1
    [4(x1)]	=	1 + 2 · x1
    [0(x1)]	=	1 + 2 · x1
    [2(x1)]	=	2 · x1
    [1(x1)]	=	1 · x1
    [2#(x1)]	=	1 · x1

    the rule

    1(4(3(4(3(4(2(4(0(4(x1))))))))))

    →

    1(4(4(3(0(4(3(2(4(4(x1))))))))))

    (115)

    could be deleted.

    1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [2#(x1)]	=	1 · x1
    [3(x1)]	=	1 + 4 · x1
    [2(x1)]	=	2 · x1
    [4(x1)]	=	3 · x1
    [0(x1)]	=	0
    [5(x1)]	=	0
    [1(x1)]	=	0

    the pair

    2#(3(2(4(4(3(0(0(3(5(x1))))))))))

    →

    2#(3(2(4(3(4(0(0(3(5(x1))))))))))

    (444)

    could be deleted.

    1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [2#(x1)]	=	1 · x1
    [3(x1)]	=	1 + 2 · x1
    [1(x1)]	=	1 · x1
    [0(x1)]	=	2 + 2 · x1
    [5(x1)]	=	2 · x1
    [2(x1)]	=	0
    [4(x1)]	=	0

    the pair

    2#(3(1(0(5(5(0(1(0(1(x1))))))))))

    →

    2#(3(1(0(5(0(5(1(0(1(x1))))))))))

    (211)

    could be deleted.

    1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1 P is empty

    There are no pairs anymore.

• The 3^rd component contains the pair

  5#(5(1(0(3(2(5(1(0(0(x1))))))))))	→	5#(0(5(2(1(3(1(0(0(x1)))))))))	(154)
  5#(5(1(0(3(2(5(1(0(0(x1))))))))))	→	5#(5(0(5(2(1(3(1(0(0(x1))))))))))	(153)
  5#(5(1(0(3(2(5(1(0(0(x1))))))))))	→	0#(5(2(1(3(1(0(0(x1))))))))	(155)
  0#(3(5(1(5(4(3(5(0(0(x1))))))))))	→	0#(3(5(5(4(5(1(3(0(0(x1))))))))))	(164)
  0#(3(5(1(5(4(3(5(0(0(x1))))))))))	→	3#(5(5(4(5(1(3(0(0(x1)))))))))	(165)
  3#(3(0(2(0(1(0(4(0(0(x1))))))))))	→	3#(3(2(0(0(1(0(4(0(0(x1))))))))))	(160)
  3#(3(0(2(0(1(0(4(0(0(x1))))))))))	→	3#(2(0(0(1(0(4(0(0(x1)))))))))	(161)
  3#(5(3(4(0(5(0(2(3(0(x1))))))))))	→	3#(5(3(4(5(0(0(2(3(0(x1))))))))))	(177)
  3#(5(3(4(0(5(0(2(3(0(x1))))))))))	→	5#(3(4(5(0(0(2(3(0(x1)))))))))	(178)
  5#(5(1(0(3(2(5(1(0(0(x1))))))))))	→	5#(2(1(3(1(0(0(x1)))))))	(156)
  5#(3(0(0(4(2(2(5(4(1(x1))))))))))	→	5#(2(4(0(3(1(0(5(2(4(x1))))))))))	(245)
  5#(2(2(1(2(3(3(0(1(2(x1))))))))))	→	5#(2(2(1(3(2(3(0(1(2(x1))))))))))	(282)
  5#(4(1(5(3(0(4(1(0(3(x1))))))))))	→	5#(4(0(5(4(3(1(1(0(3(x1))))))))))	(333)
  5#(4(0(3(5(1(2(5(3(3(x1))))))))))	→	5#(0(3(5(5(2(4(1(3(3(x1))))))))))	(354)
  5#(4(0(3(5(1(2(5(3(3(x1))))))))))	→	0#(3(5(5(2(4(1(3(3(x1)))))))))	(355)
  0#(3(5(1(5(4(3(5(0(0(x1))))))))))	→	5#(5(4(5(1(3(0(0(x1))))))))	(166)
  5#(4(0(3(5(1(2(5(3(3(x1))))))))))	→	3#(5(5(2(4(1(3(3(x1))))))))	(356)
  3#(5(3(4(0(5(0(2(3(0(x1))))))))))	→	3#(4(5(0(0(2(3(0(x1))))))))	(179)
  3#(3(4(2(1(5(3(3(4(0(x1))))))))))	→	3#(3(4(2(5(1(3(3(4(0(x1))))))))))	(183)
  3#(3(4(2(1(5(3(3(4(0(x1))))))))))	→	3#(4(2(5(1(3(3(4(0(x1)))))))))	(184)
  3#(4(3(1(0(0(4(2(5(0(x1))))))))))	→	3#(4(3(1(0(0(2(4(5(0(x1))))))))))	(189)
  3#(4(1(0(4(3(5(0(0(1(x1))))))))))	→	3#(4(4(0(3(1(5(0(0(1(x1))))))))))	(205)
  3#(4(3(1(5(1(5(1(2(1(x1))))))))))	→	1#(3(1(5(4(5(3(1(2(1(x1))))))))))	(226)
  1#(2(1(3(0(4(0(5(5(0(x1))))))))))	→	1#(2(1(3(0(0(5(4(5(0(x1))))))))))	(197)
  1#(4(2(5(3(4(4(2(1(2(x1))))))))))	→	5#(2(4(4(1(3(4(2(1(2(x1))))))))))	(288)
  5#(4(0(3(5(1(2(5(3(3(x1))))))))))	→	5#(5(2(4(1(3(3(x1)))))))	(357)
  5#(4(0(3(5(1(2(5(3(3(x1))))))))))	→	5#(2(4(1(3(3(x1))))))	(358)
  5#(0(2(3(5(5(1(3(5(5(x1))))))))))	→	5#(0(3(2(5(5(1(3(5(5(x1))))))))))	(450)
  5#(0(2(3(5(5(1(3(5(5(x1))))))))))	→	0#(3(2(5(5(1(3(5(5(x1)))))))))	(451)
  0#(3(5(1(5(4(3(5(0(0(x1))))))))))	→	5#(4(5(1(3(0(0(x1)))))))	(167)
  5#(0(2(3(5(5(1(3(5(5(x1))))))))))	→	3#(2(5(5(1(3(5(5(x1))))))))	(452)
  3#(4(3(1(5(1(5(1(2(1(x1))))))))))	→	3#(1(5(4(5(3(1(2(1(x1)))))))))	(227)
  3#(4(3(1(5(1(5(1(2(1(x1))))))))))	→	1#(5(4(5(3(1(2(1(x1))))))))	(228)
  1#(2(4(2(3(1(4(3(1(2(x1))))))))))	→	1#(2(4(3(2(1(4(3(1(2(x1))))))))))	(294)
  1#(4(3(4(3(4(2(4(0(4(x1))))))))))	→	1#(4(4(3(0(4(3(2(4(4(x1))))))))))	(382)
  1#(5(5(5(3(4(0(1(5(4(x1))))))))))	→	1#(5(5(5(4(3(0(1(5(4(x1))))))))))	(420)
  1#(5(5(5(3(4(0(1(5(4(x1))))))))))	→	5#(5(5(4(3(0(1(5(4(x1)))))))))	(421)
  1#(5(5(5(3(4(0(1(5(4(x1))))))))))	→	5#(5(4(3(0(1(5(4(x1))))))))	(422)
  1#(5(5(5(3(4(0(1(5(4(x1))))))))))	→	5#(4(3(0(1(5(4(x1)))))))	(423)
  3#(4(3(1(5(1(5(1(2(1(x1))))))))))	→	5#(4(5(3(1(2(1(x1)))))))	(229)
  3#(3(3(2(3(1(4(0(5(1(x1))))))))))	→	3#(3(3(2(1(3(0(4(5(1(x1))))))))))	(255)
  3#(3(3(2(3(1(4(0(5(1(x1))))))))))	→	3#(3(2(1(3(0(4(5(1(x1)))))))))	(256)
  3#(3(3(2(3(1(4(0(5(1(x1))))))))))	→	3#(2(1(3(0(4(5(1(x1))))))))	(257)
  3#(5(0(0(4(2(1(0(2(3(x1))))))))))	→	0#(5(0(3(2(4(1(0(2(3(x1))))))))))	(348)
  0#(0(4(3(0(1(0(2(2(0(x1))))))))))	→	0#(4(0(0(3(1(0(2(2(0(x1))))))))))	(172)
  0#(2(1(4(2(3(2(0(3(1(x1))))))))))	→	0#(2(4(2(3(2(1(0(3(1(x1))))))))))	(233)
  0#(1(0(2(5(2(2(2(3(1(x1))))))))))	→	0#(1(0(5(2(2(2(2(3(1(x1))))))))))	(240)
  0#(1(0(2(5(2(2(2(3(1(x1))))))))))	→	1#(0(5(2(2(2(2(3(1(x1)))))))))	(241)
  0#(1(0(2(5(2(2(2(3(1(x1))))))))))	→	0#(5(2(2(2(2(3(1(x1))))))))	(242)
  0#(1(0(2(5(2(2(2(3(1(x1))))))))))	→	5#(2(2(2(2(3(1(x1)))))))	(243)
  0#(4(0(4(5(5(0(3(5(1(x1))))))))))	→	0#(4(0(5(4(5(0(3(5(1(x1))))))))))	(270)
  0#(5(5(4(3(5(0(4(1(2(x1))))))))))	→	0#(5(5(4(3(0(5(1(4(2(x1))))))))))	(299)
  0#(5(5(4(3(5(0(4(1(2(x1))))))))))	→	5#(5(4(3(0(5(1(4(2(x1)))))))))	(300)
  0#(5(5(4(3(5(0(4(1(2(x1))))))))))	→	5#(4(3(0(5(1(4(2(x1))))))))	(301)
  0#(5(3(4(3(0(3(1(5(2(x1))))))))))	→	0#(5(3(0(4(3(3(1(5(2(x1))))))))))	(327)
  0#(5(3(4(3(0(3(1(5(2(x1))))))))))	→	5#(3(0(4(3(3(1(5(2(x1)))))))))	(328)
  0#(5(3(4(3(0(3(1(5(2(x1))))))))))	→	3#(0(4(3(3(1(5(2(x1))))))))	(329)
  3#(5(0(0(4(2(1(0(2(3(x1))))))))))	→	5#(0(3(2(4(1(0(2(3(x1)))))))))	(349)
  3#(5(0(0(4(2(1(0(2(3(x1))))))))))	→	0#(3(2(4(1(0(2(3(x1))))))))	(350)
  0#(5(3(4(3(0(3(1(5(2(x1))))))))))	→	0#(4(3(3(1(5(2(x1)))))))	(330)
  0#(2(5(0(4(2(3(4(0(3(x1))))))))))	→	3#(2(4(0(5(4(2(0(0(3(x1))))))))))	(340)
  3#(5(0(0(4(2(1(0(2(3(x1))))))))))	→	3#(2(4(1(0(2(3(x1)))))))	(351)
  3#(5(0(4(1(5(0(5(4(3(x1))))))))))	→	3#(5(0(1(4(5(0(5(4(3(x1))))))))))	(362)
  3#(5(0(4(1(5(0(5(4(3(x1))))))))))	→	5#(0(1(4(5(0(5(4(3(x1)))))))))	(363)
  3#(5(0(4(1(5(0(5(4(3(x1))))))))))	→	0#(1(4(5(0(5(4(3(x1))))))))	(364)
  0#(3(0(0(2(4(3(0(5(3(x1))))))))))	→	0#(3(0(2(4(0(3(0(5(3(x1))))))))))	(367)
  0#(3(0(0(2(4(3(0(5(3(x1))))))))))	→	3#(0(2(4(0(3(0(5(3(x1)))))))))	(368)
  3#(5(0(4(1(5(0(5(4(3(x1))))))))))	→	1#(4(5(0(5(4(3(x1)))))))	(365)
  3#(1(0(2(0(2(2(3(1(4(x1))))))))))	→	3#(1(0(2(0(1(2(3(2(4(x1))))))))))	(391)
  3#(1(0(2(0(2(2(3(1(4(x1))))))))))	→	1#(0(2(0(1(2(3(2(4(x1)))))))))	(392)
  3#(1(0(2(0(2(2(3(1(4(x1))))))))))	→	0#(2(0(1(2(3(2(4(x1))))))))	(393)
  0#(3(0(0(2(4(3(0(5(3(x1))))))))))	→	0#(2(4(0(3(0(5(3(x1))))))))	(369)
  0#(1(3(0(1(5(0(5(2(4(x1))))))))))	→	0#(1(3(1(0(5(0(5(2(4(x1))))))))))	(400)
  0#(1(3(0(1(5(0(5(2(4(x1))))))))))	→	1#(3(1(0(5(0(5(2(4(x1)))))))))	(401)
  0#(1(3(0(1(5(0(5(2(4(x1))))))))))	→	3#(1(0(5(0(5(2(4(x1))))))))	(402)
  3#(0(0(3(0(0(0(2(3(4(x1))))))))))	→	3#(0(0(3(0(0(2(0(3(4(x1))))))))))	(405)
  3#(0(0(3(0(0(0(2(3(4(x1))))))))))	→	0#(0(3(0(0(2(0(3(4(x1)))))))))	(406)
  0#(1(3(0(1(5(0(5(2(4(x1))))))))))	→	1#(0(5(0(5(2(4(x1)))))))	(403)
  0#(1(3(0(1(5(0(5(2(4(x1))))))))))	→	0#(5(0(5(2(4(x1))))))	(404)
  3#(0(0(3(0(0(0(2(3(4(x1))))))))))	→	0#(3(0(0(2(0(3(4(x1))))))))	(407)
  3#(0(0(3(0(0(0(2(3(4(x1))))))))))	→	3#(0(0(2(0(3(4(x1)))))))	(408)
  3#(0(0(3(0(0(0(2(3(4(x1))))))))))	→	0#(0(2(0(3(4(x1))))))	(409)
  3#(0(0(3(0(0(0(2(3(4(x1))))))))))	→	0#(2(0(3(4(x1)))))	(410)
  3#(4(1(1(5(1(5(1(1(5(x1))))))))))	→	1#(4(5(1(3(1(5(1(1(5(x1))))))))))	(439)

  1.1.1.1.1.3 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [5#(x1)]	=	1 + 1 · x1
  [5(x1)]	=	1 + 1 · x1
  [1(x1)]	=	1 · x1
  [0(x1)]	=	1 · x1
  [3(x1)]	=	1 + 1 · x1
  [2(x1)]	=	1 · x1
  [0#(x1)]	=	1 · x1
  [4(x1)]	=	1 + 1 · x1
  [3#(x1)]	=	1 + 1 · x1
  [1#(x1)]	=	1 · x1

  the pairs

  5#(5(1(0(3(2(5(1(0(0(x1))))))))))	→	5#(0(5(2(1(3(1(0(0(x1)))))))))	(154)
  5#(5(1(0(3(2(5(1(0(0(x1))))))))))	→	0#(5(2(1(3(1(0(0(x1))))))))	(155)
  3#(3(0(2(0(1(0(4(0(0(x1))))))))))	→	3#(2(0(0(1(0(4(0(0(x1)))))))))	(161)
  3#(5(3(4(0(5(0(2(3(0(x1))))))))))	→	5#(3(4(5(0(0(2(3(0(x1)))))))))	(178)
  5#(5(1(0(3(2(5(1(0(0(x1))))))))))	→	5#(2(1(3(1(0(0(x1)))))))	(156)
  5#(4(0(3(5(1(2(5(3(3(x1))))))))))	→	0#(3(5(5(2(4(1(3(3(x1)))))))))	(355)
  0#(3(5(1(5(4(3(5(0(0(x1))))))))))	→	5#(5(4(5(1(3(0(0(x1))))))))	(166)
  5#(4(0(3(5(1(2(5(3(3(x1))))))))))	→	3#(5(5(2(4(1(3(3(x1))))))))	(356)
  3#(5(3(4(0(5(0(2(3(0(x1))))))))))	→	3#(4(5(0(0(2(3(0(x1))))))))	(179)
  3#(3(4(2(1(5(3(3(4(0(x1))))))))))	→	3#(4(2(5(1(3(3(4(0(x1)))))))))	(184)
  5#(4(0(3(5(1(2(5(3(3(x1))))))))))	→	5#(5(2(4(1(3(3(x1)))))))	(357)
  5#(4(0(3(5(1(2(5(3(3(x1))))))))))	→	5#(2(4(1(3(3(x1))))))	(358)
  5#(0(2(3(5(5(1(3(5(5(x1))))))))))	→	0#(3(2(5(5(1(3(5(5(x1)))))))))	(451)
  0#(3(5(1(5(4(3(5(0(0(x1))))))))))	→	5#(4(5(1(3(0(0(x1)))))))	(167)
  5#(0(2(3(5(5(1(3(5(5(x1))))))))))	→	3#(2(5(5(1(3(5(5(x1))))))))	(452)
  3#(4(3(1(5(1(5(1(2(1(x1))))))))))	→	1#(5(4(5(3(1(2(1(x1))))))))	(228)
  1#(5(5(5(3(4(0(1(5(4(x1))))))))))	→	5#(5(4(3(0(1(5(4(x1))))))))	(422)
  1#(5(5(5(3(4(0(1(5(4(x1))))))))))	→	5#(4(3(0(1(5(4(x1)))))))	(423)
  3#(4(3(1(5(1(5(1(2(1(x1))))))))))	→	5#(4(5(3(1(2(1(x1)))))))	(229)
  3#(3(3(2(3(1(4(0(5(1(x1))))))))))	→	3#(3(2(1(3(0(4(5(1(x1)))))))))	(256)
  3#(3(3(2(3(1(4(0(5(1(x1))))))))))	→	3#(2(1(3(0(4(5(1(x1))))))))	(257)
  0#(5(5(4(3(5(0(4(1(2(x1))))))))))	→	5#(4(3(0(5(1(4(2(x1))))))))	(301)
  0#(5(3(4(3(0(3(1(5(2(x1))))))))))	→	3#(0(4(3(3(1(5(2(x1))))))))	(329)
  3#(5(0(0(4(2(1(0(2(3(x1))))))))))	→	0#(3(2(4(1(0(2(3(x1))))))))	(350)
  0#(5(3(4(3(0(3(1(5(2(x1))))))))))	→	0#(4(3(3(1(5(2(x1)))))))	(330)
  3#(5(0(0(4(2(1(0(2(3(x1))))))))))	→	3#(2(4(1(0(2(3(x1)))))))	(351)
  3#(5(0(4(1(5(0(5(4(3(x1))))))))))	→	5#(0(1(4(5(0(5(4(3(x1)))))))))	(363)
  3#(5(0(4(1(5(0(5(4(3(x1))))))))))	→	0#(1(4(5(0(5(4(3(x1))))))))	(364)
  3#(5(0(4(1(5(0(5(4(3(x1))))))))))	→	1#(4(5(0(5(4(3(x1)))))))	(365)
  3#(1(0(2(0(2(2(3(1(4(x1))))))))))	→	1#(0(2(0(1(2(3(2(4(x1)))))))))	(392)
  3#(1(0(2(0(2(2(3(1(4(x1))))))))))	→	0#(2(0(1(2(3(2(4(x1))))))))	(393)
  0#(3(0(0(2(4(3(0(5(3(x1))))))))))	→	0#(2(4(0(3(0(5(3(x1))))))))	(369)
  3#(0(0(3(0(0(0(2(3(4(x1))))))))))	→	0#(0(3(0(0(2(0(3(4(x1)))))))))	(406)
  0#(1(3(0(1(5(0(5(2(4(x1))))))))))	→	1#(0(5(0(5(2(4(x1)))))))	(403)
  0#(1(3(0(1(5(0(5(2(4(x1))))))))))	→	0#(5(0(5(2(4(x1))))))	(404)
  3#(0(0(3(0(0(0(2(3(4(x1))))))))))	→	0#(3(0(0(2(0(3(4(x1))))))))	(407)
  3#(0(0(3(0(0(0(2(3(4(x1))))))))))	→	3#(0(0(2(0(3(4(x1)))))))	(408)
  3#(0(0(3(0(0(0(2(3(4(x1))))))))))	→	0#(0(2(0(3(4(x1))))))	(409)
  3#(0(0(3(0(0(0(2(3(4(x1))))))))))	→	0#(2(0(3(4(x1)))))	(410)

  could be deleted.

  1.1.1.1.1.3.1 Dependency Graph Processor

  The dependency pairs are split into 3 components.

  • The 1^st component contains the pair

    3#(3(0(2(0(1(0(4(0(0(x1))))))))))	→	3#(3(2(0(0(1(0(4(0(0(x1))))))))))	(160)
    3#(5(3(4(0(5(0(2(3(0(x1))))))))))	→	3#(5(3(4(5(0(0(2(3(0(x1))))))))))	(177)
    3#(3(4(2(1(5(3(3(4(0(x1))))))))))	→	3#(3(4(2(5(1(3(3(4(0(x1))))))))))	(183)
    3#(4(3(1(0(0(4(2(5(0(x1))))))))))	→	3#(4(3(1(0(0(2(4(5(0(x1))))))))))	(189)
    3#(4(1(0(4(3(5(0(0(1(x1))))))))))	→	3#(4(4(0(3(1(5(0(0(1(x1))))))))))	(205)
    3#(4(3(1(5(1(5(1(2(1(x1))))))))))	→	3#(1(5(4(5(3(1(2(1(x1)))))))))	(227)
    3#(3(3(2(3(1(4(0(5(1(x1))))))))))	→	3#(3(3(2(1(3(0(4(5(1(x1))))))))))	(255)
    3#(5(0(0(4(2(1(0(2(3(x1))))))))))	→	0#(5(0(3(2(4(1(0(2(3(x1))))))))))	(348)
    0#(3(5(1(5(4(3(5(0(0(x1))))))))))	→	0#(3(5(5(4(5(1(3(0(0(x1))))))))))	(164)
    0#(3(5(1(5(4(3(5(0(0(x1))))))))))	→	3#(5(5(4(5(1(3(0(0(x1)))))))))	(165)
    3#(5(0(4(1(5(0(5(4(3(x1))))))))))	→	3#(5(0(1(4(5(0(5(4(3(x1))))))))))	(362)
    3#(1(0(2(0(2(2(3(1(4(x1))))))))))	→	3#(1(0(2(0(1(2(3(2(4(x1))))))))))	(391)
    3#(0(0(3(0(0(0(2(3(4(x1))))))))))	→	3#(0(0(3(0(0(2(0(3(4(x1))))))))))	(405)
    0#(0(4(3(0(1(0(2(2(0(x1))))))))))	→	0#(4(0(0(3(1(0(2(2(0(x1))))))))))	(172)
    0#(2(1(4(2(3(2(0(3(1(x1))))))))))	→	0#(2(4(2(3(2(1(0(3(1(x1))))))))))	(233)
    0#(1(0(2(5(2(2(2(3(1(x1))))))))))	→	0#(1(0(5(2(2(2(2(3(1(x1))))))))))	(240)
    0#(1(0(2(5(2(2(2(3(1(x1))))))))))	→	0#(5(2(2(2(2(3(1(x1))))))))	(242)
    0#(4(0(4(5(5(0(3(5(1(x1))))))))))	→	0#(4(0(5(4(5(0(3(5(1(x1))))))))))	(270)
    0#(5(5(4(3(5(0(4(1(2(x1))))))))))	→	0#(5(5(4(3(0(5(1(4(2(x1))))))))))	(299)
    0#(5(3(4(3(0(3(1(5(2(x1))))))))))	→	0#(5(3(0(4(3(3(1(5(2(x1))))))))))	(327)
    0#(2(5(0(4(2(3(4(0(3(x1))))))))))	→	3#(2(4(0(5(4(2(0(0(3(x1))))))))))	(340)
    0#(3(0(0(2(4(3(0(5(3(x1))))))))))	→	0#(3(0(2(4(0(3(0(5(3(x1))))))))))	(367)
    0#(3(0(0(2(4(3(0(5(3(x1))))))))))	→	3#(0(2(4(0(3(0(5(3(x1)))))))))	(368)
    0#(1(3(0(1(5(0(5(2(4(x1))))))))))	→	0#(1(3(1(0(5(0(5(2(4(x1))))))))))	(400)
    0#(1(3(0(1(5(0(5(2(4(x1))))))))))	→	3#(1(0(5(0(5(2(4(x1))))))))	(402)

    1.1.1.1.1.3.1.1 Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [3#(x1)]	=	1 · x1
    [3(x1)]	=	0
    [0(x1)]	=	0
    [2(x1)]	=	0
    [1(x1)]	=	0
    [4(x1)]	=	1
    [5(x1)]	=	0
    [0#(x1)]	=	0

    the pair

    3#(4(3(1(5(1(5(1(2(1(x1))))))))))

    →

    3#(1(5(4(5(3(1(2(1(x1)))))))))

    (227)

    could be deleted.

    1.1.1.1.1.3.1.1.1 Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [3#(x1)]	=	1 · x1
    [3(x1)]	=	1 · x1
    [0(x1)]	=	0
    [2(x1)]	=	1 · x1
    [1(x1)]	=	1 · x1
    [4(x1)]	=	1 · x1
    [5(x1)]	=	1
    [0#(x1)]	=	1

    the pairs

    0#(2(5(0(4(2(3(4(0(3(x1))))))))))	→	3#(2(4(0(5(4(2(0(0(3(x1))))))))))	(340)
    0#(3(0(0(2(4(3(0(5(3(x1))))))))))	→	3#(0(2(4(0(3(0(5(3(x1)))))))))	(368)
    0#(1(3(0(1(5(0(5(2(4(x1))))))))))	→	3#(1(0(5(0(5(2(4(x1))))))))	(402)

    could be deleted.

    1.1.1.1.1.3.1.1.1.1 Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [3#(x1)]	=	1 · x1
    [3(x1)]	=	1 + 1 · x1
    [0(x1)]	=	1
    [2(x1)]	=	0
    [1(x1)]	=	0
    [4(x1)]	=	0
    [5(x1)]	=	0
    [0#(x1)]	=	0

    the pair

    3#(3(0(2(0(1(0(4(0(0(x1))))))))))

    →

    3#(3(2(0(0(1(0(4(0(0(x1))))))))))

    (160)

    could be deleted.

    1.1.1.1.1.3.1.1.1.1.1 Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [3#(x1)]	=	1 + 1 · x1
    [5(x1)]	=	1 + 1 · x1
    [3(x1)]	=	1 · x1
    [4(x1)]	=	1 · x1
    [0(x1)]	=	1 + 1 · x1
    [2(x1)]	=	1 · x1
    [1(x1)]	=	1 · x1
    [0#(x1)]	=	1 + 1 · x1

    the pairs

    3#(5(0(0(4(2(1(0(2(3(x1))))))))))	→	0#(5(0(3(2(4(1(0(2(3(x1))))))))))	(348)
    0#(1(0(2(5(2(2(2(3(1(x1))))))))))	→	0#(5(2(2(2(2(3(1(x1))))))))	(242)

    could be deleted.

    1.1.1.1.1.3.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 2 components.

    • The 1^st component contains the pair

      0#(0(4(3(0(1(0(2(2(0(x1))))))))))	→	0#(4(0(0(3(1(0(2(2(0(x1))))))))))	(172)
      0#(3(5(1(5(4(3(5(0(0(x1))))))))))	→	0#(3(5(5(4(5(1(3(0(0(x1))))))))))	(164)
      0#(2(1(4(2(3(2(0(3(1(x1))))))))))	→	0#(2(4(2(3(2(1(0(3(1(x1))))))))))	(233)
      0#(1(0(2(5(2(2(2(3(1(x1))))))))))	→	0#(1(0(5(2(2(2(2(3(1(x1))))))))))	(240)
      0#(4(0(4(5(5(0(3(5(1(x1))))))))))	→	0#(4(0(5(4(5(0(3(5(1(x1))))))))))	(270)
      0#(5(5(4(3(5(0(4(1(2(x1))))))))))	→	0#(5(5(4(3(0(5(1(4(2(x1))))))))))	(299)
      0#(5(3(4(3(0(3(1(5(2(x1))))))))))	→	0#(5(3(0(4(3(3(1(5(2(x1))))))))))	(327)
      0#(3(0(0(2(4(3(0(5(3(x1))))))))))	→	0#(3(0(2(4(0(3(0(5(3(x1))))))))))	(367)
      0#(1(3(0(1(5(0(5(2(4(x1))))))))))	→	0#(1(3(1(0(5(0(5(2(4(x1))))))))))	(400)

      1.1.1.1.1.3.1.1.1.1.1.1.1 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [0#(x1)]	=	1 · x1
      [0(x1)]	=	1
      [4(x1)]	=	0
      [3(x1)]	=	1
      [1(x1)]	=	0
      [2(x1)]	=	0
      [5(x1)]	=	0

      the pair

      0#(0(4(3(0(1(0(2(2(0(x1))))))))))

      →

      0#(4(0(0(3(1(0(2(2(0(x1))))))))))

      (172)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.1.1 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [0#(x1)]	=	1 · x1
      [3(x1)]	=	0
      [5(x1)]	=	0
      [1(x1)]	=	0
      [4(x1)]	=	1 + 1 · x1
      [0(x1)]	=	1 · x1
      [2(x1)]	=	0

      the pair

      0#(4(0(4(5(5(0(3(5(1(x1))))))))))

      →

      0#(4(0(5(4(5(0(3(5(1(x1))))))))))

      (270)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [0#(x1)]	=	1 · x1
      [3(x1)]	=	1
      [5(x1)]	=	0
      [1(x1)]	=	1
      [4(x1)]	=	0
      [0(x1)]	=	1
      [2(x1)]	=	1 · x1

      the pair

      0#(2(1(4(2(3(2(0(3(1(x1))))))))))

      →

      0#(2(4(2(3(2(1(0(3(1(x1))))))))))

      (233)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [0#(x1)]	=	1 · x1
      [3(x1)]	=	1
      [5(x1)]	=	0
      [1(x1)]	=	1 · x1
      [4(x1)]	=	0
      [0(x1)]	=	1 · x1
      [2(x1)]	=	1

      the pair

      0#(1(0(2(5(2(2(2(3(1(x1))))))))))

      →

      0#(1(0(5(2(2(2(2(3(1(x1))))))))))

      (240)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [0#(x1)]	=	1 · x1
      [3(x1)]	=	1 + 1 · x1
      [5(x1)]	=	0
      [1(x1)]	=	0
      [4(x1)]	=	0
      [0(x1)]	=	1 + 1 · x1
      [2(x1)]	=	0

      the pair

      0#(3(0(0(2(4(3(0(5(3(x1))))))))))

      →

      0#(3(0(2(4(0(3(0(5(3(x1))))))))))

      (367)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [0#(x1)]	=	1 · x1
      [3(x1)]	=	1 + 1 · x1
      [5(x1)]	=	1 · x1
      [1(x1)]	=	1 · x1
      [4(x1)]	=	1 · x1
      [0(x1)]	=	1
      [2(x1)]	=	0

      the pair

      0#(5(3(4(3(0(3(1(5(2(x1))))))))))

      →

      0#(5(3(0(4(3(3(1(5(2(x1))))))))))

      (327)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [5(x1)]	=	2 + 2 · x1
      [1(x1)]	=	1 · x1
      [0(x1)]	=	1 · x1
      [3(x1)]	=	2 + 2 · x1
      [2(x1)]	=	2 · x1
      [4(x1)]	=	1 · x1
      [0#(x1)]	=	2 · x1

      the rules

      4(3(3(5(0(1(2(3(1(1(x1))))))))))	→	4(3(3(5(0(1(3(2(1(1(x1))))))))))	(89)
      0(1(0(2(5(2(2(2(3(1(x1))))))))))	→	0(1(0(5(2(2(2(2(3(1(x1))))))))))	(92)
      5(3(0(0(4(2(2(5(4(1(x1))))))))))	→	5(2(4(0(3(1(0(5(2(4(x1))))))))))	(93)
      5(2(2(1(2(3(3(0(1(2(x1))))))))))	→	5(2(2(1(3(2(3(0(1(2(x1))))))))))	(98)
      1(4(2(5(3(4(4(2(1(2(x1))))))))))	→	5(2(4(4(1(3(4(2(1(2(x1))))))))))	(99)
      1(2(4(2(3(1(4(3(1(2(x1))))))))))	→	1(2(4(3(2(1(4(3(1(2(x1))))))))))	(100)
      0(2(5(0(4(2(3(4(0(3(x1))))))))))	→	3(2(4(0(5(4(2(0(0(3(x1))))))))))	(108)
      5(4(0(3(5(1(2(5(3(3(x1))))))))))	→	5(0(3(5(5(2(4(1(3(3(x1))))))))))	(110)
      3(2(4(2(0(2(3(1(5(3(x1))))))))))	→	2(3(2(0(5(4(1(3(2(3(x1))))))))))	(114)
      3(1(0(2(0(2(2(3(1(4(x1))))))))))	→	3(1(0(2(0(1(2(3(2(4(x1))))))))))	(116)
      0(2(0(1(2(3(1(0(1(5(x1))))))))))	→	2(0(0(1(3(2(1(0(1(5(x1))))))))))	(123)
      5(0(2(3(5(5(1(3(5(5(x1))))))))))	→	5(0(3(2(5(5(1(3(5(5(x1))))))))))	(126)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [0#(x1)]	=	1 · x1
      [3(x1)]	=	1 + 1 · x1
      [5(x1)]	=	1 · x1
      [1(x1)]	=	1
      [4(x1)]	=	0
      [0(x1)]	=	0
      [2(x1)]	=	1 · x1

      the pair

      0#(3(5(1(5(4(3(5(0(0(x1))))))))))

      →

      0#(3(5(5(4(5(1(3(0(0(x1))))))))))

      (164)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [5(x1)]	=	1 · x1
      [1(x1)]	=	1 · x1
      [0(x1)]	=	2 + 2 · x1
      [3(x1)]	=	2 · x1
      [2(x1)]	=	2 + 2 · x1
      [4(x1)]	=	1 · x1
      [0#(x1)]	=	2 · x1

      the rules

      5(5(1(0(3(2(5(1(0(0(x1))))))))))	→	5(5(0(5(2(1(3(1(0(0(x1))))))))))	(78)
      0(0(4(3(0(1(0(2(2(0(x1))))))))))	→	0(4(0(0(3(1(0(2(2(0(x1))))))))))	(81)
      0(5(3(4(3(0(3(1(5(2(x1))))))))))	→	0(5(3(0(4(3(3(1(5(2(x1))))))))))	(105)
      5(4(1(5(3(0(4(1(0(3(x1))))))))))	→	5(4(0(5(4(3(1(1(0(3(x1))))))))))	(106)
      3(5(0(0(4(2(1(0(2(3(x1))))))))))	→	0(5(0(3(2(4(1(0(2(3(x1))))))))))	(109)
      1(4(3(4(3(4(2(4(0(4(x1))))))))))	→	1(4(4(3(0(4(3(2(4(4(x1))))))))))	(115)
      4(4(3(0(3(2(5(3(0(5(x1))))))))))	→	4(4(0(3(3(2(5(3(0(5(x1))))))))))	(122)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 2 components.

      • The 1^st component contains the pair

        0#(1(3(0(1(5(0(5(2(4(x1))))))))))

        →

        0#(1(3(1(0(5(0(5(2(4(x1))))))))))

        (400)

        1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

        Using the linear polynomial interpretation over the naturals

        [3(x1)]	=	1 · x1
        [0(x1)]	=	2 · x1
        [2(x1)]	=	2 + 2 · x1
        [1(x1)]	=	1 · x1
        [4(x1)]	=	1 · x1
        [5(x1)]	=	2 · x1
        [0#(x1)]	=	2 · x1

        the rules

        3(3(0(2(0(1(0(4(0(0(x1))))))))))	→	3(3(2(0(0(1(0(4(0(0(x1))))))))))	(79)
        2(3(1(5(2(0(2(1(5(1(x1))))))))))	→	2(3(1(5(2(2(0(1(5(1(x1))))))))))	(95)
        0(3(0(0(2(4(3(0(5(3(x1))))))))))	→	0(3(0(2(4(0(3(0(5(3(x1))))))))))	(113)
        3(0(0(3(0(0(0(2(3(4(x1))))))))))	→	3(0(0(3(0(0(2(0(3(4(x1))))))))))	(118)

        could be deleted.

        1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

        Using the linear polynomial interpretation over the naturals

        [0(x1)]	=	1 · x1
        [3(x1)]	=	2 · x1
        [5(x1)]	=	1 · x1
        [1(x1)]	=	2 · x1
        [4(x1)]	=	2 · x1
        [2(x1)]	=	1 + 2 · x1
        [0#(x1)]	=	2 · x1

        the rules

        3(4(3(1(0(0(4(2(5(0(x1))))))))))	→	3(4(3(1(0(0(2(4(5(0(x1))))))))))	(84)
        0(2(1(4(2(3(2(0(3(1(x1))))))))))	→	0(2(4(2(3(2(1(0(3(1(x1))))))))))	(91)

        could be deleted.

        1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

        Using the linear polynomial interpretation over the naturals

        [0(x1)]	=	1 · x1
        [3(x1)]	=	2 · x1
        [5(x1)]	=	1 + 1 · x1
        [1(x1)]	=	2 · x1
        [4(x1)]	=	1 · x1
        [2(x1)]	=	2 · x1
        [0#(x1)]	=	2 · x1

        the rules

        0(3(5(1(5(4(3(5(0(0(x1))))))))))	→	0(3(5(5(4(5(1(3(0(0(x1))))))))))	(80)
        3(3(4(2(1(5(3(3(4(0(x1))))))))))	→	3(3(4(2(5(1(3(3(4(0(x1))))))))))	(83)
        3(4(3(1(5(1(5(1(2(1(x1))))))))))	→	1(3(1(5(4(5(3(1(2(1(x1))))))))))	(90)
        3(4(1(1(5(1(5(1(1(5(x1))))))))))	→	1(4(5(1(3(1(5(1(1(5(x1))))))))))	(124)

        could be deleted.

        1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

        Using the linear polynomial interpretation over the naturals

        [0#(x1)]	=	4 · x1
        [1(x1)]	=	2 + 4 · x1
        [3(x1)]	=	1 + 4 · x1
        [0(x1)]	=	2 · x1
        [5(x1)]	=	0
        [2(x1)]	=	1 · x1
        [4(x1)]	=	0

        together with the usable rules

        4(3(3(4(1(4(1(0(4(2(x1))))))))))	→	4(3(1(1(3(4(4(0(4(2(x1))))))))))	(103)
        4(3(3(1(2(3(4(0(0(2(x1))))))))))	→	4(3(3(1(2(4(3(0(0(2(x1))))))))))	(97)
        0(5(5(4(3(5(0(4(1(2(x1))))))))))	→	0(5(5(4(3(0(5(1(4(2(x1))))))))))	(101)
        1(2(1(3(0(4(0(5(5(0(x1))))))))))	→	1(2(1(3(0(0(5(4(5(0(x1))))))))))	(85)
        1(5(5(5(3(4(0(1(5(4(x1))))))))))	→	1(5(5(5(4(3(0(1(5(4(x1))))))))))	(120)
        3(5(3(4(0(5(0(2(3(0(x1))))))))))	→	3(5(3(4(5(0(0(2(3(0(x1))))))))))	(82)
        3(4(1(0(4(3(5(0(0(1(x1))))))))))	→	3(4(4(0(3(1(5(0(0(1(x1))))))))))	(87)
        3(3(3(2(3(1(4(0(5(1(x1))))))))))	→	3(3(3(2(1(3(0(4(5(1(x1))))))))))	(94)
        3(5(0(4(1(5(0(5(4(3(x1))))))))))	→	3(5(0(1(4(5(0(5(4(3(x1))))))))))	(112)

        (w.r.t. the implicit argument filter of the reduction pair), the pair

        0#(1(3(0(1(5(0(5(2(4(x1))))))))))

        →

        0#(1(3(1(0(5(0(5(2(4(x1))))))))))

        (400)

        could be deleted.

        1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 P is empty

        There are no pairs anymore.

      • The 2^nd component contains the pair

        0#(5(5(4(3(5(0(4(1(2(x1))))))))))

        →

        0#(5(5(4(3(0(5(1(4(2(x1))))))))))

        (299)

        1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2 Monotonic Reduction Pair Processor

        Using the linear polynomial interpretation over the naturals

        [3(x1)]	=	1 · x1
        [0(x1)]	=	1 · x1
        [2(x1)]	=	2 + 2 · x1
        [1(x1)]	=	2 · x1
        [4(x1)]	=	1 · x1
        [5(x1)]	=	2 · x1
        [0#(x1)]	=	1 · x1

        the rule

        0(2(1(4(2(3(2(0(3(1(x1))))))))))

        →

        0(2(4(2(3(2(1(0(3(1(x1))))))))))

        (91)

        could be deleted.

        1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1 Monotonic Reduction Pair Processor

        Using the linear polynomial interpretation over the naturals

        [3(x1)]	=	2 · x1
        [0(x1)]	=	2 · x1
        [2(x1)]	=	2 + 2 · x1
        [1(x1)]	=	2 · x1
        [4(x1)]	=	2 · x1
        [5(x1)]	=	2 · x1
        [0#(x1)]	=	1 · x1

        the rules

        3(3(0(2(0(1(0(4(0(0(x1))))))))))	→	3(3(2(0(0(1(0(4(0(0(x1))))))))))	(79)
        3(4(3(1(0(0(4(2(5(0(x1))))))))))	→	3(4(3(1(0(0(2(4(5(0(x1))))))))))	(84)
        2(3(1(5(2(0(2(1(5(1(x1))))))))))	→	2(3(1(5(2(2(0(1(5(1(x1))))))))))	(95)
        0(3(0(0(2(4(3(0(5(3(x1))))))))))	→	0(3(0(2(4(0(3(0(5(3(x1))))))))))	(113)
        3(0(0(3(0(0(0(2(3(4(x1))))))))))	→	3(0(0(3(0(0(2(0(3(4(x1))))))))))	(118)

        could be deleted.

        1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1 Monotonic Reduction Pair Processor

        Using the linear polynomial interpretation over the naturals

        [0(x1)]	=	1 + 2 · x1
        [3(x1)]	=	2 · x1
        [5(x1)]	=	2 + 3 · x1
        [1(x1)]	=	1 · x1
        [4(x1)]	=	1 · x1
        [2(x1)]	=	2 · x1
        [0#(x1)]	=	1 · x1

        the rules

        0(3(5(1(5(4(3(5(0(0(x1))))))))))	→	0(3(5(5(4(5(1(3(0(0(x1))))))))))	(80)
        3(4(3(1(5(1(5(1(2(1(x1))))))))))	→	1(3(1(5(4(5(3(1(2(1(x1))))))))))	(90)
        3(4(1(1(5(1(5(1(1(5(x1))))))))))	→	1(4(5(1(3(1(5(1(1(5(x1))))))))))	(124)

        could be deleted.

        1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1 Monotonic Reduction Pair Processor

        Using the linear polynomial interpretation over the naturals

        [3(x1)]	=	1 · x1
        [5(x1)]	=	1 + 1 · x1
        [4(x1)]	=	1 · x1
        [0(x1)]	=	1 · x1
        [2(x1)]	=	2 · x1
        [1(x1)]	=	2 + 2 · x1
        [0#(x1)]	=	1 · x1

        the rule

        3(3(4(2(1(5(3(3(4(0(x1))))))))))

        →

        3(3(4(2(5(1(3(3(4(0(x1))))))))))

        (83)

        could be deleted.

        1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1 Reduction Pair Processor

        Using the linear polynomial interpretation over the naturals

        [0(x1)]	=	-1 + 2 · x1
        [0#(x1)]	=	2 + 2 · x1
        [3(x1)]	=	-2 + x1
        [4(x1)]	=	-2 + 2 · x1
        [5(x1)]	=	-2 + 2 · x1
        [1(x1)]	=	2
        [2(x1)]	=	-2

        the pair

        0#(5(5(4(3(5(0(4(1(2(x1))))))))))

        →

        0#(5(5(4(3(0(5(1(4(2(x1))))))))))

        (299)

        could be deleted.

        1.1.1.1.1.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1 P is empty

        There are no pairs anymore.

    • The 2^nd component contains the pair

      3#(3(4(2(1(5(3(3(4(0(x1))))))))))	→	3#(3(4(2(5(1(3(3(4(0(x1))))))))))	(183)
      3#(5(3(4(0(5(0(2(3(0(x1))))))))))	→	3#(5(3(4(5(0(0(2(3(0(x1))))))))))	(177)
      3#(4(3(1(0(0(4(2(5(0(x1))))))))))	→	3#(4(3(1(0(0(2(4(5(0(x1))))))))))	(189)
      3#(4(1(0(4(3(5(0(0(1(x1))))))))))	→	3#(4(4(0(3(1(5(0(0(1(x1))))))))))	(205)
      3#(3(3(2(3(1(4(0(5(1(x1))))))))))	→	3#(3(3(2(1(3(0(4(5(1(x1))))))))))	(255)
      3#(5(0(4(1(5(0(5(4(3(x1))))))))))	→	3#(5(0(1(4(5(0(5(4(3(x1))))))))))	(362)
      3#(1(0(2(0(2(2(3(1(4(x1))))))))))	→	3#(1(0(2(0(1(2(3(2(4(x1))))))))))	(391)
      3#(0(0(3(0(0(0(2(3(4(x1))))))))))	→	3#(0(0(3(0(0(2(0(3(4(x1))))))))))	(405)

      1.1.1.1.1.3.1.1.1.1.1.1.2 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [3#(x1)]	=	1 · x1
      [3(x1)]	=	1 · x1
      [4(x1)]	=	1 · x1
      [2(x1)]	=	0
      [1(x1)]	=	1
      [5(x1)]	=	0
      [0(x1)]	=	0

      the pair

      3#(4(1(0(4(3(5(0(0(1(x1))))))))))

      →

      3#(4(4(0(3(1(5(0(0(1(x1))))))))))

      (205)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.2.1 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [3#(x1)]	=	1 · x1
      [3(x1)]	=	0
      [4(x1)]	=	1
      [2(x1)]	=	0
      [1(x1)]	=	0
      [5(x1)]	=	1 · x1
      [0(x1)]	=	1 · x1

      the pair

      3#(5(0(4(1(5(0(5(4(3(x1))))))))))

      →

      3#(5(0(1(4(5(0(5(4(3(x1))))))))))

      (362)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.2.1.1 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [3#(x1)]	=	1 · x1
      [3(x1)]	=	1 + 1 · x1
      [4(x1)]	=	0
      [2(x1)]	=	0
      [1(x1)]	=	0
      [5(x1)]	=	0
      [0(x1)]	=	1 + 1 · x1

      the pair

      3#(0(0(3(0(0(0(2(3(4(x1))))))))))

      →

      3#(0(0(3(0(0(2(0(3(4(x1))))))))))

      (405)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.2.1.1.1 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [3#(x1)]	=	1 · x1
      [3(x1)]	=	1 · x1
      [4(x1)]	=	1 + 1 · x1
      [2(x1)]	=	0
      [1(x1)]	=	1 · x1
      [5(x1)]	=	0
      [0(x1)]	=	1 · x1

      the pair

      3#(4(3(1(0(0(4(2(5(0(x1))))))))))

      →

      3#(4(3(1(0(0(2(4(5(0(x1))))))))))

      (189)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.2.1.1.1.1 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [3#(x1)]	=	1 · x1
      [3(x1)]	=	1 · x1
      [4(x1)]	=	1 · x1
      [2(x1)]	=	1 · x1
      [1(x1)]	=	1
      [5(x1)]	=	0
      [0(x1)]	=	0

      the pair

      3#(3(4(2(1(5(3(3(4(0(x1))))))))))

      →

      3#(3(4(2(5(1(3(3(4(0(x1))))))))))

      (183)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.2.1.1.1.1.1 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [3#(x1)]	=	2 · x1
      [5(x1)]	=	0
      [3(x1)]	=	4 · x1
      [4(x1)]	=	1 · x1
      [0(x1)]	=	0
      [2(x1)]	=	4 · x1
      [1(x1)]	=	2

      the pair

      3#(3(3(2(3(1(4(0(5(1(x1))))))))))

      →

      3#(3(3(2(1(3(0(4(5(1(x1))))))))))

      (255)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.2.1.1.1.1.1.1 Monotonic Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [5(x1)]	=	1 + 1 · x1
      [1(x1)]	=	1 · x1
      [0(x1)]	=	1 · x1
      [3(x1)]	=	2 + 2 · x1
      [2(x1)]	=	2 · x1
      [4(x1)]	=	1 · x1
      [3#(x1)]	=	1 · x1

      the pair

      3#(1(0(2(0(2(2(3(1(4(x1))))))))))

      →

      3#(1(0(2(0(1(2(3(2(4(x1))))))))))

      (391)

      and the rules

      5(5(1(0(3(2(5(1(0(0(x1))))))))))	→	5(5(0(5(2(1(3(1(0(0(x1))))))))))	(78)
      0(3(5(1(5(4(3(5(0(0(x1))))))))))	→	0(3(5(5(4(5(1(3(0(0(x1))))))))))	(80)
      4(3(3(5(0(1(2(3(1(1(x1))))))))))	→	4(3(3(5(0(1(3(2(1(1(x1))))))))))	(89)
      3(4(3(1(5(1(5(1(2(1(x1))))))))))	→	1(3(1(5(4(5(3(1(2(1(x1))))))))))	(90)
      0(1(0(2(5(2(2(2(3(1(x1))))))))))	→	0(1(0(5(2(2(2(2(3(1(x1))))))))))	(92)
      5(3(0(0(4(2(2(5(4(1(x1))))))))))	→	5(2(4(0(3(1(0(5(2(4(x1))))))))))	(93)
      5(2(2(1(2(3(3(0(1(2(x1))))))))))	→	5(2(2(1(3(2(3(0(1(2(x1))))))))))	(98)
      1(4(2(5(3(4(4(2(1(2(x1))))))))))	→	5(2(4(4(1(3(4(2(1(2(x1))))))))))	(99)
      1(2(4(2(3(1(4(3(1(2(x1))))))))))	→	1(2(4(3(2(1(4(3(1(2(x1))))))))))	(100)
      0(2(5(0(4(2(3(4(0(3(x1))))))))))	→	3(2(4(0(5(4(2(0(0(3(x1))))))))))	(108)
      3(5(0(0(4(2(1(0(2(3(x1))))))))))	→	0(5(0(3(2(4(1(0(2(3(x1))))))))))	(109)
      5(4(0(3(5(1(2(5(3(3(x1))))))))))	→	5(0(3(5(5(2(4(1(3(3(x1))))))))))	(110)
      3(2(4(2(0(2(3(1(5(3(x1))))))))))	→	2(3(2(0(5(4(1(3(2(3(x1))))))))))	(114)
      3(1(0(2(0(2(2(3(1(4(x1))))))))))	→	3(1(0(2(0(1(2(3(2(4(x1))))))))))	(116)
      0(2(0(1(2(3(1(0(1(5(x1))))))))))	→	2(0(0(1(3(2(1(0(1(5(x1))))))))))	(123)
      3(4(1(1(5(1(5(1(1(5(x1))))))))))	→	1(4(5(1(3(1(5(1(1(5(x1))))))))))	(124)
      5(0(2(3(5(5(1(3(5(5(x1))))))))))	→	5(0(3(2(5(5(1(3(5(5(x1))))))))))	(126)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.2.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [3(x1)]	=	2 · x1
      [0(x1)]	=	1 · x1
      [2(x1)]	=	2 + 2 · x1
      [1(x1)]	=	2 · x1
      [4(x1)]	=	1 · x1
      [5(x1)]	=	2 · x1
      [3#(x1)]	=	1 · x1

      the rule

      0(2(1(4(2(3(2(0(3(1(x1))))))))))

      →

      0(2(4(2(3(2(1(0(3(1(x1))))))))))

      (91)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [3(x1)]	=	2 · x1
      [0(x1)]	=	1 · x1
      [2(x1)]	=	1 + 2 · x1
      [1(x1)]	=	2 · x1
      [4(x1)]	=	2 · x1
      [5(x1)]	=	2 · x1
      [3#(x1)]	=	1 · x1

      the rule

      3(4(3(1(0(0(4(2(5(0(x1))))))))))

      →

      3(4(3(1(0(0(2(4(5(0(x1))))))))))

      (84)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [3(x1)]	=	2 · x1
      [0(x1)]	=	1 + 2 · x1
      [2(x1)]	=	1 · x1
      [1(x1)]	=	1 · x1
      [4(x1)]	=	1 · x1
      [5(x1)]	=	1 · x1
      [3#(x1)]	=	1 · x1

      the rules

      0(0(4(3(0(1(0(2(2(0(x1))))))))))	→	0(4(0(0(3(1(0(2(2(0(x1))))))))))	(81)
      0(5(3(4(3(0(3(1(5(2(x1))))))))))	→	0(5(3(0(4(3(3(1(5(2(x1))))))))))	(105)
      5(4(1(5(3(0(4(1(0(3(x1))))))))))	→	5(4(0(5(4(3(1(1(0(3(x1))))))))))	(106)
      1(4(3(4(3(4(2(4(0(4(x1))))))))))	→	1(4(4(3(0(4(3(2(4(4(x1))))))))))	(115)
      4(4(3(0(3(2(5(3(0(5(x1))))))))))	→	4(4(0(3(3(2(5(3(0(5(x1))))))))))	(122)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [3(x1)]	=	2 · x1
      [0(x1)]	=	2 · x1
      [2(x1)]	=	2 + 2 · x1
      [1(x1)]	=	2 · x1
      [4(x1)]	=	2 · x1
      [5(x1)]	=	2 · x1
      [3#(x1)]	=	1 · x1

      the rules

      3(3(0(2(0(1(0(4(0(0(x1))))))))))	→	3(3(2(0(0(1(0(4(0(0(x1))))))))))	(79)
      2(3(1(5(2(0(2(1(5(1(x1))))))))))	→	2(3(1(5(2(2(0(1(5(1(x1))))))))))	(95)
      0(3(0(0(2(4(3(0(5(3(x1))))))))))	→	0(3(0(2(4(0(3(0(5(3(x1))))))))))	(113)
      3(0(0(3(0(0(0(2(3(4(x1))))))))))	→	3(0(0(3(0(0(2(0(3(4(x1))))))))))	(118)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [3(x1)]	=	1 · x1
      [5(x1)]	=	2 + 2 · x1
      [4(x1)]	=	1 · x1
      [0(x1)]	=	1 · x1
      [2(x1)]	=	1 · x1
      [1(x1)]	=	2 · x1
      [3#(x1)]	=	2 · x1

      the rule

      3(3(4(2(1(5(3(3(4(0(x1))))))))))

      →

      3(3(4(2(5(1(3(3(4(0(x1))))))))))

      (83)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

      Using the linear polynomial interpretation over the naturals

      [3(x1)]	=	-2 + 2 · x1
      [3#(x1)]	=	-2 + 2 · x1
      [4(x1)]	=	2 · x1
      [5(x1)]	=	-2 + 2 · x1
      [0(x1)]	=	1
      [1(x1)]	=	-2
      [2(x1)]	=	0

      together with the usable rules

      0(4(0(4(5(5(0(3(5(1(x1))))))))))	→	0(4(0(5(4(5(0(3(5(1(x1))))))))))	(96)
      0(5(5(4(3(5(0(4(1(2(x1))))))))))	→	0(5(5(4(3(0(5(1(4(2(x1))))))))))	(101)
      0(1(3(0(1(5(0(5(2(4(x1))))))))))	→	0(1(3(1(0(5(0(5(2(4(x1))))))))))	(117)

      (w.r.t. the implicit argument filter of the reduction pair), the pair

      3#(5(3(4(0(5(0(2(3(0(x1))))))))))

      →

      3#(5(3(4(5(0(0(2(3(0(x1))))))))))

      (177)

      could be deleted.

      1.1.1.1.1.3.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1 P is empty

      There are no pairs anymore.

  • The 2^nd component contains the pair

    1#(2(4(2(3(1(4(3(1(2(x1))))))))))	→	1#(2(4(3(2(1(4(3(1(2(x1))))))))))	(294)
    1#(2(1(3(0(4(0(5(5(0(x1))))))))))	→	1#(2(1(3(0(0(5(4(5(0(x1))))))))))	(197)
    1#(4(3(4(3(4(2(4(0(4(x1))))))))))	→	1#(4(4(3(0(4(3(2(4(4(x1))))))))))	(382)
    1#(5(5(5(3(4(0(1(5(4(x1))))))))))	→	1#(5(5(5(4(3(0(1(5(4(x1))))))))))	(420)

    1.1.1.1.1.3.1.2 Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [1#(x1)]	=	1 · x1
    [2(x1)]	=	0
    [4(x1)]	=	1 + 1 · x1
    [3(x1)]	=	1 + 1 · x1
    [1(x1)]	=	1 + 1 · x1
    [0(x1)]	=	1
    [5(x1)]	=	0

    the pair

    1#(4(3(4(3(4(2(4(0(4(x1))))))))))

    →

    1#(4(4(3(0(4(3(2(4(4(x1))))))))))

    (382)

    could be deleted.

    1.1.1.1.1.3.1.2.1 Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [1#(x1)]	=	1 · x1
    [2(x1)]	=	0
    [4(x1)]	=	1
    [3(x1)]	=	1 + 1 · x1
    [1(x1)]	=	1
    [0(x1)]	=	1
    [5(x1)]	=	1 · x1

    the pair

    1#(5(5(5(3(4(0(1(5(4(x1))))))))))

    →

    1#(5(5(5(4(3(0(1(5(4(x1))))))))))

    (420)

    could be deleted.

    1.1.1.1.1.3.1.2.1.1 Monotonic Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [5(x1)]	=	3 + 1 · x1
    [1(x1)]	=	1 · x1
    [0(x1)]	=	1 · x1
    [3(x1)]	=	2 + 1 · x1
    [2(x1)]	=	2 · x1
    [4(x1)]	=	1 · x1
    [1#(x1)]	=	2 · x1

    the pair

    1#(2(4(2(3(1(4(3(1(2(x1))))))))))

    →

    1#(2(4(3(2(1(4(3(1(2(x1))))))))))

    (294)

    and the rules

    5(5(1(0(3(2(5(1(0(0(x1))))))))))	→	5(5(0(5(2(1(3(1(0(0(x1))))))))))	(78)
    4(3(3(5(0(1(2(3(1(1(x1))))))))))	→	4(3(3(5(0(1(3(2(1(1(x1))))))))))	(89)
    0(1(0(2(5(2(2(2(3(1(x1))))))))))	→	0(1(0(5(2(2(2(2(3(1(x1))))))))))	(92)
    5(3(0(0(4(2(2(5(4(1(x1))))))))))	→	5(2(4(0(3(1(0(5(2(4(x1))))))))))	(93)
    5(2(2(1(2(3(3(0(1(2(x1))))))))))	→	5(2(2(1(3(2(3(0(1(2(x1))))))))))	(98)
    1(4(2(5(3(4(4(2(1(2(x1))))))))))	→	5(2(4(4(1(3(4(2(1(2(x1))))))))))	(99)
    1(2(4(2(3(1(4(3(1(2(x1))))))))))	→	1(2(4(3(2(1(4(3(1(2(x1))))))))))	(100)
    0(2(5(0(4(2(3(4(0(3(x1))))))))))	→	3(2(4(0(5(4(2(0(0(3(x1))))))))))	(108)
    5(4(0(3(5(1(2(5(3(3(x1))))))))))	→	5(0(3(5(5(2(4(1(3(3(x1))))))))))	(110)
    3(2(4(2(0(2(3(1(5(3(x1))))))))))	→	2(3(2(0(5(4(1(3(2(3(x1))))))))))	(114)
    3(1(0(2(0(2(2(3(1(4(x1))))))))))	→	3(1(0(2(0(1(2(3(2(4(x1))))))))))	(116)
    0(2(0(1(2(3(1(0(1(5(x1))))))))))	→	2(0(0(1(3(2(1(0(1(5(x1))))))))))	(123)
    5(0(2(3(5(5(1(3(5(5(x1))))))))))	→	5(0(3(2(5(5(1(3(5(5(x1))))))))))	(126)

    could be deleted.

    1.1.1.1.1.3.1.2.1.1.1 Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [1#(x1)]	=	1 · x1
    [2(x1)]	=	1 · x1
    [1(x1)]	=	1 · x1
    [3(x1)]	=	1 + 1 · x1
    [0(x1)]	=	1 · x1
    [4(x1)]	=	1
    [5(x1)]	=	0

    the pair

    1#(2(1(3(0(4(0(5(5(0(x1))))))))))

    →

    1#(2(1(3(0(0(5(4(5(0(x1))))))))))

    (197)

    could be deleted.

    1.1.1.1.1.3.1.2.1.1.1.1 P is empty

    There are no pairs anymore.

  • The 3^rd component contains the pair

    5#(3(0(0(4(2(2(5(4(1(x1))))))))))	→	5#(2(4(0(3(1(0(5(2(4(x1))))))))))	(245)
    5#(5(1(0(3(2(5(1(0(0(x1))))))))))	→	5#(5(0(5(2(1(3(1(0(0(x1))))))))))	(153)
    5#(2(2(1(2(3(3(0(1(2(x1))))))))))	→	5#(2(2(1(3(2(3(0(1(2(x1))))))))))	(282)
    5#(4(1(5(3(0(4(1(0(3(x1))))))))))	→	5#(4(0(5(4(3(1(1(0(3(x1))))))))))	(333)
    5#(4(0(3(5(1(2(5(3(3(x1))))))))))	→	5#(0(3(5(5(2(4(1(3(3(x1))))))))))	(354)
    5#(0(2(3(5(5(1(3(5(5(x1))))))))))	→	5#(0(3(2(5(5(1(3(5(5(x1))))))))))	(450)

    1.1.1.1.1.3.1.3 Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [5#(x1)]	=	1 · x1
    [3(x1)]	=	0
    [0(x1)]	=	0
    [4(x1)]	=	1
    [2(x1)]	=	0
    [5(x1)]	=	0
    [1(x1)]	=	0

    the pair

    5#(4(0(3(5(1(2(5(3(3(x1))))))))))

    →

    5#(0(3(5(5(2(4(1(3(3(x1))))))))))

    (354)

    could be deleted.

    1.1.1.1.1.3.1.3.1 Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [5#(x1)]	=	1 · x1
    [3(x1)]	=	1
    [0(x1)]	=	1
    [4(x1)]	=	0
    [2(x1)]	=	0
    [5(x1)]	=	0
    [1(x1)]	=	0

    the pair

    5#(3(0(0(4(2(2(5(4(1(x1))))))))))

    →

    5#(2(4(0(3(1(0(5(2(4(x1))))))))))

    (245)

    could be deleted.

    1.1.1.1.1.3.1.3.1.1 Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [5#(x1)]	=	1 · x1
    [5(x1)]	=	1 · x1
    [1(x1)]	=	1
    [0(x1)]	=	0
    [3(x1)]	=	1 · x1
    [2(x1)]	=	0
    [4(x1)]	=	1 + 1 · x1

    the pairs

    5#(5(1(0(3(2(5(1(0(0(x1))))))))))	→	5#(5(0(5(2(1(3(1(0(0(x1))))))))))	(153)
    5#(4(1(5(3(0(4(1(0(3(x1))))))))))	→	5#(4(0(5(4(3(1(1(0(3(x1))))))))))	(333)

    could be deleted.

    1.1.1.1.1.3.1.3.1.1.1 Monotonic Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [5(x1)]	=	2 + 1 · x1
    [1(x1)]	=	1 · x1
    [0(x1)]	=	1 · x1
    [3(x1)]	=	3 + 2 · x1
    [2(x1)]	=	3 + 3 · x1
    [4(x1)]	=	1 · x1
    [5#(x1)]	=	3 · x1

    the pairs

    5#(2(2(1(2(3(3(0(1(2(x1))))))))))	→	5#(2(2(1(3(2(3(0(1(2(x1))))))))))	(282)
    5#(0(2(3(5(5(1(3(5(5(x1))))))))))	→	5#(0(3(2(5(5(1(3(5(5(x1))))))))))	(450)

    and the rules

    5(5(1(0(3(2(5(1(0(0(x1))))))))))	→	5(5(0(5(2(1(3(1(0(0(x1))))))))))	(78)
    0(3(5(1(5(4(3(5(0(0(x1))))))))))	→	0(3(5(5(4(5(1(3(0(0(x1))))))))))	(80)
    4(3(3(5(0(1(2(3(1(1(x1))))))))))	→	4(3(3(5(0(1(3(2(1(1(x1))))))))))	(89)
    3(4(3(1(5(1(5(1(2(1(x1))))))))))	→	1(3(1(5(4(5(3(1(2(1(x1))))))))))	(90)
    0(1(0(2(5(2(2(2(3(1(x1))))))))))	→	0(1(0(5(2(2(2(2(3(1(x1))))))))))	(92)
    5(3(0(0(4(2(2(5(4(1(x1))))))))))	→	5(2(4(0(3(1(0(5(2(4(x1))))))))))	(93)
    5(2(2(1(2(3(3(0(1(2(x1))))))))))	→	5(2(2(1(3(2(3(0(1(2(x1))))))))))	(98)
    1(4(2(5(3(4(4(2(1(2(x1))))))))))	→	5(2(4(4(1(3(4(2(1(2(x1))))))))))	(99)
    1(2(4(2(3(1(4(3(1(2(x1))))))))))	→	1(2(4(3(2(1(4(3(1(2(x1))))))))))	(100)
    0(2(5(0(4(2(3(4(0(3(x1))))))))))	→	3(2(4(0(5(4(2(0(0(3(x1))))))))))	(108)
    3(5(0(0(4(2(1(0(2(3(x1))))))))))	→	0(5(0(3(2(4(1(0(2(3(x1))))))))))	(109)
    5(4(0(3(5(1(2(5(3(3(x1))))))))))	→	5(0(3(5(5(2(4(1(3(3(x1))))))))))	(110)
    3(2(4(2(0(2(3(1(5(3(x1))))))))))	→	2(3(2(0(5(4(1(3(2(3(x1))))))))))	(114)
    3(1(0(2(0(2(2(3(1(4(x1))))))))))	→	3(1(0(2(0(1(2(3(2(4(x1))))))))))	(116)
    0(2(0(1(2(3(1(0(1(5(x1))))))))))	→	2(0(0(1(3(2(1(0(1(5(x1))))))))))	(123)
    3(4(1(1(5(1(5(1(1(5(x1))))))))))	→	1(4(5(1(3(1(5(1(1(5(x1))))))))))	(124)
    5(0(2(3(5(5(1(3(5(5(x1))))))))))	→	5(0(3(2(5(5(1(3(5(5(x1))))))))))	(126)

    could be deleted.

    1.1.1.1.1.3.1.3.1.1.1.1 P is empty

    There are no pairs anymore.