Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-547)

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

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

  1.1.1 Reduction Pair Processor

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

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

  the pair

  c#(b(x1))

  →

  c#(a(x1))

  (6)

  could be deleted.

  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

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

  the pair

  c#(b(x1))

  →

  c#(c(a(x1)))

  (5)

  could be deleted.

  1.1.1.1.1 P is empty

  There are no pairs anymore.