Certification Problem

Input (TPDB SRS_Relative/Waldmann_19/random-82)

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

  b#(c(a(x1)))	→	a#(a(c(x1)))	(15)
  a#(c(c(x1)))	→	a#(a(a(x1)))	(18)
  a#(c(a(x1)))	→	b#(a(c(x1)))	(20)
  a#(b(a(x1)))	→	a#(c(b(x1)))	(24)

  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)]	=	4 4 4 4 · x1 + -∞ -∞
  [b(x1)]	=	4 4 2 2 · x1 + -∞ -∞
  [a(x1)]	=	4 4 2 2 · x1 + -∞ -∞
  [b#(x1)]	=	1 3 1 3 · x1 + -∞ -∞
  [a#(x1)]	=	3 3 3 3 · x1 + -∞ -∞

  together with the usable rules

  b(c(a(x1)))	→	a(a(c(x1)))	(1)
  c(b(b(x1)))	→	b(b(a(x1)))	(2)
  a(c(c(x1)))	→	a(a(a(x1)))	(3)
  c(c(a(x1)))	→	a(a(b(x1)))	(4)
  a(c(a(x1)))	→	b(a(c(x1)))	(5)
  a(b(a(x1)))	→	a(c(b(x1)))	(6)

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

  a#(c(a(x1)))

  →

  b#(a(c(x1)))

  (20)

  could be deleted.

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

  together with the usable rules

  b(c(a(x1)))	→	a(a(c(x1)))	(1)
  c(b(b(x1)))	→	b(b(a(x1)))	(2)
  a(c(c(x1)))	→	a(a(a(x1)))	(3)
  c(c(a(x1)))	→	a(a(b(x1)))	(4)
  a(c(a(x1)))	→	b(a(c(x1)))	(5)
  a(b(a(x1)))	→	a(c(b(x1)))	(6)

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

  b#(c(a(x1)))

  →

  a#(a(c(x1)))

  (15)

  and no rules could be deleted.

  1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    a#(c(c(x1)))	→	a#(a(a(x1)))	(18)
    a#(b(a(x1)))	→	a#(c(b(x1)))	(24)

    1.1.1.1.1.1.1 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)]	=	3 3 6 0 3 3 0 3 3 · x1 + -∞ -∞ -∞
    [b(x1)]	=	0 3 3 0 3 3 -3 0 0 · x1 + -∞ -∞ -∞
    [a(x1)]	=	0 3 3 0 3 3 0 0 0 · x1 + -∞ -∞ -∞
    [a#(x1)]	=	6 7 7 6 7 7 6 7 7 · x1 + -∞ -∞ -∞

    together with the usable rules

    b(c(a(x1)))	→	a(a(c(x1)))	(1)
    c(b(b(x1)))	→	b(b(a(x1)))	(2)
    a(c(c(x1)))	→	a(a(a(x1)))	(3)
    c(c(a(x1)))	→	a(a(b(x1)))	(4)
    a(c(a(x1)))	→	b(a(c(x1)))	(5)
    a(b(a(x1)))	→	a(c(b(x1)))	(6)

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

    a#(c(c(x1)))

    →

    a#(a(a(x1)))

    (18)

    could be deleted.

    1.1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 1 component.

    • The 1^st component contains the pair

      a#(b(a(x1)))

      →

      a#(c(b(x1)))

      (24)

      1.1.1.1.1.1.1.1.1 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)]	=	-∞ -∞ -∞ -∞ 1 -∞ -∞ 0 -∞ · x1 + 15 1 0
      [b(x1)]	=	-∞ 16 -∞ -∞ -∞ -∞ 0 -∞ -∞ · x1 + 0 -∞ -∞
      [a(x1)]	=	-∞ -∞ -∞ -∞ -∞ -∞ -∞ 0 -∞ · x1 + 1 -∞ 1
      [a#(x1)]	=	-∞ -∞ 1 -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 0 -∞ -∞

      together with the usable rules

      b(c(a(x1)))	→	a(a(c(x1)))	(1)
      c(b(b(x1)))	→	b(b(a(x1)))	(2)
      a(c(c(x1)))	→	a(a(a(x1)))	(3)
      c(c(a(x1)))	→	a(a(b(x1)))	(4)
      a(c(a(x1)))	→	b(a(c(x1)))	(5)
      a(b(a(x1)))	→	a(c(b(x1)))	(6)

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

      a#(b(a(x1)))

      →

      a#(c(b(x1)))

      (24)

      could be deleted.

      1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.