Certification Problem

Input (TPDB SRS_Standard/Trafo_06/un09)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by matchbox @ 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#(b(b(a(b(x1)))))	→	b#(a(b(b(a(b(x1))))))	(4)
  b#(a(b(b(x1))))	→	b#(b(a(b(a(b(x1))))))	(5)
  b#(a(b(b(x1))))	→	b#(a(b(x1)))	(6)
  b#(a(b(b(x1))))	→	b#(a(b(a(b(x1)))))	(7)
  b#(a(b(a(a(b(b(x1)))))))	→	b#(b(b(a(b(x1)))))	(8)
  b#(a(b(a(a(b(b(x1)))))))	→	b#(b(a(b(x1))))	(9)
  b#(a(b(a(a(b(b(x1)))))))	→	b#(a(b(x1)))	(10)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 4 with strict dimension 1 over the arctic semiring over the naturals

  [b(x1)]	=	0 0 0 4 0 0 0 0 -4 -4 -4 0 -4 -4 -4 0 · x1 + -∞ -∞ -∞ -∞
  [a(x1)]	=	0 0 0 0 0 0 0 0 -4 0 0 0 -4 -4 0 0 · x1 + -∞ -∞ -∞ -∞
  [b#(x1)]	=	13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 · x1 + -∞ -∞ -∞ -∞

  together with the usable rules

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

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

  b#(a(b(a(a(b(b(x1)))))))	→	b#(b(b(a(b(x1)))))	(8)
  b#(a(b(a(a(b(b(x1)))))))	→	b#(b(a(b(x1))))	(9)
  b#(a(b(a(a(b(b(x1)))))))	→	b#(a(b(x1)))	(10)

  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(b(a(b(x1)))))	→	b#(a(b(b(a(b(x1))))))	(4)
    b#(a(b(b(x1))))	→	b#(b(a(b(a(b(x1))))))	(5)
    b#(a(b(b(x1))))	→	b#(a(b(x1)))	(6)
    b#(a(b(b(x1))))	→	b#(a(b(a(b(x1)))))	(7)

    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

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

    together with the usable rules

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

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

    b#(b(b(a(b(x1)))))	→	b#(a(b(b(a(b(x1))))))	(4)
    b#(a(b(b(x1))))	→	b#(a(b(x1)))	(6)
    b#(a(b(b(x1))))	→	b#(a(b(a(b(x1)))))	(7)

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 1 component.

    • The 1^st component contains the pair

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

      →

      b#(b(a(b(a(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 arctic semiring over the naturals

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

      together with the usable rules

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

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

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

      →

      b#(b(a(b(a(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.