Certification Problem

Input (TPDB SRS_Standard/Trafo_06/un14)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

1.1 Dependency Pair Transformation

1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  b#(a(a(b(a(a(a(b(x1))))))))	→	b#(a(a(b(x1))))	(15)
  b#(a(a(b(a(a(a(b(x1))))))))	→	b#(a(a(b(a(a(b(x1)))))))	(14)

  1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

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

  the pairs

  b#(a(a(b(a(a(a(b(x1))))))))	→	b#(a(a(b(x1))))	(15)
  b#(a(a(b(a(a(a(b(x1))))))))	→	b#(a(a(b(a(a(b(x1)))))))	(14)

  could be deleted.

  1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  b#(b(b(a(a(a(b(x1)))))))	→	b#(b(x1))	(18)
  b#(b(b(a(a(a(b(x1)))))))	→	b#(b(b(x1)))	(17)

  1.1.1.2 Reduction Pair Processor

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

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

  the pairs

  b#(b(b(a(a(a(b(x1)))))))	→	b#(b(x1))	(18)
  b#(b(b(a(a(a(b(x1)))))))	→	b#(b(b(x1)))	(17)

  could be deleted.

  1.1.1.2.1 P is empty

  There are no pairs anymore.