Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/abacaabaca-aabacabacaabac.srs)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by matchbox @ termCOMP 2023)

1 String Reversal

1.1 Dependency Pair Transformation

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  a#(c(a(b(a(a(c(a(b(a(x1))))))))))	→	a#(c(a(b(a(c(a(b(a(a(x1))))))))))	(3)
  a#(c(a(b(a(a(c(a(b(a(x1))))))))))	→	a#(c(a(b(a(a(x1))))))	(4)
  a#(c(a(b(a(a(c(a(b(a(x1))))))))))	→	a#(a(x1))	(8)
  a#(c(a(b(a(a(c(a(b(a(x1))))))))))	→	a#(a(c(a(b(a(c(a(b(a(a(x1)))))))))))	(9)

  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

  [c(x1)]	=	0 0 0 0 -4 -4 -4 0 -4 -4 -4 -4 -4 -4 -4 -4 · x1 + -∞ -∞ -∞ -∞
  [b(x1)]	=	0 0 0 0 -4 -4 -4 0 -4 -4 -4 0 -4 -4 -4 -4 · x1 + -∞ -∞ -∞ -∞
  [a(x1)]	=	0 0 0 0 0 0 0 0 0 0 0 0 -4 -4 0 0 · x1 + -∞ -∞ -∞ -∞
  [a#(x1)]	=	30 31 31 31 30 31 31 31 30 31 31 31 30 31 31 31 · x1 + -∞ -∞ -∞ -∞

  together with the usable rule

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

  →

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

  (2)

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

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

  →

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

  (3)

  could be deleted.

  1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    a#(c(a(b(a(a(c(a(b(a(x1))))))))))	→	a#(c(a(b(a(a(x1))))))	(4)
    a#(c(a(b(a(a(c(a(b(a(x1))))))))))	→	a#(a(x1))	(8)
    a#(c(a(b(a(a(c(a(b(a(x1))))))))))	→	a#(a(c(a(b(a(c(a(b(a(a(x1)))))))))))	(9)

    1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

    [c(x1)]	=	-∞ 0 0 0 -∞ -∞ 0 -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 0 -∞ 2 -∞
    [b(x1)]	=	0 -∞ -∞ -∞ 0 -∞ -∞ -∞ 0 -∞ -∞ 0 -∞ -∞ 0 -∞ · x1 + 6 -∞ -∞ 4
    [a(x1)]	=	-∞ 0 -∞ -∞ -∞ 0 -∞ -∞ -∞ -∞ -∞ 0 0 0 0 -∞ · x1 + 0 -∞ 4 -∞
    [a#(x1)]	=	1 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + -∞ -∞ -∞ -∞

    together with the usable rule

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

    →

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

    (2)

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

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

    →

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

    (9)

    could be deleted.

    1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 1 component.

    • The 1^st component contains the pair

      a#(c(a(b(a(a(c(a(b(a(x1))))))))))	→	a#(c(a(b(a(a(x1))))))	(4)
      a#(c(a(b(a(a(c(a(b(a(x1))))))))))	→	a#(a(x1))	(8)

      1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

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

      together with the usable rule

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

      →

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

      (2)

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

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

      →

      a#(a(x1))

      (8)

      could be deleted.

      1.1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 1 component.

      • The 1^st component contains the pair

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

        →

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

        (4)

        1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

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

        together with the usable rule

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

        →

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

        (2)

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

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

        →

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

        (4)

        could be deleted.

        1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

        The dependency pairs are split into 0 components.