Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z088)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by matchbox @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  a#(b(b(c(x1))))	→	a#(b(x1))	(7)
  a#(b(b(c(x1))))	→	a#(a(b(x1)))	(8)
  a#(b(b(c(x1))))	→	a#(a(a(b(x1))))	(9)
  a#(a(x1))	→	a#(b(b(b(x1))))	(13)

  1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 4 with strict dimension 1 over the arctic semiring over the naturals

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

  together with the usable rules

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

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

  a#(a(x1))

  →

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

  (13)

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

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

    together with the usable rules

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

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

    a#(b(b(c(x1))))	→	a#(a(b(x1)))	(8)
    a#(b(b(c(x1))))	→	a#(a(a(b(x1))))	(9)

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

      →

      a#(b(x1))

      (7)

      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 + 1
      [b(x1)]	=	x1 + 1
      [a(x1)]	=	x1 + 3
      [a#(x1)]	=	x1 + 0

      together with the usable rule

      b(a(x1))

      →

      b(b(c(x1)))

      (2)

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

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

      →

      a#(b(x1))

      (7)

      and the rule

      b(a(x1))

      →

      b(b(c(x1)))

      (2)

      could be deleted.

      1.1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.