Certification Problem

Input (TPDB SRS_Standard/Waldmann_19/random-390)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  a#(b(b(a(x1))))	→	a#(a(a(x1)))	(16)
  b#(a(b(b(x1))))	→	a#(b(x1))	(9)
  a#(a(a(b(x1))))	→	b#(a(x1))	(15)
  b#(b(b(b(x1))))	→	b#(a(b(b(x1))))	(14)
  a#(b(b(a(x1))))	→	b#(a(a(a(x1))))	(13)
  b#(a(b(b(x1))))	→	b#(a(b(x1)))	(8)
  a#(a(a(b(x1))))	→	a#(x1)	(7)
  a#(a(a(b(x1))))	→	a#(b(b(a(x1))))	(6)
  a#(b(b(a(x1))))	→	a#(a(x1))	(12)
  b#(a(b(b(x1))))	→	b#(b(a(b(x1))))	(11)
  a#(a(a(b(x1))))	→	b#(b(a(x1)))	(10)
  b#(b(b(b(x1))))	→	a#(b(b(x1)))	(5)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [a(x1)]	=	x1 + 11799
  [b(x1)]	=	x1 + 11799
  [a#(x1)]	=	x1 + 11798
  [b#(x1)]	=	x1 + 0

  together with the usable rules

  b(a(b(b(x1))))	→	b(b(a(b(x1))))	(4)
  a(a(a(b(x1))))	→	a(b(b(a(x1))))	(1)
  a(b(b(a(x1))))	→	b(a(a(a(x1))))	(3)
  b(b(b(b(x1))))	→	b(a(b(b(x1))))	(2)

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

  a#(b(b(a(x1))))	→	a#(a(a(x1)))	(16)
  b#(a(b(b(x1))))	→	a#(b(x1))	(9)
  a#(a(a(b(x1))))	→	b#(a(x1))	(15)
  a#(b(b(a(x1))))	→	b#(a(a(a(x1))))	(13)
  b#(a(b(b(x1))))	→	b#(a(b(x1)))	(8)
  a#(a(a(b(x1))))	→	a#(x1)	(7)
  a#(b(b(a(x1))))	→	a#(a(x1))	(12)
  a#(a(a(b(x1))))	→	b#(b(a(x1)))	(10)
  b#(b(b(b(x1))))	→	a#(b(b(x1)))	(5)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 2 components.

  • The 1^st component contains the pair

    a#(a(a(b(x1))))

    →

    a#(b(b(a(x1))))

    (6)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [a(x1)]	=	x1 + 5854
    [b(x1)]	=	30613
    [a#(x1)]	=	x1 + 11798
    [b#(x1)]	=	x1 + 0

    together with the usable rules

    b(a(b(b(x1))))	→	b(b(a(b(x1))))	(4)
    a(a(a(b(x1))))	→	a(b(b(a(x1))))	(1)
    a(b(b(a(x1))))	→	b(a(a(a(x1))))	(3)
    b(b(b(b(x1))))	→	b(a(b(b(x1))))	(2)

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

    a#(a(a(b(x1))))

    →

    a#(b(b(a(x1))))

    (6)

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.

  • The 2^nd component contains the pair

    b#(a(b(b(x1))))	→	b#(b(a(b(x1))))	(11)
    b#(b(b(b(x1))))	→	b#(a(b(b(x1))))	(14)

    1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

    [a(x1)]	=	1 1 1 1 · x1 + 1 13359
    [b(x1)]	=	1 1 1 1 · x1 + 2 13360
    [a#(x1)]	=	0 0
    [b#(x1)]	=	1 0 1 1 · x1 + 0 0

    together with the usable rules

    b(a(b(b(x1))))	→	b(b(a(b(x1))))	(4)
    a(a(a(b(x1))))	→	a(b(b(a(x1))))	(1)
    a(b(b(a(x1))))	→	b(a(a(a(x1))))	(3)
    b(b(b(b(x1))))	→	b(a(b(b(x1))))	(2)

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

    b#(a(b(b(x1))))	→	b#(b(a(b(x1))))	(11)
    b#(b(b(b(x1))))	→	b#(a(b(b(x1))))	(14)

    could be deleted.

    1.1.1.1.2.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.