Certification Problem

Input (TPDB SRS_Standard/Waldmann_19/random-268)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ 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(b(x1))))	→	a#(x1)	(7)
  a#(b(b(b(x1))))	→	a#(b(a(x1)))	(5)
  a#(a(a(b(x1))))	→	a#(b(a(a(x1))))	(12)
  a#(a(a(b(x1))))	→	a#(a(x1))	(14)
  a#(a(a(b(x1))))	→	a#(x1)	(15)

  1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

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

  the pairs

  a#(b(b(b(x1))))	→	a#(x1)	(7)
  a#(b(b(b(x1))))	→	a#(b(a(x1)))	(5)
  a#(a(a(b(x1))))	→	a#(a(x1))	(14)
  a#(a(a(b(x1))))	→	a#(x1)	(15)

  could be deleted.

  1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rule

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

  →

  b(b(b(b(x1))))

  (2)

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

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

  →

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

  (12)

  could be deleted.

  1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  b#(a(b(a(x1))))	→	b#(b(b(x1)))	(9)
  b#(a(b(a(x1))))	→	b#(b(b(b(x1))))	(8)
  b#(a(b(a(x1))))	→	b#(b(x1))	(10)
  b#(a(b(a(x1))))	→	b#(x1)	(11)

  1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [b(x1)]	=	1 · x1
  [a(x1)]	=	1 · x1
  [b#(x1)]	=	1 · x1

  together with the usable rule

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

  →

  b(b(b(b(x1))))

  (2)

  (w.r.t. the implicit argument filter of the reduction pair), the rule 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#(a(b(a(x1))))

    →

    b#(x1)

    (11)

    1.1.2.1.1 Monotonic Reduction Pair Processor with Usable Rules

    Using the linear polynomial interpretation over the naturals

    [a(x1)]	=	1 · x1
    [b(x1)]	=	1 · x1
    [b#(x1)]	=	1 · x1

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

    1.1.2.1.1.1 Size-Change Termination

    Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

    b#(a(b(a(x1))))	→	b#(x1)	(11)
    1	>	1

    As there is no critical graph in the transitive closure, there are no infinite chains.