Certification Problem

Input (TPDB SRS_Standard/Waldmann_06_SRS/z086-variant)

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#(c(x1))	→	b#(x1)	(4)
  b#(b(x1))	→	c#(x1)	(6)
  c#(c(x1))	→	a#(b(x1))	(5)
  a#(a(x1))	→	c#(x1)	(10)
  a#(a(x1))	→	b#(c(x1))	(11)
  b#(b(x1))	→	c#(c(x1))	(7)
  b#(b(x1))	→	c#(c(c(x1)))	(8)
  b#(b(x1))	→	a#(c(c(c(x1))))	(9)

  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

  [c(x1)]	=	-∞ 2 -∞ -∞ 0 -∞ -∞ 1 -∞ · x1 + 4 12 -∞
  [b(x1)]	=	-∞ 0 -∞ -∞ 1 -∞ -∞ 0 -∞ · x1 + 12 13 -∞
  [a(x1)]	=	-∞ 1 0 0 -∞ -∞ -∞ 0 0 · x1 + 13 12 -∞
  [c#(x1)]	=	-∞ 2 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 7 -∞ -∞
  [b#(x1)]	=	-∞ 2 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 0 -∞ -∞
  [a#(x1)]	=	1 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 14 -∞ -∞

  together with the usable rules

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

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

  b#(b(x1))	→	c#(x1)	(6)
  b#(b(x1))	→	c#(c(x1))	(7)
  b#(b(x1))	→	c#(c(c(x1)))	(8)

  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#(c(x1))	→	b#(x1)	(4)
    b#(b(x1))	→	a#(c(c(c(x1))))	(9)
    a#(a(x1))	→	c#(x1)	(10)
    c#(c(x1))	→	a#(b(x1))	(5)
    a#(a(x1))	→	b#(c(x1))	(11)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

    [c(x1)]	=	-∞ 0 4 -∞ 0 -∞ -∞ 4 0 · x1 + 9 1 0
    [b(x1)]	=	-∞ 0 -∞ -∞ 4 0 -∞ 2 -∞ · x1 + 1 5 -∞
    [a(x1)]	=	-∞ 4 0 0 -∞ -∞ -∞ 0 2 · x1 + 5 0 1
    [c#(x1)]	=	-∞ 7 3 -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 6 -∞ -∞
    [b#(x1)]	=	-∞ 7 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 3 -∞ -∞
    [a#(x1)]	=	3 2 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + -∞ -∞ -∞

    together with the usable rules

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

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

    c#(c(x1))

    →

    a#(b(x1))

    (5)

    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#(c(x1))	→	b#(x1)	(4)
      b#(b(x1))	→	a#(c(c(c(x1))))	(9)
      a#(a(x1))	→	c#(x1)	(10)
      a#(a(x1))	→	b#(c(x1))	(11)

      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 arctic semiring over the naturals

      [c(x1)]	=	-∞ 0 8 -∞ -∞ 0 -∞ 0 0 · x1 + 12 0 -∞
      [b(x1)]	=	-∞ 0 -∞ -∞ 4 4 -∞ 0 -∞ · x1 + 0 8 -∞
      [a(x1)]	=	-∞ 4 4 0 -∞ 0 -∞ -∞ 0 · x1 + 8 -∞ 0
      [c#(x1)]	=	7 -∞ 6 -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 9 -∞ -∞
      [b#(x1)]	=	-∞ 4 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 9 -∞ -∞
      [a#(x1)]	=	-∞ 8 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 9 -∞ -∞

      together with the usable rules

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

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

      c#(c(x1))

      →

      b#(x1)

      (4)

      could be deleted.

      1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

      Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

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

      together with the usable rules

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

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

      a#(a(x1))

      →

      c#(x1)

      (10)

      and no rules 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

        b#(b(x1))	→	a#(c(c(c(x1))))	(9)
        a#(a(x1))	→	b#(c(x1))	(11)

        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 arctic semiring over the naturals

        [c(x1)]	=	-∞ -∞ 0 2 -∞ 10 0 -∞ 0 · x1 + 0 4 1
        [b(x1)]	=	5 -∞ 5 0 -∞ -∞ 0 -∞ -∞ · x1 + 6 0 -∞
        [a(x1)]	=	-∞ 0 0 5 -∞ 5 -∞ -∞ 0 · x1 + 0 6 0
        [b#(x1)]	=	8 2 2 -∞ -∞ -∞ -∞ -∞ -∞ · x1 + -∞ -∞ -∞
        [a#(x1)]	=	13 0 9 -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 12 -∞ -∞

        together with the usable rules

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

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

        a#(a(x1))

        →

        b#(c(x1))

        (11)

        could be deleted.

        1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

        Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

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

        together with the usable rules

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

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

        b#(b(x1))

        →

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

        (9)

        and no rules could be deleted.

        1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

        The dependency pairs are split into 0 components.