Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-427)

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 1 component.

• The 1^st component contains the pair

  b#(b(c(x1)))	→	b#(c(a(x1)))	(8)
  b#(b(c(x1)))	→	a#(x1)	(6)
  a#(x1)	→	b#(x1)	(4)
  a#(x1)	→	b#(b(x1))	(5)

  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

  [b#(x1)]	=	2 0 -∞ -∞ · x1 + 0 -∞ -∞ -∞
  [b(x1)]	=	-∞ 0 0 -∞ · x1 + 0 -∞ 0 -∞
  [a#(x1)]	=	2 3 -∞ -∞ · x1 + 3 -∞ -∞ -∞
  [a(x1)]	=	0 0 -∞ 0 · x1 + 0 -∞ 0 -∞
  [c(x1)]	=	-∞ 0 0 1 · x1 + 0 -∞ 1 -∞

  together with the usable rules

  a(x1)	→	b(b(x1))	(1)
  b(b(c(x1)))	→	c(b(c(a(x1))))	(2)
  c(c(x1))	→	x1	(3)

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

  b#(b(c(x1)))	→	b#(c(a(x1)))	(8)
  a#(x1)	→	b#(b(x1))	(5)

  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.

  b#(b(c(x1)))	→	a#(x1)	(6)
  1	>	1
  a#(x1)	→	b#(x1)	(4)
  1	≥	1

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