Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/211915)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

There are 125 ruless (increase limit for explicit display).

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  0#(0(5(4(x1))))	→	0#(0(x1))	(83)
  0#(1(2(x1)))	→	0#(x1)	(54)
  0#(0(5(4(x1))))	→	0#(x1)	(84)
  0#(1(0(1(x1))))	→	0#(x1)	(88)
  0#(1(2(5(x1))))	→	0#(x1)	(97)
  0#(1(0(5(4(x1)))))	→	0#(1(5(x1)))	(142)

  1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

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

  the pairs

  0#(0(5(4(x1))))	→	0#(x1)	(84)
  0#(1(0(1(x1))))	→	0#(x1)	(88)
  0#(1(0(5(4(x1)))))	→	0#(1(5(x1)))	(142)

  could be deleted.

  1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rules

  0(1(2(x1)))	→	1(3(0(2(x1))))	(1)
  0(1(2(x1)))	→	3(1(2(0(x1))))	(2)
  0(1(2(x1)))	→	2(0(4(1(3(x1)))))	(3)
  0(1(2(x1)))	→	3(0(2(1(3(x1)))))	(4)
  0(1(2(x1)))	→	3(3(0(2(1(x1)))))	(5)
  0(1(2(x1)))	→	1(3(3(0(2(3(x1))))))	(6)
  0(1(2(x1)))	→	2(0(4(3(1(3(x1))))))	(7)
  0(1(2(x1)))	→	3(0(1(3(1(2(x1))))))	(8)
  0(1(2(x1)))	→	3(0(5(3(1(2(x1))))))	(9)
  0(5(2(x1)))	→	1(5(0(2(x1))))	(10)
  0(5(2(x1)))	→	3(5(0(2(x1))))	(11)
  0(5(2(x1)))	→	3(0(5(1(2(x1)))))	(12)
  0(5(2(x1)))	→	3(5(3(0(2(x1)))))	(13)
  0(5(2(x1)))	→	1(5(5(3(0(2(x1))))))	(14)
  0(5(2(x1)))	→	5(0(4(3(1(2(x1))))))	(15)
  0(0(5(4(x1))))	→	5(3(0(4(0(0(x1))))))	(18)
  0(1(0(1(x1))))	→	1(1(3(0(3(0(x1))))))	(19)
  0(1(1(2(x1))))	→	2(1(1(3(0(2(x1))))))	(20)
  0(1(2(5(x1))))	→	3(0(5(1(2(x1)))))	(21)
  0(1(2(5(x1))))	→	3(5(3(1(2(0(x1))))))	(22)
  0(1(3(2(x1))))	→	3(0(2(1(3(5(x1))))))	(23)
  0(1(4(2(x1))))	→	3(0(0(4(1(2(x1))))))	(24)
  0(1(4(2(x1))))	→	4(0(1(3(1(2(x1))))))	(25)
  0(5(2(4(x1))))	→	5(5(3(0(2(4(x1))))))	(26)
  0(5(2(5(x1))))	→	0(2(3(5(5(x1)))))	(27)
  0(5(3(2(x1))))	→	3(0(2(1(5(x1)))))	(28)
  0(5(3(2(x1))))	→	5(3(1(4(0(2(x1))))))	(29)
  0(0(1(3(2(x1)))))	→	0(3(1(2(0(3(x1))))))	(36)
  0(0(5(3(2(x1)))))	→	0(3(0(3(2(5(x1))))))	(37)
  0(1(0(3(4(x1)))))	→	4(0(1(3(0(3(x1))))))	(38)
  0(1(0(5(4(x1)))))	→	3(0(4(0(1(5(x1))))))	(39)
  0(1(2(4(4(x1)))))	→	0(4(1(2(1(4(x1))))))	(40)
  0(1(5(3(2(x1)))))	→	1(2(3(0(4(5(x1))))))	(41)
  0(5(4(2(4(x1)))))	→	4(5(3(0(4(2(x1))))))	(42)
  0(5(4(4(2(x1)))))	→	0(4(4(3(5(2(x1))))))	(43)

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

  0#(1(2(x1)))	→	0#(x1)	(54)
  0#(1(2(5(x1))))	→	0#(x1)	(97)

  could be deleted.

  1.1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rules

  0(1(2(x1)))	→	1(3(0(2(x1))))	(1)
  0(1(2(x1)))	→	3(1(2(0(x1))))	(2)
  0(1(2(x1)))	→	2(0(4(1(3(x1)))))	(3)
  0(1(2(x1)))	→	3(0(2(1(3(x1)))))	(4)
  0(1(2(x1)))	→	3(3(0(2(1(x1)))))	(5)
  0(1(2(x1)))	→	1(3(3(0(2(3(x1))))))	(6)
  0(1(2(x1)))	→	2(0(4(3(1(3(x1))))))	(7)
  0(1(2(x1)))	→	3(0(1(3(1(2(x1))))))	(8)
  0(1(2(x1)))	→	3(0(5(3(1(2(x1))))))	(9)
  0(5(2(x1)))	→	1(5(0(2(x1))))	(10)
  0(5(2(x1)))	→	3(5(0(2(x1))))	(11)
  0(5(2(x1)))	→	3(0(5(1(2(x1)))))	(12)
  0(5(2(x1)))	→	3(5(3(0(2(x1)))))	(13)
  0(5(2(x1)))	→	1(5(5(3(0(2(x1))))))	(14)
  0(5(2(x1)))	→	5(0(4(3(1(2(x1))))))	(15)
  0(0(5(4(x1))))	→	5(3(0(4(0(0(x1))))))	(18)
  0(1(0(1(x1))))	→	1(1(3(0(3(0(x1))))))	(19)
  0(1(1(2(x1))))	→	2(1(1(3(0(2(x1))))))	(20)
  0(1(2(5(x1))))	→	3(0(5(1(2(x1)))))	(21)
  0(1(2(5(x1))))	→	3(5(3(1(2(0(x1))))))	(22)
  0(1(3(2(x1))))	→	3(0(2(1(3(5(x1))))))	(23)
  0(1(4(2(x1))))	→	3(0(0(4(1(2(x1))))))	(24)
  0(1(4(2(x1))))	→	4(0(1(3(1(2(x1))))))	(25)
  0(5(2(4(x1))))	→	5(5(3(0(2(4(x1))))))	(26)
  0(5(2(5(x1))))	→	0(2(3(5(5(x1)))))	(27)
  0(5(3(2(x1))))	→	3(0(2(1(5(x1)))))	(28)
  0(5(3(2(x1))))	→	5(3(1(4(0(2(x1))))))	(29)
  0(0(1(3(2(x1)))))	→	0(3(1(2(0(3(x1))))))	(36)
  0(0(5(3(2(x1)))))	→	0(3(0(3(2(5(x1))))))	(37)
  0(1(0(3(4(x1)))))	→	4(0(1(3(0(3(x1))))))	(38)
  0(1(0(5(4(x1)))))	→	3(0(4(0(1(5(x1))))))	(39)
  0(1(2(4(4(x1)))))	→	0(4(1(2(1(4(x1))))))	(40)
  0(1(5(3(2(x1)))))	→	1(2(3(0(4(5(x1))))))	(41)
  0(5(4(2(4(x1)))))	→	4(5(3(0(4(2(x1))))))	(42)
  0(5(4(4(2(x1)))))	→	0(4(4(3(5(2(x1))))))	(43)

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

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

  →

  0#(0(x1))

  (83)

  could be deleted.

  1.1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  5#(1(1(2(1(x1)))))	→	5#(1(1(x1)))	(174)
  5#(1(0(5(2(x1)))))	→	5#(x1)	(172)

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

  together with the usable rules

  1(5(2(x1)))	→	5(3(1(2(x1))))	(16)
  1(5(2(x1)))	→	1(5(3(1(2(x1)))))	(17)
  1(0(3(4(x1))))	→	1(0(4(1(3(x1)))))	(30)
  1(5(3(2(x1))))	→	3(1(3(5(2(x1)))))	(31)
  1(0(1(3(4(x1)))))	→	4(1(1(3(0(3(x1))))))	(44)
  1(0(3(4(5(x1)))))	→	2(1(3(0(4(5(x1))))))	(45)
  1(1(3(4(2(x1)))))	→	2(5(1(3(1(4(x1))))))	(46)
  1(5(3(2(2(x1)))))	→	2(3(5(3(1(2(x1))))))	(47)
  5(0(1(2(x1))))	→	1(5(0(4(1(2(x1))))))	(32)
  5(1(0(2(x1))))	→	3(1(5(0(2(x1)))))	(33)
  5(1(1(2(x1))))	→	1(3(5(2(1(3(x1))))))	(34)
  5(1(5(2(x1))))	→	1(3(5(5(2(x1)))))	(35)
  5(0(5(2(4(x1)))))	→	5(1(5(0(4(2(x1))))))	(48)
  5(1(0(5(2(x1)))))	→	1(5(0(2(3(5(x1))))))	(49)
  5(1(1(2(1(x1)))))	→	3(1(2(5(1(1(x1))))))	(50)

  (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

  1.1.2.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rules

  1(5(2(x1)))	→	5(3(1(2(x1))))	(16)
  1(5(2(x1)))	→	1(5(3(1(2(x1)))))	(17)
  1(0(3(4(x1))))	→	1(0(4(1(3(x1)))))	(30)
  1(5(3(2(x1))))	→	3(1(3(5(2(x1)))))	(31)
  1(0(1(3(4(x1)))))	→	4(1(1(3(0(3(x1))))))	(44)
  1(0(3(4(5(x1)))))	→	2(1(3(0(4(5(x1))))))	(45)
  1(1(3(4(2(x1)))))	→	2(5(1(3(1(4(x1))))))	(46)
  1(5(3(2(2(x1)))))	→	2(3(5(3(1(2(x1))))))	(47)

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

  5#(1(1(2(1(x1)))))	→	5#(1(1(x1)))	(174)
  5#(1(0(5(2(x1)))))	→	5#(x1)	(172)

  could be deleted.

  1.1.2.1.1 P is empty

  There are no pairs anymore.