Certification Problem

Input (TPDB TRS_Standard/AG01/#3.48)

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

  g#(x,c(y))	→	g#(s(x),y)	(13)
  g#(x,c(y))	→	g#(x,y)	(9)

  1.1.1 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [c(x1)]	=	1 · x1
  [s(x1)]	=	1 · x1
  [g#(x1, x2)]	=	1 · x1 + 1 · x2

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

  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.

  g#(x,c(y))	→	g#(s(x),y)	(13)
  2	>	2
  g#(x,c(y))	→	g#(x,y)	(9)
  1	≥	1
  2	>	2

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

• The 2^nd component contains the pair

  f#(s(x))

  →

  f#(x)

  (8)

  1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [f#(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 Size-Change Termination

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

  f#(s(x))	→	f#(x)	(8)
  1	>	1

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