Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/aprove5)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by matchbox @ termCOMP 2023)

1 Rule Removal

1.1 Dependency Pair Transformation

1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

• The 1^st component contains the pair

  e#(e(f(f(x1))))

  →

  e#(e(x1))

  (11)

  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 naturals

  [f(x1)]	=	1 · x1 + 1
  [e(x1)]	=	8 · x1 + 3
  [e#(x1)]	=	1 · x1 + 2

  together with the usable rule

  e(e(f(f(x1))))

  →

  f(f(f(e(e(x1)))))

  (7)

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

  e#(e(f(f(x1))))

  →

  e#(e(x1))

  (11)

  and the rule

  e(e(f(f(x1))))

  →

  f(f(f(e(e(x1)))))

  (7)

  could be deleted.

  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))	→	b#(x1)	(15)
  b#(c(x1))	→	b#(b(x1))	(16)

  1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules

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

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

  together with the usable rules

  b(c(x1))	→	c(b(b(x1)))	(2)
  d(a(x1))	→	d(d(c(x1)))	(5)
  b(d(x1))	→	d(d(x1))	(9)

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

  b#(c(x1))	→	b#(x1)	(15)
  b#(c(x1))	→	b#(b(x1))	(16)

  and the rule

  d(a(x1))

  →

  d(d(c(x1)))

  (5)

  could be deleted.

  1.1.1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 3^rd component contains the pair

  a#(d(d(c(x1))))	→	a#(d(x1))	(18)
  a#(d(d(c(x1))))	→	a#(a(d(x1)))	(19)
  a#(d(d(c(x1))))	→	a#(a(a(d(x1))))	(20)
  a#(c(x1))	→	a#(b(x1))	(22)
  a#(b(x1))	→	a#(x1)	(24)
  a#(a(x1))	→	a#(d(d(d(x1))))	(28)

  1.1.1.1.3 Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 4 with strict dimension 1 over the arctic semiring over the naturals

  [d(x1)]	=	0 0 0 0 -4 -4 0 0 -4 -4 -4 0 -4 -4 -4 -4 · x1 + -∞ -∞ -∞ -∞
  [c(x1)]	=	0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 · x1 + -∞ -∞ -∞ -∞
  [b(x1)]	=	0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 · x1 + -∞ -∞ -∞ -∞
  [a(x1)]	=	0 0 0 0 0 0 0 0 0 0 0 0 -4 -4 -4 0 · x1 + -∞ -∞ -∞ -∞
  [a#(x1)]	=	2 5 5 6 2 5 5 6 2 5 5 6 2 5 5 6 · x1 + -∞ -∞ -∞ -∞

  together with the usable rules

  a(b(x1))	→	b(c(a(x1)))	(1)
  b(c(x1))	→	c(b(b(x1)))	(2)
  a(c(x1))	→	c(a(b(x1)))	(3)
  a(a(x1))	→	a(d(d(d(x1))))	(4)
  d(a(x1))	→	d(d(c(x1)))	(5)
  a(d(d(c(x1))))	→	a(a(a(d(x1))))	(6)
  b(d(x1))	→	d(d(x1))	(9)

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

  a#(a(x1))

  →

  a#(d(d(d(x1))))

  (28)

  could be deleted.

  1.1.1.1.3.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    a#(d(d(c(x1))))	→	a#(d(x1))	(18)
    a#(d(d(c(x1))))	→	a#(a(d(x1)))	(19)
    a#(d(d(c(x1))))	→	a#(a(a(d(x1))))	(20)
    a#(c(x1))	→	a#(b(x1))	(22)
    a#(b(x1))	→	a#(x1)	(24)

    1.1.1.1.3.1.1 Reduction Pair Processor with Usable Rules

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

    [d(x1)]	=	0 0 0 0 · x1 + -∞ -∞
    [c(x1)]	=	0 0 -2 0 · x1 + -∞ -∞
    [b(x1)]	=	0 0 -2 0 · x1 + -∞ -∞
    [a(x1)]	=	0 0 -2 -2 · x1 + -∞ -∞
    [a#(x1)]	=	6 8 6 8 · x1 + -∞ -∞

    together with the usable rules

    a(b(x1))	→	b(c(a(x1)))	(1)
    b(c(x1))	→	c(b(b(x1)))	(2)
    a(c(x1))	→	c(a(b(x1)))	(3)
    a(a(x1))	→	a(d(d(d(x1))))	(4)
    d(a(x1))	→	d(d(c(x1)))	(5)
    a(d(d(c(x1))))	→	a(a(a(d(x1))))	(6)
    b(d(x1))	→	d(d(x1))	(9)

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

    a#(d(d(c(x1))))	→	a#(a(d(x1)))	(19)
    a#(d(d(c(x1))))	→	a#(a(a(d(x1))))	(20)

    could be deleted.

    1.1.1.1.3.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 1 component.

    • The 1^st component contains the pair

      a#(d(d(c(x1))))	→	a#(d(x1))	(18)
      a#(c(x1))	→	a#(b(x1))	(22)
      a#(b(x1))	→	a#(x1)	(24)

      1.1.1.1.3.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

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

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

      together with the usable rules

      b(c(x1))	→	c(b(b(x1)))	(2)
      d(a(x1))	→	d(d(c(x1)))	(5)
      b(d(x1))	→	d(d(x1))	(9)

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

      a#(d(d(c(x1))))	→	a#(d(x1))	(18)
      a#(c(x1))	→	a#(b(x1))	(22)

      and the rule

      d(a(x1))

      →

      d(d(c(x1)))

      (5)

      could be deleted.

      1.1.1.1.3.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 1 component.

      • The 1^st component contains the pair

        a#(b(x1))

        →

        a#(x1)

        (24)

        1.1.1.1.3.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

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

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

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

        a#(b(x1))

        →

        a#(x1)

        (24)

        and no rules could be deleted.

        1.1.1.1.3.1.1.1.1.1.1.1 Dependency Graph Processor

        The dependency pairs are split into 0 components.