Certification Problem

Input (TPDB SRS_Standard/Waldmann_06_SRS/jw4)

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

  b#(b(a(b(x1))))	→	b#(b(x1))	(7)
  b#(b(a(b(x1))))	→	b#(b(b(x1)))	(8)
  b#(b(a(a(b(x1)))))	→	b#(b(a(b(x1))))	(10)
  b#(b(a(a(a(b(x1))))))	→	b#(b(a(a(b(x1)))))	(13)

  1.1.1.1 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 -3 -3 -3 -3 -3 -3 · x1 + -∞ -∞ -∞
  [a(x1)]	=	0 0 3 0 0 0 -3 0 0 · x1 + -∞ -∞ -∞
  [b#(x1)]	=	20 20 20 20 20 20 20 20 20 · x1 + -∞ -∞ -∞

  together with the usable rules

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

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

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

  →

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

  (13)

  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

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

    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

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

    together with the usable rules

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

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

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

    →

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

    (10)

    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

      b#(b(a(b(x1))))	→	b#(b(x1))	(7)
      b#(b(a(b(x1))))	→	b#(b(b(x1)))	(8)

      1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

      Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals

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

      together with the usable rules

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

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

      b#(b(a(b(x1))))	→	b#(b(x1))	(7)
      b#(b(a(b(x1))))	→	b#(b(b(x1)))	(8)

      could be deleted.

      1.1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.