Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-2-num-3)

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

  b#(b(a(x1)))	→	b#(x1)	(4)
  b#(b(a(x1)))	→	b#(b(x1))	(5)
  b#(b(a(x1)))	→	b#(a(b(b(x1))))	(6)

  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 + 23 -∞
  [a(x1)]	=	7 0 0 -∞ · x1 + 30 -∞
  [b#(x1)]	=	1 6 -∞ -∞ · x1 + 3 -∞

  together with the usable rules

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

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

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

  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(a(x1)))

    →

    b#(x1)

    (4)

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

    having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair

    b#(b(a(x1)))

    →

    b#(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.