Certification Problem

Input (TPDB SRS_Standard/Waldmann_06_SRS/uni-5)

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#(c(x1))	→	c#(b(x1))	(9)
  c#(c(x1))	→	b#(x1)	(10)
  b#(b(x1))	→	c#(c(c(c(x1))))	(4)
  b#(b(x1))	→	c#(c(c(x1)))	(5)
  b#(b(x1))	→	c#(c(x1))	(6)
  b#(b(x1))	→	c#(x1)	(7)

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

  the pairs

  b#(b(x1))	→	c#(c(c(c(x1))))	(4)
  b#(b(x1))	→	c#(c(c(x1)))	(5)
  b#(b(x1))	→	c#(c(x1))	(6)
  b#(b(x1))	→	c#(x1)	(7)

  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

    c#(c(x1))

    →

    c#(b(x1))

    (9)

    1.1.1.1.1 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
    [c(x1)]	=	0 -∞ 1 + -∞ -∞ -∞ -∞ -∞ 1 1 -∞ 0 · x1
    [b(x1)]	=	1 0 -∞ + 1 -∞ 0 1 -∞ 0 0 -∞ -∞ · x1
    [a(x1)]	=	-∞ -∞ 1 + 0 -∞ -∞ 0 0 0 0 0 -∞ · x1

    the pair

    c#(c(x1))

    →

    c#(b(x1))

    (9)

    could be deleted.

    1.1.1.1.1.1 P is empty

    There are no pairs anymore.