Certification Problem

Input (TPDB SRS_Relative/Waldmann_06_relative/r8)

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

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

  together with the usable rules

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

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

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

  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(c(x1)))	→	a#(x1)	(5)
    a#(b(a(x1)))	→	c#(c(c(x1)))	(10)
    c#(c(c(x1)))	→	a#(c(a(x1)))	(6)

    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)]	=	-∞ 6 12 9 · x1 + 0 13
    [b(x1)]	=	16 -∞ -∞ -∞ · x1 + -∞ -∞
    [a(x1)]	=	9 0 12 0 · x1 + 16 11
    [c#(x1)]	=	7 7 -∞ -∞ · x1 + -∞ -∞
    [a#(x1)]	=	10 -∞ -∞ -∞ · x1 + 5 -∞

    together with the usable rules

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

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

    c#(c(c(x1)))	→	a#(x1)	(5)
    a#(b(a(x1)))	→	c#(c(c(x1)))	(10)

    could be deleted.

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

    together with the usable rules

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

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

    c#(c(c(x1)))

    →

    a#(c(a(x1)))

    (6)

    and no rules could be deleted.

    1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.