Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/aprove3)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by ttt2 @ termCOMP 2023)

1 Rule Removal

1.1 Dependency Pair Transformation

1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  g#(x1)	→	f#(p(p(x1)))	(15)
  f#(s(x1))	→	g#(q(i(x1)))	(12)

  1.1.1.1 Reduction Pair Processor with Usable Rules

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

  [i(x1)]	=	1 · x1 + 1
  [f#(x1)]	=	0 · x1 + 0
  [p(x1)]	=	-1 · x1 + 0
  [q(x1)]	=	1 · x1 + -16
  [0(x1)]	=	0 · x1 + 0
  [g#(x1)]	=	-1 · x1 + 1
  [s(x1)]	=	1 · x1 + 1

  together with the usable rules

  p(s(0(x1)))	→	0(x1)	(2)
  p(s(s(x1)))	→	s(p(s(x1)))	(3)
  i(x1)	→	s(x1)	(8)
  q(i(x1))	→	q(s(x1))	(6)
  q(s(x1))	→	s(s(x1))	(7)

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

  g#(x1)

  →

  f#(p(p(x1)))

  (15)

  could be deleted.

  1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

  p#(s(s(x1)))

  →

  p#(s(x1))

  (9)

  1.1.1.2 Size-Change Termination

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

  p#(s(s(x1)))	→	p#(s(x1))	(9)
  1	>	1

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