Certification Problem

Input (TPDB TRS_Standard/Der95/21)

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 3 components.

• The 1^st component contains the pair

  fact#(s(x))

  →

  fact#(p(s(x)))

  (9)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the arctic semiring over the integers

  [p(x1)]	=	-9 · x1 + 9
  [s(x1)]	=	10 · x1 + 13
  [fact#(x1)]	=	-9 · x1 + 0

  together with the usable rule

  p(s(x))

  →

  x

  (1)

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

  fact#(s(x))

  →

  fact#(p(s(x)))

  (9)

  could be deleted.

  1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  *#(s(x),y)

  →

  *#(x,y)

  (11)

  1.1.2 Size-Change Termination

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

  *#(s(x),y)	→	*#(x,y)	(11)
  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

  +#(x,s(y))

  →

  +#(x,y)

  (13)

  1.1.3 Size-Change Termination

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

  +#(x,s(y))	→	+#(x,y)	(13)
  2	>	2
  1	≥	1

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