Certification Problem

Input (TPDB SRS_Standard/Trafo_06/dup05)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by NaTT @ 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

  a#(a(b(b(a(a(a(a(x1))))))))	→	a#(b(b(a(a(x1)))))	(24)
  a#(a(s(s(x1))))	→	a#(x1)	(11)
  a#(a(b(b(a(a(a(a(x1))))))))	→	b#(b(a(a(b(b(a(a(x1))))))))	(22)
  b#(b(a(a(b(b(b(b(x1))))))))	→	a#(b(b(x1)))	(10)
  b#(b(a(a(b(b(b(b(x1))))))))	→	a#(b(b(a(a(b(b(x1)))))))	(21)
  a#(a(b(b(a(a(a(a(x1))))))))	→	b#(a(a(b(b(a(a(x1)))))))	(18)
  b#(b(a(a(b(b(b(b(x1))))))))	→	a#(a(b(b(x1))))	(9)
  b#(b(a(a(b(b(b(b(x1))))))))	→	b#(b(a(a(b(b(x1))))))	(8)
  a#(a(s(s(x1))))	→	a#(a(x1))	(7)
  a#(a(b(b(a(a(a(a(x1))))))))	→	b#(b(a(a(x1))))	(17)
  a#(a(b(b(a(a(a(a(x1))))))))	→	a#(a(b(b(a(a(x1))))))	(16)
  b#(b(a(a(b(b(s(s(x1))))))))	→	a#(x1)	(6)
  b#(b(a(a(b(b(s(s(x1))))))))	→	a#(a(x1))	(15)
  b#(b(a(a(b(b(b(b(x1))))))))	→	b#(a(a(b(b(x1)))))	(14)
  b#(b(a(a(b(b(b(b(x1))))))))	→	a#(a(b(b(a(a(b(b(x1))))))))	(13)
  a#(a(b(b(a(a(a(a(x1))))))))	→	b#(a(a(x1)))	(12)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [a(x1)]	=	x1 + 20538
  [s(x1)]	=	x1 + 16203
  [b(x1)]	=	x1 + 20538
  [a#(x1)]	=	x1 + 0
  [b#(x1)]	=	x1 + 0

  together with the usable rules

  a(a(b(b(a(a(a(a(x1))))))))	→	b(b(a(a(b(b(a(a(x1))))))))	(4)
  a(a(s(s(x1))))	→	s(s(a(a(x1))))	(1)
  b(b(a(a(b(b(b(b(x1))))))))	→	a(a(b(b(a(a(b(b(x1))))))))	(3)
  b(b(a(a(b(b(s(s(x1))))))))	→	a(a(b(b(s(s(a(a(x1))))))))	(2)

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

  a#(a(b(b(a(a(a(a(x1))))))))	→	a#(b(b(a(a(x1)))))	(24)
  a#(a(s(s(x1))))	→	a#(x1)	(11)
  b#(b(a(a(b(b(b(b(x1))))))))	→	a#(b(b(x1)))	(10)
  b#(b(a(a(b(b(b(b(x1))))))))	→	a#(b(b(a(a(b(b(x1)))))))	(21)
  a#(a(b(b(a(a(a(a(x1))))))))	→	b#(a(a(b(b(a(a(x1)))))))	(18)
  b#(b(a(a(b(b(b(b(x1))))))))	→	a#(a(b(b(x1))))	(9)
  b#(b(a(a(b(b(b(b(x1))))))))	→	b#(b(a(a(b(b(x1))))))	(8)
  a#(a(s(s(x1))))	→	a#(a(x1))	(7)
  a#(a(b(b(a(a(a(a(x1))))))))	→	b#(b(a(a(x1))))	(17)
  a#(a(b(b(a(a(a(a(x1))))))))	→	a#(a(b(b(a(a(x1))))))	(16)
  b#(b(a(a(b(b(s(s(x1))))))))	→	a#(x1)	(6)
  b#(b(a(a(b(b(s(s(x1))))))))	→	a#(a(x1))	(15)
  b#(b(a(a(b(b(b(b(x1))))))))	→	b#(a(a(b(b(x1)))))	(14)
  a#(a(b(b(a(a(a(a(x1))))))))	→	b#(a(a(x1)))	(12)

  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

    a#(a(b(b(a(a(a(a(x1))))))))	→	b#(b(a(a(b(b(a(a(x1))))))))	(22)
    b#(b(a(a(b(b(b(b(x1))))))))	→	a#(a(b(b(a(a(b(b(x1))))))))	(13)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals

    [a(x1)]	=	0 1 0 1 · x1 + 1 28734
    [s(x1)]	=	988 27553
    [b(x1)]	=	1 0 1 0 · x1 + 28734 1
    [a#(x1)]	=	x1 + 0 0
    [b#(x1)]	=	0 1 0 1 · x1 + 28732 0

    together with the usable rules

    a(a(b(b(a(a(a(a(x1))))))))	→	b(b(a(a(b(b(a(a(x1))))))))	(4)
    a(a(s(s(x1))))	→	s(s(a(a(x1))))	(1)
    b(b(a(a(b(b(b(b(x1))))))))	→	a(a(b(b(a(a(b(b(x1))))))))	(3)
    b(b(a(a(b(b(s(s(x1))))))))	→	a(a(b(b(s(s(a(a(x1))))))))	(2)

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

    a#(a(b(b(a(a(a(a(x1))))))))	→	b#(b(a(a(b(b(a(a(x1))))))))	(22)
    b#(b(a(a(b(b(b(b(x1))))))))	→	a#(a(b(b(a(a(b(b(x1))))))))	(13)

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.