Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size11/size-11-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 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  b#(a(a(b(x1))))	→	b#(b(a(x1)))	(3)
  b#(a(a(b(x1))))	→	b#(a(x1))	(4)
  b#(a(a(b(x1))))	→	a#(x1)	(5)
  a#(x1)	→	b#(x1)	(8)
  b#(a(a(b(x1))))	→	a#(b(b(a(x1))))	(6)
  b#(a(a(b(x1))))	→	a#(a(b(b(a(x1)))))	(7)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

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

  together with the usable rules

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

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

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

  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#(a(a(b(x1))))	→	a#(x1)	(5)
    a#(x1)	→	b#(x1)	(8)
    b#(a(a(b(x1))))	→	a#(b(b(a(x1))))	(6)
    b#(a(a(b(x1))))	→	a#(a(b(b(a(x1)))))	(7)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

    [b(x1)]	=	0 -∞ -∞ 0 -∞ -∞ -∞ 0 -∞ · x1 + -∞ 1 9
    [a(x1)]	=	0 0 -∞ 1 -∞ 1 0 0 0 · x1 + 8 2 9
    [b#(x1)]	=	4 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 8 -∞ -∞
    [a#(x1)]	=	5 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 12 -∞ -∞

    together with the usable rules

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

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

    a#(x1)

    →

    b#(x1)

    (8)

    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

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

    together with the usable rules

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

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

    b#(a(a(b(x1))))	→	a#(x1)	(5)
    b#(a(a(b(x1))))	→	a#(b(b(a(x1))))	(6)
    b#(a(a(b(x1))))	→	a#(a(b(b(a(x1)))))	(7)

    and no rules could be deleted.

    1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.