Certification Problem

Input (TPDB SRS_Standard/Secret_06_SRS/5)

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(b(b(x1))))	→	b#(a(a(x1)))	(3)
  b#(b(b(b(x1))))	→	a#(x1)	(4)
  a#(a(a(x1)))	→	b#(x1)	(7)
  b#(b(b(b(x1))))	→	a#(b(a(a(x1))))	(5)
  a#(a(a(x1)))	→	b#(b(x1))	(8)
  b#(b(b(b(x1))))	→	a#(a(x1))	(6)
  a#(a(a(x1)))	→	a#(b(b(x1)))	(9)
  a#(a(a(x1)))	→	a#(a(b(b(x1))))	(10)

  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 0 0 0 · x1 + -∞ -∞
  [a(x1)]	=	0 0 -2 -2 · x1 + -∞ -∞
  [b#(x1)]	=	5 7 5 7 · x1 + -∞ -∞
  [a#(x1)]	=	7 7 7 7 · x1 + -∞ -∞

  together with the usable rules

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

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

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

  →

  b#(a(a(x1)))

  (3)

  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

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

    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 0 0 -∞ · x1 + 3 0
    [a(x1)]	=	-∞ -∞ 0 0 · x1 + 0 3
    [b#(x1)]	=	-∞ 25 -∞ -∞ · x1 + 28 -∞
    [a#(x1)]	=	-∞ 25 -∞ -∞ · x1 + 25 -∞

    together with the usable rules

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

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

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

    →

    a#(b(a(a(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

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

      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

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

      together with the usable rules

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

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

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

      →

      a#(a(x1))

      (6)

      could be deleted.

      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(b(b(x1))))	→	a#(x1)	(4)
        a#(a(a(x1)))	→	b#(x1)	(7)
        a#(a(a(x1)))	→	b#(b(x1))	(8)
        a#(a(a(x1)))	→	a#(b(b(x1)))	(9)
        a#(a(a(x1)))	→	a#(a(b(b(x1))))	(10)

        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

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

        together with the usable rules

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

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

        a#(a(a(x1)))

        →

        b#(x1)

        (7)

        could be deleted.

        1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

        The dependency pairs are split into 1 component.

        • The 1^st component contains the pair

          b#(b(b(b(x1))))	→	a#(x1)	(4)
          a#(a(a(x1)))	→	b#(b(x1))	(8)
          a#(a(a(x1)))	→	a#(b(b(x1)))	(9)
          a#(a(a(x1)))	→	a#(a(b(b(x1))))	(10)

          1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

          [b(x1)]	=	-∞ 3 0 -∞ · x1 + 8 -∞
          [a(x1)]	=	-∞ -∞ -∞ 11 · x1 + 0 11
          [b#(x1)]	=	23 -∞ -∞ -∞ · x1 + 31 -∞
          [a#(x1)]	=	-∞ 5 -∞ -∞ · x1 + 31 -∞

          together with the usable rules

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

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

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

          →

          a#(x1)

          (4)

          could be deleted.

          1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

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

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

          together with the usable rules

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

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

          a#(a(a(x1)))

          →

          b#(b(x1))

          (8)

          and no rules could be deleted.

          1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

          The dependency pairs are split into 1 component.

          • The 1^st component contains the pair

            a#(a(a(x1)))	→	a#(b(b(x1)))	(9)
            a#(a(a(x1)))	→	a#(a(b(b(x1))))	(10)

            1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

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

            together with the usable rules

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

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

            a#(a(a(x1)))

            →

            a#(b(b(x1)))

            (9)

            could be deleted.

            1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

            The dependency pairs are split into 1 component.

            • The 1^st component contains the pair

              a#(a(a(x1)))

              →

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

              (10)

              1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

              [b(x1)]	=	-∞ 15 -∞ 1 · x1 + 0 0
              [a(x1)]	=	3 17 -∞ -∞ · x1 + 16 -∞
              [a#(x1)]	=	0 23 -∞ -∞ · x1 + 10 -∞

              together with the usable rules

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

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

              a#(a(a(x1)))

              →

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

              (10)

              could be deleted.

              1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

              The dependency pairs are split into 0 components.