Certification Problem

Input (TPDB TRS_Standard/Endrullis_06/linear1)

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 1 component.

• The 1^st component contains the pair

  a#(a(f(b,a(x))))	→	a#(a(x))	(6)
  a#(a(f(b,a(x))))	→	a#(a(a(x)))	(5)

  1.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

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

  the pair

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

  →

  a#(a(x))

  (6)

  and no rules could be deleted.

  1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers

  [a#(x1)]	=	0 0 + 0 1 0 0 · x1
  [a(x1)]	=	0 1 + 0 1 1 0 · x1
  [f(x1, x2)]	=	1 0 + 1 0 0 0 · x1 + 1 0 0 0 · x2
  [b]	=	0 0

  the pair

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

  →

  a#(a(a(x)))

  (5)

  could be deleted.

  1.1.1.1.1 P is empty

  There are no pairs anymore.