Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/aabaaaa-aaaaaabaaab.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#(a(b(a(a(a(a(x1)))))))	→	a#(a(a(a(a(a(b(a(a(a(b(x1)))))))))))	(10)
  a#(a(b(a(a(a(a(x1)))))))	→	a#(a(b(x1)))	(4)
  a#(a(b(a(a(a(a(x1)))))))	→	a#(a(b(a(a(a(b(x1)))))))	(5)
  a#(a(b(a(a(a(a(x1)))))))	→	a#(a(a(b(x1))))	(6)
  a#(a(b(a(a(a(a(x1)))))))	→	a#(a(a(b(a(a(a(b(x1))))))))	(7)
  a#(a(b(a(a(a(a(x1)))))))	→	a#(a(a(a(b(a(a(a(b(x1)))))))))	(8)
  a#(a(b(a(a(a(a(x1)))))))	→	a#(a(a(a(a(b(a(a(a(b(x1))))))))))	(9)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

  [b(x1)]	=	0 0 0 0 0 0 0 0 0 · x1 + -∞ -∞ -∞
  [a(x1)]	=	0 0 0 -3 -3 0 -3 -3 -3 · x1 + -∞ -∞ -∞
  [a#(x1)]	=	6 9 9 6 9 9 6 9 9 · x1 + -∞ -∞ -∞

  together with the usable rule

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

  →

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

  (1)

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

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

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

    1.1.1.1.1 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 0 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 · x1 + -∞ -∞ -∞ -∞
    [a(x1)]	=	0 0 0 4 0 0 0 0 -4 0 0 0 -4 -4 0 0 · x1 + -∞ -∞ -∞ -∞
    [a#(x1)]	=	1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 · x1 + -∞ -∞ -∞ -∞

    together with the usable rule

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

    →

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

    (1)

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

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

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.