Certification Problem

Input (TPDB SRS_Standard/Zantema_06/16)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Closure Under Flat Contexts

{a(☐), b(☐)}

1.1 Semantic Labeling

Root-labeling is applied.

1.1.1 Rule Removal

1.1.1.1 Dependency Pair Transformation

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  aa#(aa(ab(x1)))	→	aa#(ab(ba(aa(ab(x1)))))	(53)
  aa#(aa(aa(x1)))	→	aa#(ab(ba(aa(aa(x1)))))	(50)

  1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [aa#(x1)]	=	1 · x1
  [aa(x1)]	=	1 + 1 · x1
  [ab(x1)]	=	1
  [ba(x1)]	=	0
  [bb(x1)]	=	0

  together with the usable rule

  ab(ba(ab(x1)))

  →

  ab(bb(ba(ab(bb(x1)))))

  (29)

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

  aa#(aa(ab(x1)))	→	aa#(ab(ba(aa(ab(x1)))))	(53)
  aa#(aa(aa(x1)))	→	aa#(ab(ba(aa(aa(x1)))))	(50)

  could be deleted.

  1.1.1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  bb#(ba(ab(x1)))

  →

  bb#(x1)

  (68)

  1.1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [ba(x1)]	=	1 · x1
  [ab(x1)]	=	1 · x1
  [bb#(x1)]	=	1 · x1

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

  1.1.1.1.1.2.1 Size-Change Termination

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

  bb#(ba(ab(x1)))	→	bb#(x1)	(68)
  1	>	1

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