Certification Problem

Input (TPDB TRS_Standard/Strategy_removed_AG01/#4.27)

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 2 components.

• The 1^st component contains the pair

  minus#(x,s(y))

  →

  minus#(x,p(s(y)))

  (12)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the naturals

  [p(x1)]	=	0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 · x1 + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  [minus#(x1, x2)]	=	0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 · x1 + 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 · x2 + 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  [s(x1)]	=	1 1 1 1 1 0 0 1 0 1 1 0 1 0 0 1 · x1 + 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

  together with the usable rule

  p(s(x))

  →

  x

  (2)

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

  minus#(x,s(y))

  →

  minus#(x,p(s(y)))

  (12)

  could be deleted.

  1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  le#(s(x),s(y))

  →

  le#(x,y)

  (10)

  1.1.2 Size-Change Termination

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

  le#(s(x),s(y))	→	le#(x,y)	(10)
  2	>	2
  1	>	1

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