Certification Problem

Input (TPDB SRS_Standard/Trafo_06/un06)

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

  b#(b(x1))	→	b#(a(b(a(a(b(x1))))))	(5)
  b#(a(a(a(b(a(a(b(x1))))))))	→	b#(b(x1))	(11)
  b#(b(x1))	→	b#(a(a(b(x1))))	(6)
  b#(a(a(a(b(a(a(b(x1))))))))	→	b#(a(a(a(b(b(x1))))))	(12)
  b#(b(a(b(x1))))	→	b#(b(x1))	(7)
  b#(b(a(b(x1))))	→	b#(a(b(b(x1))))	(8)
  b#(b(a(a(b(x1)))))	→	b#(b(x1))	(9)
  b#(b(a(a(b(x1)))))	→	b#(a(a(b(b(x1)))))	(10)

  1.1.1 Reduction Pair Processor with Usable Rules

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

  [b(x1)]	=	-∞ · x1 + 12
  [a(x1)]	=	1 · x1 + 15
  [b#(x1)]	=	31 · x1 + 47

  together with the usable rules

  b(b(x1))	→	b(a(b(a(a(b(x1))))))	(1)
  b(b(a(b(x1))))	→	b(a(b(b(x1))))	(2)
  b(a(a(a(b(a(a(b(x1))))))))	→	b(a(a(a(b(b(x1))))))	(3)
  b(b(a(a(b(x1)))))	→	b(a(a(b(b(x1)))))	(4)

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

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

  →

  b#(b(x1))

  (11)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 2 components.

  • The 1^st component contains the pair

    b#(b(a(a(b(x1)))))	→	b#(b(x1))	(9)
    b#(b(a(b(x1))))	→	b#(b(x1))	(7)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

    [b(x1)]	=	-∞ 0 -∞ 8 · x1 + 0 2
    [a(x1)]	=	2 0 0 -∞ · x1 + -∞ 2
    [b#(x1)]	=	0 -∞ -∞ -∞ · x1 + 0 -∞

    together with the usable rules

    b(b(x1))	→	b(a(b(a(a(b(x1))))))	(1)
    b(b(a(b(x1))))	→	b(a(b(b(x1))))	(2)
    b(a(a(a(b(a(a(b(x1))))))))	→	b(a(a(a(b(b(x1))))))	(3)
    b(b(a(a(b(x1)))))	→	b(a(a(b(b(x1)))))	(4)

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

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

    →

    b#(b(x1))

    (9)

    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

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

      →

      b#(b(x1))

      (7)

      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 arctic semiring over the naturals

      [b(x1)]	=	-∞ -∞ 3 14 · x1 + 17 12
      [a(x1)]	=	-∞ 0 -∞ -∞ · x1 + 7 7
      [b#(x1)]	=	7 12 -∞ -∞ · x1 + 24 -∞

      together with the usable rules

      b(b(x1))	→	b(a(b(a(a(b(x1))))))	(1)
      b(b(a(b(x1))))	→	b(a(b(b(x1))))	(2)
      b(a(a(a(b(a(a(b(x1))))))))	→	b(a(a(a(b(b(x1))))))	(3)
      b(b(a(a(b(x1)))))	→	b(a(a(b(b(x1)))))	(4)

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

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

      →

      b#(b(x1))

      (7)

      could be deleted.

      1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.

  • The 2^nd component contains the pair

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

    →

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

    (12)

    1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

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

    together with the usable rules

    b(b(x1))	→	b(a(b(a(a(b(x1))))))	(1)
    b(b(a(b(x1))))	→	b(a(b(b(x1))))	(2)
    b(a(a(a(b(a(a(b(x1))))))))	→	b(a(a(a(b(b(x1))))))	(3)
    b(b(a(a(b(x1)))))	→	b(a(a(b(b(x1)))))	(4)

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

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

    →

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

    (12)

    could be deleted.

    1.1.1.1.2.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.