Certification Problem

Input (TPDB SRS_Standard/Bouchare_06/10)

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(b(x1)))	→	b#(a(a(x1)))	(6)
  b#(b(b(x1)))	→	a#(x1)	(7)
  a#(a(x1))	→	b#(x1)	(9)
  b#(b(b(x1)))	→	a#(a(x1))	(8)
  a#(a(x1))	→	b#(a(b(x1)))	(10)
  a#(a(x1))	→	a#(b(x1))	(11)
  a#(a(b(x1)))	→	b#(a(a(x1)))	(12)
  a#(a(b(x1)))	→	a#(x1)	(13)
  a#(a(b(x1)))	→	a#(a(x1))	(14)

  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 -2 -2 · x1 + -∞ -∞
  [a(x1)]	=	0 0 0 0 · x1 + -∞ -∞
  [b#(x1)]	=	5 5 5 5 · x1 + -∞ -∞
  [a#(x1)]	=	3 5 3 5 · x1 + -∞ -∞

  together with the usable rules

  b(b(b(x1)))	→	b(a(a(x1)))	(4)
  a(a(b(x1)))	→	b(a(a(x1)))	(5)
  a(a(x1))	→	b(a(b(x1)))	(3)

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

  a#(a(x1))

  →

  a#(b(x1))

  (11)

  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(b(x1)))	→	b#(a(a(x1)))	(6)
    b#(b(b(x1)))	→	a#(x1)	(7)
    a#(a(x1))	→	b#(x1)	(9)
    b#(b(b(x1)))	→	a#(a(x1))	(8)
    a#(a(x1))	→	b#(a(b(x1)))	(10)
    a#(a(b(x1)))	→	b#(a(a(x1)))	(12)
    a#(a(b(x1)))	→	a#(x1)	(13)
    a#(a(b(x1)))	→	a#(a(x1))	(14)

    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 10 · x1 + 1 11
    [a(x1)]	=	0 10 10 0 · x1 + 11 11
    [b#(x1)]	=	0 10 -∞ -∞ · x1 + -∞ -∞
    [a#(x1)]	=	16 10 -∞ -∞ · x1 + -∞ -∞

    together with the usable rules

    b(b(b(x1)))	→	b(a(a(x1)))	(4)
    a(a(b(x1)))	→	b(a(a(x1)))	(5)
    a(a(x1))	→	b(a(b(x1)))	(3)

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

    b#(b(b(x1)))	→	a#(x1)	(7)
    a#(a(x1))	→	b#(x1)	(9)
    a#(a(x1))	→	b#(a(b(x1)))	(10)
    a#(a(b(x1)))	→	b#(a(a(x1)))	(12)
    a#(a(b(x1)))	→	a#(x1)	(13)
    a#(a(b(x1)))	→	a#(a(x1))	(14)

    could be deleted.

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

    together with the usable rules

    b(b(b(x1)))	→	b(a(a(x1)))	(4)
    a(a(b(x1)))	→	b(a(a(x1)))	(5)
    a(a(x1))	→	b(a(b(x1)))	(3)

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

    b#(b(b(x1)))

    →

    a#(a(x1))

    (8)

    and no rules 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(x1)))

      →

      b#(a(a(x1)))

      (6)

      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 naturals

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

      together with the usable rules

      b(b(b(x1)))	→	b(a(a(x1)))	(4)
      a(a(b(x1)))	→	b(a(a(x1)))	(5)
      a(a(x1))	→	b(a(b(x1)))	(3)

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

      b#(b(b(x1)))

      →

      b#(a(a(x1)))

      (6)

      could be deleted.

      1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.