Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-503)

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#(c(b(x1)))	→	b#(c(x1))	(6)
  b#(c(b(x1)))	→	a#(b(c(x1)))	(7)
  a#(a(x1))	→	b#(c(x1))	(3)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the arctic semiring over the integers

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

  together with the usable rules

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

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

  a#(a(x1))

  →

  b#(c(x1))

  (3)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    b#(c(b(x1)))

    →

    b#(c(x1))

    (6)

    1.1.1.1.1 Subterm Criterion Processor

    We use the projection to multisets

    π(b#)	=	{ 1 }
    π(c)	=	{ 1 }

    to remove the pairs:

    b#(c(b(x1)))

    →

    b#(c(x1))

    (6)

    1.1.1.1.1.1 P is empty

    There are no pairs anymore.