Certification Problem

Input (TPDB SRS_Standard/Waldmann_19/random-266)

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#(b(b(a(x1))))	→	b#(x1)	(14)
  b#(a(a(b(x1))))	→	b#(a(a(a(x1))))	(13)
  a#(b(b(a(x1))))	→	a#(a(b(b(x1))))	(12)
  a#(b(b(a(x1))))	→	b#(b(x1))	(11)
  b#(a(a(b(x1))))	→	a#(x1)	(7)
  b#(a(a(b(x1))))	→	a#(a(a(x1)))	(6)
  b#(a(a(a(x1))))	→	b#(b(a(x1)))	(5)
  b#(a(a(a(x1))))	→	b#(b(b(a(x1))))	(10)
  b#(a(a(b(x1))))	→	a#(a(x1))	(9)
  b#(a(a(a(x1))))	→	b#(a(x1))	(8)
  a#(b(b(a(x1))))	→	a#(b(b(x1)))	(4)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [a(x1)]	=	x1 + 21239
  [b(x1)]	=	x1 + 21239
  [a#(x1)]	=	x1 + 21238
  [b#(x1)]	=	x1 + 0

  together with the usable rules

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

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

  a#(b(b(a(x1))))	→	b#(x1)	(14)
  a#(b(b(a(x1))))	→	b#(b(x1))	(11)
  b#(a(a(b(x1))))	→	a#(x1)	(7)
  b#(a(a(b(x1))))	→	a#(a(a(x1)))	(6)
  b#(a(a(a(x1))))	→	b#(b(a(x1)))	(5)
  b#(a(a(b(x1))))	→	a#(a(x1))	(9)
  b#(a(a(a(x1))))	→	b#(a(x1))	(8)
  a#(b(b(a(x1))))	→	a#(b(b(x1)))	(4)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 2 components.

  • The 1^st component contains the pair

    b#(a(a(b(x1))))	→	b#(a(a(a(x1))))	(13)
    b#(a(a(a(x1))))	→	b#(b(b(a(x1))))	(10)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [a(x1)]	=	1144
    [b(x1)]	=	1
    [a#(x1)]	=	x1 + 21238
    [b#(x1)]	=	x1 + 0

    together with the usable rules

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

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

    b#(a(a(a(x1))))

    →

    b#(b(b(a(x1))))

    (10)

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 1 component.

    • The 1^st component contains the pair

      b#(a(a(b(x1))))

      →

      b#(a(a(a(x1))))

      (13)

      1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

      Using the argument filter
      in combination with the following Weighted Path Order with the following precedence and status

      prec(a)	=	0		status(a)	=	[1]		list-extension(a)	=	Lex
      prec(b)	=	1		status(b)	=	[]		list-extension(b)	=	Lex
      prec(a#)	=	0		status(a#)	=	[]		list-extension(a#)	=	Lex
      prec(b#)	=	0		status(b#)	=	[1]		list-extension(b#)	=	Lex

      and the following Max-polynomial interpretation

      [a(x1)]	=	x1 + 21239
      [b(x1)]	=	x1 + 21239
      [a#(x1)]	=	1
      [b#(x1)]	=	x1 + 1

      together with the usable rules

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

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

      b#(a(a(b(x1))))

      →

      b#(a(a(a(x1))))

      (13)

      could be deleted.

      1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.

  • The 2^nd component contains the pair

    a#(b(b(a(x1))))

    →

    a#(a(b(b(x1))))

    (12)

    1.1.1.1.2 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [a(x1)]	=	1
    [b(x1)]	=	2
    [a#(x1)]	=	x1 + 21238
    [b#(x1)]	=	0

    together with the usable rules

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

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

    a#(b(b(a(x1))))

    →

    a#(a(b(b(x1))))

    (12)

    could be deleted.

    1.1.1.1.2.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.