Certification Problem

Input (TPDB TRS_Standard/Various_04/11)

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#(0,1,x)

  →

  f#(h(x),h(x),x)

  (4)

  1.1.1.1 Usable Rules Processor

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

  h(0)	→	0	(2)
  h(g(x,y))	→	y	(3)

  1.1.1.1.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  h(0)

  h(g(x0,x1))

  1.1.1.1.1.1 Narrowing Processor

  We consider all narrowings of the pair below position 1 to get the following set of pairs

  f#(0,1,0)	→	f#(0,h(0),0)	(6)
  f#(0,1,g(x0,x1))	→	f#(x1,h(g(x0,x1)),g(x0,x1))	(7)

  1.1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    f#(0,1,g(x0,x1))

    →

    f#(x1,h(g(x0,x1)),g(x0,x1))

    (7)

    1.1.1.1.1.1.1.1 Usable Rules Processor

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

    h(g(x,y))

    →

    y

    (3)

    1.1.1.1.1.1.1.1.1 Rewriting Processor

    We rewrite the right hand side of the pair resulting in

    →

    1.1.1.1.1.1.1.1.1.1 Usable Rules Processor

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

    There are no rules.

    1.1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

    We restrict the innermost strategy to the following left hand sides.

    There are no lhss.

    1.1.1.1.1.1.1.1.1.1.1.1 Instantiation Processor

    We instantiate the pair to the following set of pairs

    There are no rules.

    1.1.1.1.1.1.1.1.1.1.1.1.1 P is empty

    There are no pairs anymore.