Certification Problem

Input (TPDB TRS_Standard/Zantema_05/jw43)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Switch to Innermost Termination

The TRS is overlay and locally confluent:

Hence, it suffices to show innermost termination in the following.

1.1 Dependency Pair Transformation

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  f#(x,f(f(a,a),a))

  →

  f#(f(a,f(a,a)),x)

  (2)

  1.1.1.1 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  There are no rules.

  1.1.1.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [f#(x1, x2)]	=	2 · x1 + 2 · x2
  [f(x1, x2)]	=	2 · x1 + 1 · x2
  [a]	=	2

  the pair

  f#(x,f(f(a,a),a))

  →

  f#(f(a,f(a,a)),x)

  (2)

  and the rule

  f(x,f(f(a,a),a))

  →

  f(f(a,f(a,a)),x)

  (1)

  could be deleted.

  1.1.1.1.1.1 P is empty

  There are no pairs anymore.