Certification Problem

Input (TPDB TRS_Standard/SK90/2.52)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.

• The 1^st component contains the pair

  f#(s(x),s(y),s(z))	→	f#(s(x),s(y),z)	(15)
  f#(s(x),s(y),s(z))	→	f#(x,y,f(s(x),s(y),z))	(16)
  f#(s(x),0,s(z))	→	f#(x,s(0),z)	(14)
  f#(s(x),s(y),0)	→	f#(x,y,s(0))	(13)

  1.1.1 Subterm Criterion Processor

  We use the projection

  π(f#)

  =

  1

  and remove the pairs:

  f#(s(x),s(y),s(z))	→	f#(x,y,f(s(x),s(y),z))	(16)
  f#(s(x),0,s(z))	→	f#(x,s(0),z)	(14)
  f#(s(x),s(y),0)	→	f#(x,y,s(0))	(13)

  1.1.1.1 Size-Change Termination

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

  f#(s(x),s(y),s(z))	→	f#(s(x),s(y),z)	(15)
  3	>	3
  2	≥	2
  1	≥	1

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

• The 2^nd component contains the pair

  f#(0,s(s(y)),s(s(z)))	→	f#(0,y,f(0,s(s(y)),s(z)))	(20)
  f#(0,s(s(y)),s(s(z)))	→	f#(0,s(s(y)),s(z))	(19)

  1.1.2 Size-Change Termination

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

  f#(0,s(s(y)),s(s(z)))	→	f#(0,y,f(0,s(s(y)),s(z)))	(20)
  2	>	2
  1	≥	1
  f#(0,s(s(y)),s(s(z)))	→	f#(0,s(s(y)),s(z))	(19)
  3	>	3
  2	≥	2
  1	≥	1

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

• The 3^rd component contains the pair

  f#(0,s(0),s(s(z)))

  →

  f#(0,s(0),z)

  (18)

  1.1.3 Size-Change Termination

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

  f#(0,s(0),s(s(z)))	→	f#(0,s(0),z)	(18)
  3	>	3
  2	≥	2
  2	>	1
  1	≥	1

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

• The 4^th component contains the pair

  f#(0,s(s(y)),s(0))

  →

  f#(0,y,s(0))

  (17)

  1.1.4 Size-Change Termination

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

  f#(0,s(s(y)),s(0))	→	f#(0,y,s(0))	(17)
  3	≥	3
  3	>	1
  2	>	2
  1	≥	1

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