Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/abbaaaaa-aaaaaabbabb.srs)

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

  a#(b(b(a(a(a(a(a(x1))))))))	→	a#(b(b(x1)))	(2)
  a#(b(b(a(a(a(a(a(x1))))))))	→	a#(b(b(a(b(b(x1))))))	(3)
  a#(b(b(a(a(a(a(a(x1))))))))	→	a#(a(b(b(a(b(b(x1)))))))	(4)
  a#(b(b(a(a(a(a(a(x1))))))))	→	a#(a(a(b(b(a(b(b(x1))))))))	(5)
  a#(b(b(a(a(a(a(a(x1))))))))	→	a#(a(a(a(b(b(a(b(b(x1)))))))))	(6)
  a#(b(b(a(a(a(a(a(x1))))))))	→	a#(a(a(a(a(b(b(a(b(b(x1))))))))))	(7)
  a#(b(b(a(a(a(a(a(x1))))))))	→	a#(a(a(a(a(a(b(b(a(b(b(x1)))))))))))	(8)

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

  together with the usable rule

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

  →

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

  (1)

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

  a#(b(b(a(a(a(a(a(x1))))))))	→	a#(a(b(b(a(b(b(x1)))))))	(4)
  a#(b(b(a(a(a(a(a(x1))))))))	→	a#(a(a(b(b(a(b(b(x1))))))))	(5)
  a#(b(b(a(a(a(a(a(x1))))))))	→	a#(a(a(a(b(b(a(b(b(x1)))))))))	(6)
  a#(b(b(a(a(a(a(a(x1))))))))	→	a#(a(a(a(a(b(b(a(b(b(x1))))))))))	(7)
  a#(b(b(a(a(a(a(a(x1))))))))	→	a#(a(a(a(a(a(b(b(a(b(b(x1)))))))))))	(8)

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

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the matrix interpretations of dimension 5 with strict dimension 1 over the arctic semiring over the naturals

    [b(x1)]	=	0 0 0 0 0 0 0 0 0 0 -5 0 0 0 0 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 · x1 + -∞ -∞ -∞ -∞ -∞
    [a(x1)]	=	0 0 0 5 5 0 0 0 0 5 0 0 0 0 0 -5 0 0 0 0 -5 -5 0 0 0 · x1 + -∞ -∞ -∞ -∞ -∞
    [a#(x1)]	=	5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 · x1 + -∞ -∞ -∞ -∞ -∞

    together with the usable rule

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

    →

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

    (1)

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

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

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.