Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-495)

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 2 components.

• 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 3 with strict dimension 1 over the arctic semiring over the naturals

  [c(x1)]	=	0 0 0 -3 -3 0 -3 -3 -3 · x1 + -∞ -∞ -∞
  [b(x1)]	=	3 3 6 0 3 3 0 3 3 · x1 + -∞ -∞ -∞
  [a(x1)]	=	0 0 3 0 0 3 0 0 3 · x1 + -∞ -∞ -∞
  [c#(x1)]	=	15 16 18 15 16 18 15 16 18 · x1 + -∞ -∞ -∞

  together with the usable rules

  a(a(x1))	→	b(x1)	(1)
  b(c(x1))	→	a(x1)	(2)
  c(b(x1))	→	b(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 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    c#(b(x1))

    →

    c#(a(x1))

    (5)

    1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

    Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

    [c(x1)]	=	x1 + 1
    [b(x1)]	=	x1 + 3
    [a(x1)]	=	x1 + 2
    [c#(x1)]	=	x1 + 0

    together with the usable rules

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

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

    c#(b(x1))

    →

    c#(a(x1))

    (5)

    and the rules

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

    could be deleted.

    1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

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

  1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

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

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

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

  and no rules could be deleted.

  1.1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.