Certification Problem

Input (TPDB TRS_Standard/Zantema_05/z08)

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

  f#(a,f(b,f(a,x)))

  →

  f#(a,f(b,f(b,f(a,x))))

  (4)

  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]	=	0 -∞ 0 -∞
  [f#(x1, x2)]	=	0 0 -∞ -∞ · x1 + 0 0 -∞ -∞ · x2 + 0 -∞ -∞ -∞
  [a]	=	0 -∞ 2 -∞
  [f(x1, x2)]	=	0 0 -∞ 0 · x1 + -∞ -∞ 1 -∞ · x2 + 0 -∞ 0 -∞

  together with the usable rules

  f(a,f(b,f(a,x)))	→	f(a,f(b,f(b,f(a,x))))	(1)
  f(b,f(b,f(b,x)))	→	f(b,f(b,x))	(2)

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

  f#(a,f(b,f(a,x)))

  →

  f#(a,f(b,f(b,f(a,x))))

  (4)

  could be deleted.

  1.1.1.1 P is empty

  There are no pairs anymore.