Certification Problem

Input (TPDB SRS_Standard/Trafo_06/un06)

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(a(a(b(a(a(b(x1))))))))	→	b#(b(x1))	(12)
  b#(b(a(a(b(x1)))))	→	b#(b(x1))	(11)
  b#(b(a(b(x1))))	→	b#(a(b(b(x1))))	(10)
  b#(a(a(a(b(a(a(b(x1))))))))	→	b#(a(a(a(b(b(x1))))))	(9)
  b#(b(x1))	→	b#(a(a(b(x1))))	(6)
  b#(b(x1))	→	b#(a(b(a(a(b(x1))))))	(8)
  b#(b(a(a(b(x1)))))	→	b#(a(a(b(b(x1)))))	(7)
  b#(b(a(b(x1))))	→	b#(b(x1))	(5)

  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)]	=	0 1 1 0 · x1 + 0 0
  [b(x1)]	=	1 0 1 0 · x1 + 32465 0
  [b#(x1)]	=	1 0 0 0 · x1 + 0 0

  together with the usable rules

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

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

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

  →

  b#(b(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

    b#(b(a(a(b(x1)))))	→	b#(a(a(b(b(x1)))))	(7)
    b#(b(x1))	→	b#(a(a(b(x1))))	(6)
    b#(b(x1))	→	b#(a(b(a(a(b(x1))))))	(8)
    b#(a(a(a(b(a(a(b(x1))))))))	→	b#(b(x1))	(12)
    b#(a(a(a(b(a(a(b(x1))))))))	→	b#(a(a(a(b(b(x1))))))	(9)
    b#(b(a(b(x1))))	→	b#(b(x1))	(5)
    b#(b(a(b(x1))))	→	b#(a(b(b(x1))))	(10)

    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 0 1 0 · x1 + 2 27877
    [b(x1)]	=	1 27881
    [b#(x1)]	=	0 1 0 0 · x1 + 0 0

    together with the usable rules

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

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

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

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

      →

      b#(b(x1))

      (5)

      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)]	=	0 1 0 0 · x1 + 29044 16403
      [b(x1)]	=	0 0 1 1 · x1 + 107297 1
      [b#(x1)]	=	0 1 0 0 · x1 + 0 0

      together with the usable rules

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

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

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

      →

      b#(b(x1))

      (5)

      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

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

      →

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

      (9)

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

      together with the usable rules

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

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

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

      →

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

      (9)

      could be deleted.

      1.1.1.1.1.1.2.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.