Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-503)

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#(c(b(x1)))	→	b#(c(x1))	(3)
  b#(c(b(x1)))	→	a#(b(c(x1)))	(4)
  a#(a(x1))	→	b#(c(x1))	(5)
  a#(a(x1))	→	b#(b(c(x1)))	(6)
  a#(a(x1))	→	b#(b(b(c(x1))))	(7)

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

  together with the usable rules

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

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

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

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

    1.1.1.1.1 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 -4 -4 0 0 -4 -4 -4 -4 -4 -4 -4 -4 · x1 + -∞ -∞ -∞ -∞
    [b(x1)]	=	0 0 4 4 0 0 0 0 0 0 0 0 -4 0 0 0 · x1 + -∞ -∞ -∞ -∞
    [a(x1)]	=	0 0 0 4 0 0 0 4 0 0 0 4 0 0 0 0 · x1 + -∞ -∞ -∞ -∞
    [b#(x1)]	=	14 17 18 18 14 17 18 18 14 17 18 18 14 17 18 18 · x1 + -∞ -∞ -∞ -∞
    [a#(x1)]	=	17 17 17 18 17 17 17 18 17 17 17 18 17 17 17 18 · x1 + -∞ -∞ -∞ -∞

    together with the usable rules

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

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

    b#(c(b(x1)))	→	b#(c(x1))	(3)
    a#(a(x1))	→	b#(c(x1))	(5)

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

    together with the usable rules

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

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

    b#(c(b(x1)))

    →

    a#(b(c(x1)))

    (4)

    and no rules could be deleted.

    1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.