Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/86857)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Rule Removal

1.1 Dependency Pair Transformation

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

1.1.1 Reduction Pair Processor

There are 277 ruless (increase limit for explicit display).

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 5 components.

• The 1^st component contains the pair

  1#(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1)))))))))))))))	→	1#(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))))	(104)
  1#(5(2(2(5(1(1(2(4(4(5(4(x1))))))))))))	→	1#(2(2(3(3(3(3(0(1(1(3(4(x1))))))))))))	(80)
  1#(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1)))))))))))))))))))))	→	1#(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))))	(266)

  1.1.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

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

  the pair

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

  →

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

  (104)

  could be deleted.

  1.1.1.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

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

  the pair

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

  →

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

  (266)

  could be deleted.

  1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rules

  4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1)))))))))))))))))))	→	3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))	(20)
  3(0(0(1(0(4(5(4(2(x1)))))))))	→	4(4(5(3(1(2(3(4(2(x1)))))))))	(7)
  4(1(1(5(0(4(x1))))))	→	4(4(2(2(0(4(x1))))))	(3)
  4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1)))))))))))))))))))))	→	4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))	(25)
  4(1(5(3(3(5(0(4(5(1(4(4(1(x1)))))))))))))	→	5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))	(11)
  0(0(1(2(5(1(4(4(5(5(4(x1)))))))))))	→	0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))	(9)
  0(0(1(2(x1))))	→	0(3(4(2(x1))))	(1)
  0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1)))))))))))))))))	→	0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))	(16)
  0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1))))))))))))))))))))	→	0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))	(21)
  2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1)))))))))))))))	→	2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))	(13)
  2(0(1(0(5(3(x1))))))	→	2(0(3(1(3(3(x1))))))	(2)
  2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1))))))))))))))))	→	2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))	(15)
  2(3(2(5(5(0(0(0(x1))))))))	→	0(2(1(4(2(5(3(2(x1))))))))	(4)
  2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1)))))))))))))))))	→	5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))	(18)
  2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1)))))))))))))))))))))	→	4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))	(24)
  5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1)))))))))))))))	→	5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))	(14)
  5(0(0(5(3(3(4(3(x1))))))))	→	5(3(1(3(0(3(0(3(x1))))))))	(5)
  5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1))))))))))))))))))	→	5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))	(19)

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

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

  →

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

  (80)

  could be deleted.

  1.1.1.1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  5#(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1)))))))))))))))	→	5#(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))	(133)
  5#(0(0(5(3(3(4(3(x1))))))))	→	5#(3(1(3(0(3(0(3(x1))))))))	(46)
  5#(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1))))))))))))))))))	→	5#(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))	(213)

  1.1.1.1.2 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rules

  4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1)))))))))))))))))))	→	3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))	(20)
  3(0(0(1(0(4(5(4(2(x1)))))))))	→	4(4(5(3(1(2(3(4(2(x1)))))))))	(7)
  4(1(1(5(0(4(x1))))))	→	4(4(2(2(0(4(x1))))))	(3)
  4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1)))))))))))))))))))))	→	4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))	(25)
  4(1(5(3(3(5(0(4(5(1(4(4(1(x1)))))))))))))	→	5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))	(11)
  5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1)))))))))))))))	→	5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))	(14)
  5(0(0(5(3(3(4(3(x1))))))))	→	5(3(1(3(0(3(0(3(x1))))))))	(5)
  5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1))))))))))))))))))	→	5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))	(19)

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

  5#(0(0(5(3(3(4(3(x1))))))))	→	5#(3(1(3(0(3(0(3(x1))))))))	(46)
  5#(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1))))))))))))))))))	→	5#(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))	(213)

  could be deleted.

  1.1.1.1.2.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rules

  5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1)))))))))))))))	→	5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))	(14)
  5(0(0(5(3(3(4(3(x1))))))))	→	5(3(1(3(0(3(0(3(x1))))))))	(5)
  5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1))))))))))))))))))	→	5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))	(19)

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

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

  →

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

  (133)

  could be deleted.

  1.1.1.1.2.1.1 P is empty

  There are no pairs anymore.

• The 3^rd component contains the pair

  2#(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1)))))))))))))))	→	2#(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))	(118)
  2#(0(1(0(5(3(x1))))))	→	2#(0(3(1(3(3(x1))))))	(29)
  2#(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1))))))))))))))))	→	2#(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))	(146)

  1.1.1.1.3 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rules

  4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1)))))))))))))))))))	→	3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))	(20)
  3(0(0(1(0(4(5(4(2(x1)))))))))	→	4(4(5(3(1(2(3(4(2(x1)))))))))	(7)
  4(1(1(5(0(4(x1))))))	→	4(4(2(2(0(4(x1))))))	(3)
  4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1)))))))))))))))))))))	→	4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))	(25)
  4(1(5(3(3(5(0(4(5(1(4(4(1(x1)))))))))))))	→	5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))	(11)
  0(0(1(2(5(1(4(4(5(5(4(x1)))))))))))	→	0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))	(9)
  0(0(1(2(x1))))	→	0(3(4(2(x1))))	(1)
  0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1)))))))))))))))))	→	0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))	(16)
  0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1))))))))))))))))))))	→	0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))	(21)
  5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1)))))))))))))))	→	5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))	(14)
  5(0(0(5(3(3(4(3(x1))))))))	→	5(3(1(3(0(3(0(3(x1))))))))	(5)
  5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1))))))))))))))))))	→	5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))	(19)
  2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1)))))))))))))))	→	2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))	(13)
  2(0(1(0(5(3(x1))))))	→	2(0(3(1(3(3(x1))))))	(2)
  2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1))))))))))))))))	→	2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))	(15)
  2(3(2(5(5(0(0(0(x1))))))))	→	0(2(1(4(2(5(3(2(x1))))))))	(4)
  2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1)))))))))))))))))	→	5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))	(18)
  2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1)))))))))))))))))))))	→	4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))	(24)

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

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

  →

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

  (146)

  could be deleted.

  1.1.1.1.3.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rules

  4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1)))))))))))))))))))	→	3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))	(20)
  3(0(0(1(0(4(5(4(2(x1)))))))))	→	4(4(5(3(1(2(3(4(2(x1)))))))))	(7)
  4(1(1(5(0(4(x1))))))	→	4(4(2(2(0(4(x1))))))	(3)
  4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1)))))))))))))))))))))	→	4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))	(25)
  4(1(5(3(3(5(0(4(5(1(4(4(1(x1)))))))))))))	→	5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))	(11)
  0(0(1(2(5(1(4(4(5(5(4(x1)))))))))))	→	0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))	(9)
  0(0(1(2(x1))))	→	0(3(4(2(x1))))	(1)
  0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1)))))))))))))))))	→	0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))	(16)
  0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1))))))))))))))))))))	→	0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))	(21)
  5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1)))))))))))))))	→	5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))	(14)
  5(0(0(5(3(3(4(3(x1))))))))	→	5(3(1(3(0(3(0(3(x1))))))))	(5)
  5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1))))))))))))))))))	→	5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))	(19)
  2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1)))))))))))))))	→	2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))	(13)
  2(0(1(0(5(3(x1))))))	→	2(0(3(1(3(3(x1))))))	(2)
  2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1))))))))))))))))	→	2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))	(15)
  2(3(2(5(5(0(0(0(x1))))))))	→	0(2(1(4(2(5(3(2(x1))))))))	(4)
  2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1)))))))))))))))))	→	5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))	(18)
  2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1)))))))))))))))))))))	→	4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))	(24)

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

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

  →

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

  (29)

  could be deleted.

  1.1.1.1.3.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

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

  the pair

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

  →

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

  (118)

  could be deleted.

  1.1.1.1.3.1.1.1 P is empty

  There are no pairs anymore.

• The 4^th component contains the pair

  0#(0(1(2(5(1(4(4(5(5(4(x1)))))))))))	→	0#(0(0(4(1(1(3(5(3(4(2(x1)))))))))))	(69)
  0#(0(1(2(x1))))	→	0#(3(4(2(x1))))	(26)
  0#(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1)))))))))))))))))	→	0#(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))	(162)
  0#(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1))))))))))))))))))))	→	0#(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))	(248)

  1.1.1.1.4 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rules

  4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1)))))))))))))))))))	→	3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))	(20)
  3(0(0(1(0(4(5(4(2(x1)))))))))	→	4(4(5(3(1(2(3(4(2(x1)))))))))	(7)
  4(1(1(5(0(4(x1))))))	→	4(4(2(2(0(4(x1))))))	(3)
  4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1)))))))))))))))))))))	→	4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))	(25)
  4(1(5(3(3(5(0(4(5(1(4(4(1(x1)))))))))))))	→	5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))	(11)
  5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1)))))))))))))))	→	5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))	(14)
  5(0(0(5(3(3(4(3(x1))))))))	→	5(3(1(3(0(3(0(3(x1))))))))	(5)
  5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1))))))))))))))))))	→	5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))	(19)
  0(0(1(2(5(1(4(4(5(5(4(x1)))))))))))	→	0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))	(9)
  0(0(1(2(x1))))	→	0(3(4(2(x1))))	(1)
  0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1)))))))))))))))))	→	0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))	(16)
  0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1))))))))))))))))))))	→	0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))	(21)

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

  0#(0(1(2(x1))))

  →

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

  (26)

  could be deleted.

  1.1.1.1.4.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rules

  4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1)))))))))))))))))))	→	3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))	(20)
  3(0(0(1(0(4(5(4(2(x1)))))))))	→	4(4(5(3(1(2(3(4(2(x1)))))))))	(7)
  4(1(1(5(0(4(x1))))))	→	4(4(2(2(0(4(x1))))))	(3)
  4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1)))))))))))))))))))))	→	4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))	(25)
  4(1(5(3(3(5(0(4(5(1(4(4(1(x1)))))))))))))	→	5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))	(11)
  5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1)))))))))))))))	→	5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))	(14)
  5(0(0(5(3(3(4(3(x1))))))))	→	5(3(1(3(0(3(0(3(x1))))))))	(5)
  5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1))))))))))))))))))	→	5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))	(19)
  0(0(1(2(5(1(4(4(5(5(4(x1)))))))))))	→	0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))	(9)
  0(0(1(2(x1))))	→	0(3(4(2(x1))))	(1)
  0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1)))))))))))))))))	→	0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))	(16)
  0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1))))))))))))))))))))	→	0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))	(21)

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

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

  →

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

  (69)

  could be deleted.

  1.1.1.1.4.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rules

  0(0(1(2(5(1(4(4(5(5(4(x1)))))))))))	→	0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))	(9)
  0(0(1(2(x1))))	→	0(3(4(2(x1))))	(1)
  0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1)))))))))))))))))	→	0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))	(16)
  0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1))))))))))))))))))))	→	0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))	(21)
  4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1)))))))))))))))))))	→	3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))	(20)
  3(0(0(1(0(4(5(4(2(x1)))))))))	→	4(4(5(3(1(2(3(4(2(x1)))))))))	(7)
  4(1(1(5(0(4(x1))))))	→	4(4(2(2(0(4(x1))))))	(3)
  4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1)))))))))))))))))))))	→	4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))	(25)
  4(1(5(3(3(5(0(4(5(1(4(4(1(x1)))))))))))))	→	5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))	(11)
  5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1)))))))))))))))	→	5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))	(14)
  5(0(0(5(3(3(4(3(x1))))))))	→	5(3(1(3(0(3(0(3(x1))))))))	(5)
  5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1))))))))))))))))))	→	5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))	(19)
  2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1)))))))))))))))	→	2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))	(13)
  2(0(1(0(5(3(x1))))))	→	2(0(3(1(3(3(x1))))))	(2)
  2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1))))))))))))))))	→	2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))	(15)
  2(3(2(5(5(0(0(0(x1))))))))	→	0(2(1(4(2(5(3(2(x1))))))))	(4)
  2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1)))))))))))))))))	→	5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))	(18)
  2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1)))))))))))))))))))))	→	4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))	(24)

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

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

  →

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

  (162)

  could be deleted.

  1.1.1.1.4.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

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

  the pair

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

  →

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

  (248)

  could be deleted.

  1.1.1.1.4.1.1.1.1 P is empty

  There are no pairs anymore.

• The 5^th component contains the pair

  4#(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1)))))))))))))))))))	→	3#(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))	(230)
  3#(0(0(1(0(4(5(4(2(x1)))))))))	→	4#(4(5(3(1(2(3(4(2(x1)))))))))	(62)
  4#(1(1(5(0(4(x1))))))	→	4#(4(2(2(0(4(x1))))))	(34)
  4#(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1)))))))))))))))))))))	→	4#(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))	(306)

  1.1.1.1.5 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rules

  4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1)))))))))))))))))))	→	3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))	(20)
  3(0(0(1(0(4(5(4(2(x1)))))))))	→	4(4(5(3(1(2(3(4(2(x1)))))))))	(7)
  4(1(1(5(0(4(x1))))))	→	4(4(2(2(0(4(x1))))))	(3)
  4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1)))))))))))))))))))))	→	4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))	(25)
  4(1(5(3(3(5(0(4(5(1(4(4(1(x1)))))))))))))	→	5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))	(11)
  2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1)))))))))))))))	→	2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))	(13)
  2(0(1(0(5(3(x1))))))	→	2(0(3(1(3(3(x1))))))	(2)
  2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1))))))))))))))))	→	2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))	(15)
  2(3(2(5(5(0(0(0(x1))))))))	→	0(2(1(4(2(5(3(2(x1))))))))	(4)
  2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1)))))))))))))))))	→	5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))	(18)
  2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1)))))))))))))))))))))	→	4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))	(24)
  5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1)))))))))))))))	→	5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))	(14)
  5(0(0(5(3(3(4(3(x1))))))))	→	5(3(1(3(0(3(0(3(x1))))))))	(5)
  5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1))))))))))))))))))	→	5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))	(19)
  0(0(1(2(5(1(4(4(5(5(4(x1)))))))))))	→	0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))	(9)
  0(0(1(2(x1))))	→	0(3(4(2(x1))))	(1)
  0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1)))))))))))))))))	→	0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))	(16)
  0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1))))))))))))))))))))	→	0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))	(21)

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

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

  →

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

  (230)

  could be deleted.

  1.1.1.1.5.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    4#(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1)))))))))))))))))))))	→	4#(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))	(306)
    4#(1(1(5(0(4(x1))))))	→	4#(4(2(2(0(4(x1))))))	(34)

    1.1.1.1.5.1.1 Reduction Pair Processor with Usable Rules

    Using the linear polynomial interpretation over the naturals

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

    together with the usable rules

    0(0(1(2(5(1(4(4(5(5(4(x1)))))))))))	→	0(0(0(4(1(1(3(5(3(4(2(x1)))))))))))	(9)
    0(0(1(2(x1))))	→	0(3(4(2(x1))))	(1)
    0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1)))))))))))))))))	→	0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))))	(16)
    0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1))))))))))))))))))))	→	0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))))	(21)
    5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1)))))))))))))))	→	5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))))	(14)
    5(0(0(5(3(3(4(3(x1))))))))	→	5(3(1(3(0(3(0(3(x1))))))))	(5)
    5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1))))))))))))))))))	→	5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))))	(19)
    4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1)))))))))))))))))))	→	3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))))	(20)
    3(0(0(1(0(4(5(4(2(x1)))))))))	→	4(4(5(3(1(2(3(4(2(x1)))))))))	(7)
    4(1(1(5(0(4(x1))))))	→	4(4(2(2(0(4(x1))))))	(3)
    4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1)))))))))))))))))))))	→	4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))))	(25)
    4(1(5(3(3(5(0(4(5(1(4(4(1(x1)))))))))))))	→	5(4(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))))	(11)
    2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1)))))))))))))))	→	2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))))	(13)
    2(0(1(0(5(3(x1))))))	→	2(0(3(1(3(3(x1))))))	(2)
    2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1))))))))))))))))	→	2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))))	(15)
    2(3(2(5(5(0(0(0(x1))))))))	→	0(2(1(4(2(5(3(2(x1))))))))	(4)
    2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1)))))))))))))))))	→	5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))))	(18)
    2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1)))))))))))))))))))))	→	4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))))	(24)

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

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

    →

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

    (34)

    could be deleted.

    1.1.1.1.5.1.1.1 Semantic Labeling Processor

    The following interpretations form a model of the rules.

    As carrier we take the set {0,1}. Symbols are labeled by the interpretation of their arguments using the interpretations (modulo 2):

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

    We obtain the set of labeled pairs

    4#0(21(40(21(50(51(20(01(20(01(20(01(30(11(50(31(00(11(40(41(20(x1)))))))))))))))))))))	→	4#0(21(40(51(10(31(50(21(40(41(20(01(50(01(10(21(50(21(40(51(00(x1)))))))))))))))))))))	(327)
    4#1(20(41(20(51(50(21(00(21(00(21(00(31(10(51(30(01(10(41(40(21(x1)))))))))))))))))))))	→	4#1(20(41(50(11(30(51(20(41(40(21(00(51(00(11(20(51(20(41(50(01(x1)))))))))))))))))))))	(328)

    and the set of labeled rules:

    01(00(11(20(x1))))	→	01(30(41(20(x1))))	(329)
    00(01(10(21(x1))))	→	00(31(40(21(x1))))	(330)
    21(00(11(00(51(30(x1))))))	→	21(00(31(10(31(30(x1))))))	(331)
    20(01(10(01(50(31(x1))))))	→	20(01(30(11(30(31(x1))))))	(332)
    41(10(11(50(01(40(x1))))))	→	41(40(21(20(01(40(x1))))))	(333)
    40(11(10(51(00(41(x1))))))	→	40(41(20(21(00(41(x1))))))	(334)
    21(30(21(50(51(00(01(00(x1))))))))	→	01(20(11(40(21(50(31(20(x1))))))))	(335)
    20(31(20(51(50(01(00(01(x1))))))))	→	00(21(10(41(20(51(30(21(x1))))))))	(336)
    51(00(01(50(31(30(41(30(x1))))))))	→	51(30(11(30(01(30(01(30(x1))))))))	(337)
    50(01(00(51(30(31(40(31(x1))))))))	→	50(31(10(31(00(31(00(31(x1))))))))	(338)
    10(01(00(11(50(51(40(21(40(x1)))))))))	→	40(31(10(01(40(11(00(31(30(x1)))))))))	(339)
    11(00(01(10(51(50(41(20(41(x1)))))))))	→	41(30(11(00(41(10(01(30(31(x1)))))))))	(340)
    30(01(00(11(00(41(50(41(20(x1)))))))))	→	40(41(50(31(10(21(30(41(20(x1)))))))))	(341)
    31(00(01(10(01(40(51(40(21(x1)))))))))	→	41(40(51(30(11(20(31(40(21(x1)))))))))	(342)
    00(01(10(21(50(11(40(41(50(51(40(x1)))))))))))	→	00(01(00(41(10(11(30(51(30(41(20(x1)))))))))))	(343)
    01(00(11(20(51(10(41(40(51(50(41(x1)))))))))))	→	01(00(01(40(11(10(31(50(31(40(21(x1)))))))))))	(344)
    11(50(21(20(51(10(11(20(41(40(51(40(x1))))))))))))	→	11(20(21(30(31(30(31(00(11(10(31(40(x1))))))))))))	(345)
    10(51(20(21(50(11(10(21(40(41(50(41(x1))))))))))))	→	10(21(20(31(30(31(30(01(10(11(30(41(x1))))))))))))	(346)
    40(11(50(31(30(51(00(41(50(11(40(41(10(x1)))))))))))))	→	50(41(30(21(20(31(30(21(10(21(50(01(50(x1)))))))))))))	(347)
    41(10(51(30(31(50(01(40(51(10(41(40(11(x1)))))))))))))	→	51(40(31(20(21(30(31(20(11(20(51(00(51(x1)))))))))))))	(348)
    10(01(30(01(00(01(00(21(30(11(40(01(20(01(40(x1)))))))))))))))	→	10(41(40(01(40(11(40(01(30(21(40(11(00(21(40(x1)))))))))))))))	(349)
    11(00(31(00(01(00(01(20(31(10(41(00(21(00(41(x1)))))))))))))))	→	11(40(41(00(41(10(41(00(31(20(41(10(01(20(41(x1)))))))))))))))	(350)
    20(51(00(51(10(51(20(31(50(31(10(31(50(41(50(x1)))))))))))))))	→	20(21(50(41(20(01(10(51(10(11(40(11(00(31(40(x1)))))))))))))))	(351)
    21(50(01(50(11(50(21(30(51(30(11(30(51(40(51(x1)))))))))))))))	→	21(20(51(40(21(00(11(50(11(10(41(10(01(30(41(x1)))))))))))))))	(352)
    50(21(50(11(10(51(00(01(30(01(50(51(30(31(20(x1)))))))))))))))	→	50(51(30(31(10(51(00(31(10(01(40(31(10(31(20(x1)))))))))))))))	(353)
    51(20(51(10(11(50(01(00(31(00(51(50(31(30(21(x1)))))))))))))))	→	51(50(31(30(11(50(01(30(11(00(41(30(11(30(21(x1)))))))))))))))	(354)
    21(10(11(50(21(50(51(00(11(30(41(10(41(40(51(30(x1))))))))))))))))	→	21(50(21(30(31(50(41(30(51(40(51(50(01(00(31(10(x1))))))))))))))))	(355)
    20(11(10(51(20(51(50(01(10(31(40(11(40(41(50(31(x1))))))))))))))))	→	20(51(20(31(30(51(40(31(50(41(50(51(00(01(30(11(x1))))))))))))))))	(356)
    00(41(40(51(00(41(50(01(50(21(30(01(00(21(40(41(20(x1)))))))))))))))))	→	00(41(40(31(00(21(50(01(50(41(10(31(50(41(40(11(00(x1)))))))))))))))))	(357)
    01(40(41(50(01(40(51(00(51(20(31(00(01(20(41(40(21(x1)))))))))))))))))	→	01(40(41(30(01(20(51(00(51(40(11(30(51(40(41(10(01(x1)))))))))))))))))	(358)
    10(11(00(01(30(11(30(01(20(11(20(51(20(51(00(21(00(x1)))))))))))))))))	→	20(21(40(31(20(11(30(21(20(11(40(31(00(21(20(51(50(x1)))))))))))))))))	(359)
    11(10(01(00(31(10(31(00(21(10(21(50(21(50(01(20(01(x1)))))))))))))))))	→	21(20(41(30(21(10(31(20(21(10(41(30(01(20(21(50(51(x1)))))))))))))))))	(360)
    20(51(40(41(40(21(30(51(40(01(40(01(00(51(40(51(50(x1)))))))))))))))))	→	50(31(10(41(40(51(00(51(00(31(10(21(10(11(20(41(30(x1)))))))))))))))))	(361)
    21(50(41(40(41(20(31(50(41(00(41(00(01(50(41(50(51(x1)))))))))))))))))	→	51(30(11(40(41(50(01(50(01(30(11(20(11(10(21(40(31(x1)))))))))))))))))	(362)
    51(00(01(10(31(30(21(50(31(40(01(30(41(30(51(40(41(20(x1))))))))))))))))))	→	51(50(01(00(11(10(11(00(21(20(21(10(31(20(11(20(21(20(x1))))))))))))))))))	(363)
    50(01(00(11(30(31(20(51(30(41(00(31(40(31(50(41(40(21(x1))))))))))))))))))	→	50(51(00(01(10(11(10(01(20(21(20(11(30(21(10(21(20(21(x1))))))))))))))))))	(364)
    40(41(00(21(20(11(10(41(10(11(10(01(00(01(20(31(40(41(40(x1)))))))))))))))))))	→	30(11(00(31(40(21(20(11(00(11(50(21(00(51(40(41(20(11(40(x1)))))))))))))))))))	(365)
    41(40(01(20(21(10(11(40(11(10(11(00(01(00(21(30(41(40(41(x1)))))))))))))))))))	→	31(10(01(30(41(20(21(10(01(10(51(20(01(50(41(40(21(10(41(x1)))))))))))))))))))	(366)
    01(30(21(20(11(40(51(30(31(50(51(10(51(10(01(50(11(50(51(10(x1))))))))))))))))))))	→	01(30(11(40(01(20(01(00(21(40(51(00(21(30(01(50(21(10(51(10(x1))))))))))))))))))))	(367)
    00(31(20(21(10(41(50(31(30(51(50(11(50(11(00(51(10(51(50(11(x1))))))))))))))))))))	→	00(31(10(41(00(21(00(01(20(41(50(01(20(31(00(51(20(11(50(11(x1))))))))))))))))))))	(368)
    10(41(10(41(20(51(40(51(50(01(50(01(40(01(40(51(00(01(50(41(20(x1)))))))))))))))))))))	→	10(51(50(11(10(31(00(31(50(51(30(21(40(21(00(31(50(11(00(31(20(x1)))))))))))))))))))))	(369)
    11(40(11(40(21(50(41(50(51(00(51(00(41(00(41(50(01(00(51(40(21(x1)))))))))))))))))))))	→	11(50(51(10(11(30(01(30(51(50(31(20(41(20(01(30(51(10(01(30(21(x1)))))))))))))))))))))	(370)
    20(21(40(01(00(11(40(31(10(41(40(31(10(21(10(11(50(01(40(51(00(x1)))))))))))))))))))))	→	40(31(00(21(40(01(40(31(00(41(20(01(50(11(40(21(30(21(30(01(00(x1)))))))))))))))))))))	(371)
    21(20(41(00(01(10(41(30(11(40(41(30(11(20(11(10(51(00(41(50(01(x1)))))))))))))))))))))	→	41(30(01(20(41(00(41(30(01(40(21(00(51(10(41(20(31(20(31(00(01(x1)))))))))))))))))))))	(372)
    40(21(40(21(50(51(20(01(20(01(20(01(30(11(50(31(00(11(40(41(20(x1)))))))))))))))))))))	→	40(21(40(51(10(31(50(21(40(41(20(01(50(01(10(21(50(21(40(51(00(x1)))))))))))))))))))))	(373)
    41(20(41(20(51(50(21(00(21(00(21(00(31(10(51(30(01(10(41(40(21(x1)))))))))))))))))))))	→	41(20(41(50(11(30(51(20(41(40(21(00(51(00(11(20(51(20(41(50(01(x1)))))))))))))))))))))	(374)

    1.1.1.1.5.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 2 components.

    • The 1^st component contains the pair

      4#0(21(40(21(50(51(20(01(20(01(20(01(30(11(50(31(00(11(40(41(20(x1)))))))))))))))))))))

      →

      4#0(21(40(51(10(31(50(21(40(41(20(01(50(01(10(21(50(21(40(51(00(x1)))))))))))))))))))))

      (327)

      1.1.1.1.5.1.1.1.1.1 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [4#0(x1)]	=	1 · x1
      [21(x1)]	=	1 · x1
      [40(x1)]	=	1 · x1
      [50(x1)]	=	1
      [51(x1)]	=	0
      [20(x1)]	=	1
      [01(x1)]	=	0
      [30(x1)]	=	0
      [11(x1)]	=	1 · x1
      [31(x1)]	=	0
      [00(x1)]	=	0
      [41(x1)]	=	0
      [10(x1)]	=	1

      the pair

      4#0(21(40(21(50(51(20(01(20(01(20(01(30(11(50(31(00(11(40(41(20(x1)))))))))))))))))))))

      →

      4#0(21(40(51(10(31(50(21(40(41(20(01(50(01(10(21(50(21(40(51(00(x1)))))))))))))))))))))

      (327)

      could be deleted.

      1.1.1.1.5.1.1.1.1.1.1 P is empty

      There are no pairs anymore.

    • The 2^nd component contains the pair

      4#1(20(41(20(51(50(21(00(21(00(21(00(31(10(51(30(01(10(41(40(21(x1)))))))))))))))))))))

      →

      4#1(20(41(50(11(30(51(20(41(40(21(00(51(00(11(20(51(20(41(50(01(x1)))))))))))))))))))))

      (328)

      1.1.1.1.5.1.1.1.1.2 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [4#1(x1)]	=	1 · x1
      [20(x1)]	=	1 · x1
      [41(x1)]	=	1 · x1
      [51(x1)]	=	1
      [50(x1)]	=	0
      [21(x1)]	=	1
      [00(x1)]	=	0
      [31(x1)]	=	0
      [10(x1)]	=	1 · x1
      [30(x1)]	=	0
      [01(x1)]	=	0
      [40(x1)]	=	0
      [11(x1)]	=	1

      the pair

      4#1(20(41(20(51(50(21(00(21(00(21(00(31(10(51(30(01(10(41(40(21(x1)))))))))))))))))))))

      →

      4#1(20(41(50(11(30(51(20(41(40(21(00(51(00(11(20(51(20(41(50(01(x1)))))))))))))))))))))

      (328)

      could be deleted.

      1.1.1.1.5.1.1.1.1.2.1 P is empty

      There are no pairs anymore.