Certification Problem

Input (TPDB SRS_Standard/Waldmann_19/random-403)

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 2 components.

• The 1^st component contains the pair

  a#(b(b(a(x1))))	→	a#(a(b(x1)))	(12)
  a#(b(b(b(x1))))	→	a#(a(a(b(x1))))	(11)
  a#(b(b(b(x1))))	→	a#(b(x1))	(10)
  a#(b(b(a(x1))))	→	a#(b(x1))	(6)
  a#(b(b(b(x1))))	→	a#(a(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 + 0
  [b#(x1)]	=	0

  together with the usable rules

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

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

  a#(b(b(a(x1))))	→	a#(a(b(x1)))	(12)
  a#(b(b(b(x1))))	→	a#(b(x1))	(10)
  a#(b(b(a(x1))))	→	a#(b(x1))	(6)
  a#(b(b(b(x1))))	→	a#(a(b(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

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

    →

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

    (11)

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

    together with the usable rules

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

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

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

    →

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

    (11)

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

  b#(b(a(a(x1))))	→	b#(b(b(a(x1))))	(13)
  b#(b(a(a(x1))))	→	b#(a(x1))	(5)
  b#(b(a(a(x1))))	→	b#(b(a(x1)))	(9)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [a(x1)]	=	x1 + 15922
  [b(x1)]	=	x1 + 15922
  [a#(x1)]	=	0
  [b#(x1)]	=	x1 + 0

  together with the usable rules

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

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

  b#(b(a(a(x1))))	→	b#(a(x1))	(5)
  b#(b(a(a(x1))))	→	b#(b(a(x1)))	(9)

  could be deleted.

  1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

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

    →

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

    (13)

    1.1.2.1.1 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)	=	1		status(a)	=	[]		list-extension(a)	=	Lex
    prec(b)	=	0		status(b)	=	[1]		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 + 20538
    [b(x1)]	=	x1 + 20538
    [a#(x1)]	=	1
    [b#(x1)]	=	x1 + 1

    together with the usable rules

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

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

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

    →

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

    (13)

    could be deleted.

    1.1.2.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.