Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-82)

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(x1))	→	c#(x1)	(8)
  c#(b(x1))	→	a#(c(x1))	(9)
  a#(a(x1))	→	a#(b(c(x1)))	(6)
  c#(b(x1))	→	c#(x1)	(10)

  1.1.1 Reduction Pair Processor

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

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

  the pair

  a#(a(x1))

  →

  c#(x1)

  (8)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 2 components.

  • The 1^st component contains the pair

    c#(b(x1))

    →

    c#(x1)

    (10)

    1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

    Using the linear polynomial interpretation over the naturals

    [b(x1)]	=	1 · x1
    [c#(x1)]	=	1 · x1

    having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

    1.1.1.1.1.1 Size-Change Termination

    Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

    c#(b(x1))	→	c#(x1)	(10)
    1	>	1

    As there is no critical graph in the transitive closure, there are no infinite chains.

  • The 2^nd component contains the pair

    a#(a(x1))

    →

    a#(b(c(x1)))

    (6)

    1.1.1.1.2 Reduction Pair Processor

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

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

    the pair

    a#(a(x1))

    →

    a#(b(c(x1)))

    (6)

    could be deleted.

    1.1.1.1.2.1 P is empty

    There are no pairs anymore.