Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/caabccaabc-aabccaabcaabcc.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

  c#(a(a(b(c(c(a(a(b(c(x1))))))))))	→	c#(c(x1))	(2)
  c#(a(a(b(c(c(a(a(b(c(x1))))))))))	→	c#(c(a(a(b(c(a(a(b(c(c(x1)))))))))))	(3)
  c#(a(a(b(c(c(a(a(b(c(x1))))))))))	→	c#(a(a(b(c(c(x1))))))	(4)
  c#(a(a(b(c(c(a(a(b(c(x1))))))))))	→	c#(a(a(b(c(a(a(b(c(c(x1))))))))))	(5)

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

  together with the usable rule

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

  →

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

  (1)

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

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

  →

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

  (5)

  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

    c#(a(a(b(c(c(a(a(b(c(x1))))))))))	→	c#(c(x1))	(2)
    c#(a(a(b(c(c(a(a(b(c(x1))))))))))	→	c#(c(a(a(b(c(a(a(b(c(c(x1)))))))))))	(3)
    c#(a(a(b(c(c(a(a(b(c(x1))))))))))	→	c#(a(a(b(c(c(x1))))))	(4)

    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 0 -∞ -∞ 0 -∞ 0 -∞ 0 · x1 + 1 4 0 -∞
    [b(x1)]	=	-∞ -∞ -∞ -∞ -∞ -∞ -∞ 0 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 0 -∞ 4 -∞
    [a(x1)]	=	0 -∞ -∞ 0 -∞ -∞ -∞ -∞ 1 -∞ 0 -∞ 0 0 -∞ -∞ · x1 + -∞ -∞ -∞ -∞
    [c#(x1)]	=	1 -∞ 3 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 2 -∞ -∞ -∞

    together with the usable rule

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

    →

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

    (1)

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

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

    →

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

    (3)

    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

      c#(a(a(b(c(c(a(a(b(c(x1))))))))))	→	c#(c(x1))	(2)
      c#(a(a(b(c(c(a(a(b(c(x1))))))))))	→	c#(a(a(b(c(c(x1))))))	(4)

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

      together with the usable rule

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

      →

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

      (1)

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

      c#(a(a(b(c(c(a(a(b(c(x1))))))))))	→	c#(c(x1))	(2)
      c#(a(a(b(c(c(a(a(b(c(x1))))))))))	→	c#(a(a(b(c(c(x1))))))	(4)

      could be deleted.

      1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.