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 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

  Q#(x(x1))	→	Q#(x1)	(12)
  Q#(a(x1))	→	b#(a(x1))	(14)
  b#(P(x1))	→	b#(Q(x1))	(11)
  b#(P(x1))	→	Q#(x1)	(10)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers

  [Q(x1)]	=	-∞ 1 -∞ -∞ · x1 + 0 -∞ 1 -∞
  [a(x1)]	=	1 0 -∞ 0 · x1 + 0 -∞ 0 -∞
  [x(x1)]	=	0 0 -∞ 1 · x1 + 0 -∞ 1 -∞
  [b(x1)]	=	-∞ -∞ -∞ -∞ · x1 + 1 -∞ 0 -∞
  [Q#(x1)]	=	0 0 -∞ -∞ · x1 + 0 -∞ -∞ -∞
  [b#(x1)]	=	0 0 -∞ -∞ · x1 + 0 -∞ -∞ -∞
  [P(x1)]	=	0 2 -∞ -∞ · x1 + 2 -∞ -∞ -∞

  together with the usable rules

  a(b(b(x1)))	→	P(a(b(x1)))	(1)
  a(P(x1))	→	P(a(x(x1)))	(2)
  a(x(x1))	→	x(a(x1))	(3)
  b(P(x1))	→	b(Q(x1))	(4)
  Q(x(x1))	→	a(Q(x1))	(5)
  Q(a(x1))	→	b(b(a(x1)))	(6)

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

  b#(P(x1))

  →

  b#(Q(x1))

  (11)

  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.

  Q#(x(x1))	→	Q#(x1)	(12)
  1	>	1
  Q#(a(x1))	→	b#(a(x1))	(14)
  1	≥	1
  b#(P(x1))	→	Q#(x1)	(10)
  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#(x(x1))	→	a#(x1)	(9)
  a#(P(x1))	→	a#(x(x1))	(8)

  1.1.2 Subterm Criterion Processor

  We use the projection to multisets

  π(a#)	=	{ 1 }
  π(x)	=	{ 1 }

  to remove the pairs:

  a#(P(x1))

  →

  a#(x(x1))

  (8)

  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

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

• The 3^rd component contains the pair

  a#(b(b(x1)))

  →

  a#(b(x1))

  (7)

  1.1.3 Size-Change Termination

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

  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.