Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/abcbaabcba-aabcbabcbaabcb.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#(b(c(b(a(a(b(c(b(a(x1))))))))))	→	a#(b(c(b(a(b(c(b(a(a(x1))))))))))	(3)
  a#(b(c(b(a(a(b(c(b(a(x1))))))))))	→	a#(b(c(b(a(a(x1))))))	(4)
  a#(b(c(b(a(a(b(c(b(a(x1))))))))))	→	a#(a(x1))	(5)
  a#(b(c(b(a(a(b(c(b(a(x1))))))))))	→	a#(a(b(c(b(a(b(c(b(a(a(x1)))))))))))	(6)

  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 0 0 0 -4 -4 0 0 -4 -4 -4 0 · x1 + -∞ -∞ -∞ -∞
  [a(x1)]	=	0 0 0 0 0 0 0 0 0 0 0 0 -4 -4 0 0 · x1 + -∞ -∞ -∞ -∞
  [a#(x1)]	=	5 6 6 6 5 6 6 6 5 6 6 6 5 6 6 6 · x1 + -∞ -∞ -∞ -∞

  together with the usable rule

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

  →

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

  (2)

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

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

  →

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

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

    together with the usable rule

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

    →

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

    (2)

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

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

    →

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

    (6)

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

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

      together with the usable rule

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

      →

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

      (2)

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

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

      could be deleted.

      1.1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.