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 matchbox @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  c#(b(x1))	→	c#(c(a(x1)))	(4)
  c#(b(x1))	→	c#(a(x1))	(5)

  1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 2 with strict dimension 1 over the arctic semiring over the naturals

  [c(x1)]	=	2 4 0 2 · x1 + -∞ -∞
  [b(x1)]	=	0 2 0 0 · x1 + -∞ -∞
  [a(x1)]	=	0 0 -2 -2 · x1 + -∞ -∞
  [c#(x1)]	=	5 7 5 7 · x1 + -∞ -∞

  together with the usable rules

  a(b(x1))	→	x1	(1)
  a(c(x1))	→	b(b(x1))	(2)
  c(b(x1))	→	a(c(c(a(x1))))	(3)

  (w.r.t. the implicit argument filter of the reduction pair), the pair

  c#(b(x1))

  →

  c#(a(x1))

  (5)

  could be deleted.

  1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    c#(b(x1))

    →

    c#(c(a(x1)))

    (4)

    1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

    [c(x1)]	=	0 1 -∞ -∞ 0 -∞ -∞ 1 0 · x1 + 2 -∞ 1
    [b(x1)]	=	-∞ 0 -∞ 1 -∞ 0 0 0 -∞ · x1 + 0 2 -∞
    [a(x1)]	=	1 -∞ 0 0 -∞ -∞ 1 0 0 · x1 + 0 0 -∞
    [c#(x1)]	=	6 7 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + -∞ -∞ -∞

    together with the usable rules

    a(b(x1))	→	x1	(1)
    a(c(x1))	→	b(b(x1))	(2)
    c(b(x1))	→	a(c(c(a(x1))))	(3)

    (w.r.t. the implicit argument filter of the reduction pair), the pair

    c#(b(x1))

    →

    c#(c(a(x1)))

    (4)

    could be deleted.

    1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.