Certification Problem

Input (TPDB SRS_Standard/Mixed_SRS/07)

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

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

  together with the usable rules

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

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

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

  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

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

    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 0 -∞ · x1 + 0 -∞
    [a(x1)]	=	-∞ 0 0 2 · x1 + 4 9
    [b#(x1)]	=	15 22 -∞ -∞ · x1 + -∞ -∞
    [a#(x1)]	=	24 22 -∞ -∞ · x1 + 31 -∞

    together with the usable rules

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

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

    a#(x1)

    →

    b#(b(x1))

    (8)

    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

      a#(x1)	→	b#(x1)	(7)
      b#(b(a(a(x1))))	→	a#(a(b(a(x1))))	(6)
      a#(x1)	→	b#(b(b(x1)))	(9)

      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 0 -∞ · x1 + 0 0
      [a(x1)]	=	12 0 0 -∞ · x1 + 12 0
      [b#(x1)]	=	7 8 -∞ -∞ · x1 + 12 -∞
      [a#(x1)]	=	19 31 -∞ -∞ · x1 + 30 -∞

      together with the usable rules

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

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

      a#(x1)	→	b#(x1)	(7)
      b#(b(a(a(x1))))	→	a#(a(b(a(x1))))	(6)
      a#(x1)	→	b#(b(b(x1)))	(9)

      could be deleted.

      1.1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.