Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/212189)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

There are 142 ruless (increase limit for explicit display).

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  5#(0(1(2(x1))))	→	5#(x1)	(161)
  0#(1(2(x1)))	→	1#(0(x1))	(64)
  0#(0(1(2(x1))))	→	0#(x1)	(60)
  0#(1(2(5(2(x1)))))	→	0#(x1)	(170)
  0#(1(2(x1)))	→	0#(x1)	(76)
  0#(1(2(x1)))	→	1#(x1)	(94)
  0#(1(2(x1)))	→	0#(x1)	(76)
  0#(1(4(2(2(x1)))))	→	1#(1(x1))	(98)
  0#(1(2(x1)))	→	1#(x1)	(94)
  1#(0(1(2(x1))))	→	1#(x1)	(96)
  5#(0(1(2(x1))))	→	5#(x1)	(161)
  0#(1(2(x1)))	→	1#(x1)	(94)
  0#(1(2(x1)))	→	0#(x1)	(76)
  0#(1(2(x1)))	→	1#(0(x1))	(64)
  0#(1(2(0(x1))))	→	0#(0(x1))	(159)
  0#(1(2(x1)))	→	0#(x1)	(76)
  0#(0(1(2(x1))))	→	5#(0(x1))	(154)
  0#(1(2(x1)))	→	0#(x1)	(76)
  5#(0(1(2(x1))))	→	1#(5(x1))	(148)
  0#(1(2(x1)))	→	1#(x1)	(94)
  0#(1(2(x1)))	→	1#(5(x1))	(78)
  0#(1(2(x1)))	→	0#(x1)	(76)
  0#(0(1(2(x1))))	→	1#(x1)	(75)
  0#(1(4(2(2(x1)))))	→	1#(x1)	(69)
  0#(1(2(x1)))	→	1#(0(x1))	(64)
  0#(1(2(x1)))	→	5#(x1)	(53)
  1#(0(1(2(x1))))	→	0#(x1)	(133)
  0#(0(1(2(x1))))	→	0#(x1)	(60)
  0#(1(2(x1)))	→	5#(x1)	(53)
  0#(1(2(x1)))	→	5#(x1)	(53)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals

  [0#(x1)]	=	0 1 0 1 · x1 + 7436 0
  [1(x1)]	=	0 0 1 0 · x1 + 0 3718
  [4(x1)]	=	1 0 1 0 · x1 + 0 0
  [5(x1)]	=	1 0 1 0 · x1 + 0 1
  [3(x1)]	=	0 0
  [0(x1)]	=	0 1 0 1 · x1 + 0 1
  [5#(x1)]	=	0 1 1 0 · x1 + 14871 7436
  [2(x1)]	=	1 1 0 1 · x1 + 3718 1
  [1#(x1)]	=	0 1 0 1 · x1 + 0 0

  together with the usable rules

  0(1(2(x1)))	→	2(0(0(3(3(1(x1))))))	(18)
  1(3(0(1(2(x1)))))	→	1(3(4(1(0(2(x1))))))	(50)
  0(1(2(x1)))	→	0(2(1(3(3(x1)))))	(4)
  0(1(2(x1)))	→	0(5(2(3(1(4(x1))))))	(15)
  0(1(2(x1)))	→	2(1(3(4(0(x1)))))	(8)
  0(1(2(x1)))	→	0(3(1(2(x1))))	(1)
  0(1(2(x1)))	→	2(1(4(0(x1))))	(3)
  0(1(2(x1)))	→	1(0(2(2(3(4(x1))))))	(16)
  0(1(2(x1)))	→	2(5(3(0(3(1(x1))))))	(21)
  0(5(2(2(x1))))	→	3(0(2(1(5(2(x1))))))	(36)
  0(0(1(2(x1))))	→	0(2(1(5(0(4(x1))))))	(26)
  0(1(2(x1)))	→	2(0(3(4(1(5(x1))))))	(19)
  0(1(2(2(x1))))	→	3(3(0(2(1(2(x1))))))	(32)
  0(1(2(x1)))	→	1(1(4(0(3(2(x1))))))	(17)
  0(0(1(2(x1))))	→	1(0(2(3(5(0(x1))))))	(27)
  0(1(5(2(x1))))	→	0(5(5(3(1(2(x1))))))	(34)
  0(1(2(x1)))	→	3(2(5(3(1(0(x1))))))	(22)
  0(0(1(2(x1))))	→	5(1(2(0(4(0(x1))))))	(28)
  5(0(1(2(x1))))	→	5(5(1(0(2(4(x1))))))	(44)
  0(1(2(x1)))	→	0(2(4(1(3(x1)))))	(5)
  0(1(5(2(x1))))	→	2(5(1(0(3(x1)))))	(33)
  0(1(2(x1)))	→	4(0(3(1(2(x1)))))	(10)
  1(0(1(2(x1))))	→	0(3(1(4(1(2(x1))))))	(39)
  0(1(2(x1)))	→	2(1(1(0(3(x1)))))	(7)
  0(1(2(x1)))	→	2(3(2(0(4(1(x1))))))	(20)
  0(0(1(2(x1))))	→	0(0(2(4(1(x1)))))	(25)
  0(1(4(2(2(x1)))))	→	0(2(4(2(1(1(x1))))))	(49)
  0(1(2(0(x1))))	→	1(2(4(0(0(x1)))))	(30)
  0(1(2(x1)))	→	0(4(5(1(2(5(x1))))))	(14)
  0(1(2(0(x1))))	→	0(4(1(0(0(2(x1))))))	(31)
  0(1(2(x1)))	→	4(4(2(1(0(x1)))))	(12)
  0(1(0(2(2(x1)))))	→	3(0(2(1(0(2(x1))))))	(45)
  0(1(2(x1)))	→	3(4(0(5(1(2(x1))))))	(23)
  0(1(2(x1)))	→	5(5(4(2(1(0(x1))))))	(24)
  0(1(2(x1)))	→	4(1(0(3(2(x1)))))	(11)
  0(1(2(x1)))	→	2(1(4(3(0(x1)))))	(9)
  0(1(2(x1)))	→	0(0(4(1(2(5(x1))))))	(13)
  1(0(1(2(x1))))	→	4(1(4(1(0(2(x1))))))	(40)
  0(1(2(x1)))	→	1(0(2(5(4(x1)))))	(6)
  1(0(1(2(x1))))	→	1(2(1(3(0(x1)))))	(38)
  0(1(4(2(2(x1)))))	→	0(2(4(1(5(2(x1))))))	(48)
  0(1(3(0(0(x1)))))	→	0(0(4(1(3(0(x1))))))	(47)
  1(0(1(2(x1))))	→	0(4(1(2(1(x1)))))	(37)
  5(0(1(2(x1))))	→	0(2(1(3(5(x1)))))	(41)
  5(0(1(2(x1))))	→	2(0(3(1(5(x1)))))	(42)
  0(1(2(5(2(x1)))))	→	3(5(1(2(2(0(x1))))))	(46)
  0(1(5(2(x1))))	→	1(3(5(0(2(2(x1))))))	(35)
  0(0(5(2(x1))))	→	5(3(0(3(0(2(x1))))))	(29)
  5(0(1(2(x1))))	→	5(3(1(0(3(2(x1))))))	(43)
  0(1(2(x1)))	→	1(0(2(1(x1))))	(2)

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

  5#(0(1(2(x1))))	→	5#(x1)	(161)
  0#(1(2(x1)))	→	1#(0(x1))	(64)
  0#(0(1(2(x1))))	→	0#(x1)	(60)
  0#(1(2(5(2(x1)))))	→	0#(x1)	(170)
  0#(1(2(x1)))	→	0#(x1)	(76)
  0#(1(2(x1)))	→	1#(x1)	(94)
  0#(1(2(x1)))	→	0#(x1)	(76)
  0#(1(4(2(2(x1)))))	→	1#(1(x1))	(98)
  0#(1(2(x1)))	→	1#(x1)	(94)
  1#(0(1(2(x1))))	→	1#(x1)	(96)
  5#(0(1(2(x1))))	→	5#(x1)	(161)
  0#(1(2(x1)))	→	1#(x1)	(94)
  0#(1(2(x1)))	→	0#(x1)	(76)
  0#(1(2(x1)))	→	1#(0(x1))	(64)
  0#(1(2(0(x1))))	→	0#(0(x1))	(159)
  0#(1(2(x1)))	→	0#(x1)	(76)
  0#(0(1(2(x1))))	→	5#(0(x1))	(154)
  0#(1(2(x1)))	→	0#(x1)	(76)
  5#(0(1(2(x1))))	→	1#(5(x1))	(148)
  0#(1(2(x1)))	→	1#(x1)	(94)
  0#(1(2(x1)))	→	1#(5(x1))	(78)
  0#(1(2(x1)))	→	0#(x1)	(76)
  0#(0(1(2(x1))))	→	1#(x1)	(75)
  0#(1(4(2(2(x1)))))	→	1#(x1)	(69)
  0#(1(2(x1)))	→	1#(0(x1))	(64)
  0#(1(2(x1)))	→	5#(x1)	(53)
  1#(0(1(2(x1))))	→	0#(x1)	(133)
  0#(0(1(2(x1))))	→	0#(x1)	(60)
  0#(1(2(x1)))	→	5#(x1)	(53)
  0#(1(2(x1)))	→	5#(x1)	(53)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.