Certification Problem

Input (TPDB SRS_Standard/Waldmann_06_SRS/jw3)

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

  Q#(x(x1))	→	Q#(x1)	(13)
  Q#(a(x1))	→	b#(b(a(x1)))	(14)
  b#(P(x1))	→	b#(Q(x1))	(10)
  b#(P(x1))	→	Q#(x1)	(11)
  Q#(a(x1))	→	b#(a(x1))	(15)

  1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [Q#(x1)]	=	1
  [x(x1)]	=	1 · x1
  [a(x1)]	=	1
  [b#(x1)]	=	1 · x1
  [b(x1)]	=	0
  [P(x1)]	=	1
  [Q(x1)]	=	1

  the pair

  Q#(a(x1))

  →

  b#(b(a(x1)))

  (14)

  could be deleted.

  1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

  [Q#(x1)]	=	0 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [x(x1)]	=	0 1 0 + 0 0 -∞ -∞ 1 -∞ -∞ -∞ 0 · x1
  [b#(x1)]	=	0 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [P(x1)]	=	0 0 1 + 0 0 -∞ -∞ -∞ -∞ 0 1 0 · x1
  [Q(x1)]	=	0 0 0 + -∞ 0 -∞ -∞ 0 -∞ -∞ 0 -∞ · x1
  [a(x1)]	=	-∞ 0 0 + 0 0 0 -∞ 0 -∞ 0 0 1 · x1
  [b(x1)]	=	0 0 0 + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1

  the pair

  b#(P(x1))

  →

  b#(Q(x1))

  (10)

  could be deleted.

  1.1.1.1.1 Size-Change Termination

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

  Q#(x(x1))	→	Q#(x1)	(13)
  1	>	1
  Q#(a(x1))	→	b#(a(x1))	(15)
  1	≥	1
  b#(P(x1))	→	Q#(x1)	(11)
  1	>	1

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

• The 2^nd component contains the pair

  a#(P(x1))	→	a#(x(x1))	(8)
  a#(x(x1))	→	a#(x1)	(9)
  a#(b(b(x1)))	→	a#(b(x1))	(7)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [a#(x1)]	=	1 · x1
  [P(x1)]	=	1 + 1 · x1
  [x(x1)]	=	1 · x1
  [b(x1)]	=	0
  [Q(x1)]	=	0
  [a(x1)]	=	0

  together with the usable rule

  b(P(x1))

  →

  b(Q(x1))

  (4)

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

  a#(P(x1))

  →

  a#(x(x1))

  (8)

  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.

  a#(x(x1))	→	a#(x1)	(9)
  1	>	1
  a#(b(b(x1)))	→	a#(b(x1))	(7)
  1	>	1

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