Certification Problem

Input (TPDB SRS_Standard/Waldmann_19/random-82)

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

  b#(a(b(a(x1))))	→	b#(a(a(a(x1))))	(17)
  a#(b(b(a(x1))))	→	b#(a(a(x1)))	(10)
  a#(b(a(b(x1))))	→	b#(b(a(x1)))	(16)
  a#(b(b(b(x1))))	→	a#(b(a(b(x1))))	(15)
  a#(b(b(b(x1))))	→	a#(b(x1))	(14)
  a#(b(b(a(x1))))	→	b#(b(a(a(x1))))	(9)
  a#(b(a(b(x1))))	→	b#(a(x1))	(8)
  a#(b(b(a(x1))))	→	a#(a(x1))	(7)
  b#(a(b(a(x1))))	→	a#(a(x1))	(6)
  b#(a(b(a(x1))))	→	a#(a(a(x1)))	(13)
  a#(b(a(b(x1))))	→	a#(x1)	(12)
  a#(b(a(b(x1))))	→	b#(b(b(a(x1))))	(11)
  a#(b(b(b(x1))))	→	b#(a(b(x1)))	(5)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

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

  together with the usable rules

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

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

  a#(b(a(b(x1))))	→	b#(b(a(x1)))	(16)
  a#(b(b(a(x1))))	→	b#(b(a(a(x1))))	(9)
  a#(b(a(b(x1))))	→	b#(b(b(a(x1))))	(11)

  could be deleted.

  1.1.1.1 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(x1))	(7)
    a#(b(b(a(x1))))	→	b#(a(a(x1)))	(10)
    a#(b(a(b(x1))))	→	a#(x1)	(12)
    a#(b(a(b(x1))))	→	b#(a(x1))	(8)
    b#(a(b(a(x1))))	→	a#(a(x1))	(6)
    b#(a(b(a(x1))))	→	a#(a(a(x1)))	(13)
    b#(a(b(a(x1))))	→	b#(a(a(a(x1))))	(17)
    a#(b(b(b(x1))))	→	a#(b(x1))	(14)
    a#(b(b(b(x1))))	→	b#(a(b(x1)))	(5)
    a#(b(b(b(x1))))	→	a#(b(a(b(x1))))	(15)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [a(x1)]	=	x1 + 11798
    [b(x1)]	=	x1 + 11798
    [a#(x1)]	=	x1 + 2
    [b#(x1)]	=	x1 + 11799

    together with the usable rules

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

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

    a#(b(b(a(x1))))	→	a#(a(x1))	(7)
    a#(b(b(a(x1))))	→	b#(a(a(x1)))	(10)
    a#(b(a(b(x1))))	→	a#(x1)	(12)
    a#(b(a(b(x1))))	→	b#(a(x1))	(8)
    b#(a(b(a(x1))))	→	a#(a(x1))	(6)
    b#(a(b(a(x1))))	→	a#(a(a(x1)))	(13)
    a#(b(b(b(x1))))	→	a#(b(x1))	(14)
    a#(b(b(b(x1))))	→	b#(a(b(x1)))	(5)

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 2 components.

    • The 1^st component contains the pair

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

      →

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

      (17)

      1.1.1.1.1.1.1 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 12831
      [b(x1)]	=	1 1 1 1 · x1 + 2 51737
      [a#(x1)]	=	0 0
      [b#(x1)]	=	1 1 1 1 · x1 + 0 0

      together with the usable rules

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

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

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

      →

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

      (17)

      could be deleted.

      1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.

    • The 2^nd component contains the pair

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

      →

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

      (15)

      1.1.1.1.1.1.2 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 1
      [b(x1)]	=	1 1 1 1 · x1 + 1 2
      [a#(x1)]	=	1 1 1 1 · x1 + 0 0
      [b#(x1)]	=	1 1 1 1 · x1 + 0 0

      together with the usable rules

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

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

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

      →

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

      (15)

      could be deleted.

      1.1.1.1.1.1.2.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.