Certification Problem

Input (TPDB SRS_Standard/Secret_07_SRS/x07)

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

  e#(e(e(x1)))	→	d#(e(x1))	(10)
  d#(x1)	→	e#(x1)	(11)
  d#(x1)	→	b#(e(x1))	(12)
  b#(e(b(x1)))	→	e#(d(x1))	(17)
  b#(e(b(x1)))	→	d#(x1)	(18)
  d#(c(x1))	→	d#(a(x1))	(13)
  d#(c(x1))	→	a#(x1)	(15)
  a#(x1)	→	e#(b(x1))	(23)
  a#(x1)	→	b#(x1)	(24)
  b#(d(x1))	→	e#(b(x1))	(19)
  b#(d(x1))	→	b#(x1)	(20)
  b#(b(b(x1)))	→	e#(x1)	(21)
  b#(b(b(x1)))	→	e#(e(x1))	(22)
  a#(a(b(x1)))	→	d#(x1)	(25)

  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

  [e(x1)]	=	0 0 -2 -2 · x1 + -∞ -∞
  [d(x1)]	=	0 0 -2 -2 · x1 + -∞ -∞
  [c(x1)]	=	2 4 0 2 · x1 + -∞ -∞
  [b(x1)]	=	0 0 -2 -2 · x1 + -∞ -∞
  [a(x1)]	=	0 2 0 0 · x1 + -∞ -∞
  [e#(x1)]	=	1 1 1 1 · x1 + -∞ -∞
  [d#(x1)]	=	1 1 1 1 · x1 + -∞ -∞
  [b#(x1)]	=	1 1 1 1 · x1 + -∞ -∞
  [a#(x1)]	=	1 2 1 2 · x1 + -∞ -∞

  together with the usable rules

  a(a(b(x1)))	→	c(d(x1))	(1)
  b(e(b(x1)))	→	e(d(x1))	(2)
  b(d(x1))	→	e(b(x1))	(3)
  b(b(b(x1)))	→	e(e(x1))	(4)
  e(e(e(x1)))	→	d(e(x1))	(5)
  d(x1)	→	b(e(x1))	(6)
  c(d(a(x1)))	→	c(x1)	(7)
  d(c(x1))	→	c(d(a(x1)))	(8)
  a(x1)	→	e(b(x1))	(9)

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

  d#(c(x1))	→	d#(a(x1))	(13)
  d#(c(x1))	→	a#(x1)	(15)
  a#(a(b(x1)))	→	d#(x1)	(25)

  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

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

  together with the usable rules

  a(a(b(x1)))	→	c(d(x1))	(1)
  b(e(b(x1)))	→	e(d(x1))	(2)
  b(d(x1))	→	e(b(x1))	(3)
  b(b(b(x1)))	→	e(e(x1))	(4)
  e(e(e(x1)))	→	d(e(x1))	(5)
  d(x1)	→	b(e(x1))	(6)
  c(d(a(x1)))	→	c(x1)	(7)
  d(c(x1))	→	c(d(a(x1)))	(8)
  a(x1)	→	e(b(x1))	(9)

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

  a#(x1)	→	e#(b(x1))	(23)
  a#(x1)	→	b#(x1)	(24)

  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

    e#(e(e(x1)))	→	d#(e(x1))	(10)
    d#(x1)	→	e#(x1)	(11)
    d#(x1)	→	b#(e(x1))	(12)
    b#(e(b(x1)))	→	e#(d(x1))	(17)
    b#(e(b(x1)))	→	d#(x1)	(18)
    b#(d(x1))	→	e#(b(x1))	(19)
    b#(d(x1))	→	b#(x1)	(20)
    b#(b(b(x1)))	→	e#(x1)	(21)
    b#(b(b(x1)))	→	e#(e(x1))	(22)

    1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

    [e(x1)]	=	17 · x1 + 27
    [d(x1)]	=	34 · x1 + 44
    [c(x1)]	=	-∞ · x1 + 3
    [b(x1)]	=	17 · x1 + 43
    [a(x1)]	=	34 · x1 + 60
    [e#(x1)]	=	5 · x1 + 17
    [d#(x1)]	=	22 · x1 + 39
    [b#(x1)]	=	5 · x1 + 0

    together with the usable rules

    a(a(b(x1)))	→	c(d(x1))	(1)
    b(e(b(x1)))	→	e(d(x1))	(2)
    b(d(x1))	→	e(b(x1))	(3)
    b(b(b(x1)))	→	e(e(x1))	(4)
    e(e(e(x1)))	→	d(e(x1))	(5)
    d(x1)	→	b(e(x1))	(6)
    c(d(a(x1)))	→	c(x1)	(7)
    d(c(x1))	→	c(d(a(x1)))	(8)
    a(x1)	→	e(b(x1))	(9)

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

    d#(x1)	→	e#(x1)	(11)
    b#(e(b(x1)))	→	d#(x1)	(18)
    b#(d(x1))	→	e#(b(x1))	(19)
    b#(d(x1))	→	b#(x1)	(20)
    b#(b(b(x1)))	→	e#(x1)	(21)
    b#(b(b(x1)))	→	e#(e(x1))	(22)

    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

      e#(e(e(x1)))	→	d#(e(x1))	(10)
      d#(x1)	→	b#(e(x1))	(12)
      b#(e(b(x1)))	→	e#(d(x1))	(17)

      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

      [e(x1)]	=	-∞ 3 6 1 3 -∞ -∞ 1 -∞ · x1 + 0 6 -∞
      [d(x1)]	=	-∞ 7 9 2 4 7 2 4 -∞ · x1 + 6 7 7
      [c(x1)]	=	-∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 8 -∞ 10
      [b(x1)]	=	3 -∞ 6 1 1 3 -∞ 1 3 · x1 + -∞ 6 1
      [a(x1)]	=	14 7 12 16 4 16 8 4 16 · x1 + 16 16 16
      [e#(x1)]	=	5 7 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + -∞ -∞ -∞
      [d#(x1)]	=	2 8 11 -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 7 -∞ -∞
      [b#(x1)]	=	5 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1 + -∞ -∞ -∞

      together with the usable rules

      a(a(b(x1)))	→	c(d(x1))	(1)
      b(e(b(x1)))	→	e(d(x1))	(2)
      b(d(x1))	→	e(b(x1))	(3)
      b(b(b(x1)))	→	e(e(x1))	(4)
      e(e(e(x1)))	→	d(e(x1))	(5)
      d(x1)	→	b(e(x1))	(6)
      c(d(a(x1)))	→	c(x1)	(7)
      d(c(x1))	→	c(d(a(x1)))	(8)
      a(x1)	→	e(b(x1))	(9)

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

      e#(e(e(x1)))

      →

      d#(e(x1))

      (10)

      could be deleted.

      1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

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

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

      together with the usable rules

      a(a(b(x1)))	→	c(d(x1))	(1)
      b(e(b(x1)))	→	e(d(x1))	(2)
      b(d(x1))	→	e(b(x1))	(3)
      b(b(b(x1)))	→	e(e(x1))	(4)
      e(e(e(x1)))	→	d(e(x1))	(5)
      d(x1)	→	b(e(x1))	(6)
      c(d(a(x1)))	→	c(x1)	(7)
      d(c(x1))	→	c(d(a(x1)))	(8)
      a(x1)	→	e(b(x1))	(9)

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

      d#(x1)	→	b#(e(x1))	(12)
      b#(e(b(x1)))	→	e#(d(x1))	(17)

      and no rules could be deleted.

      1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

  c#(d(a(x1)))

  →

  c#(x1)

  (16)

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

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

  c#(d(a(x1)))

  →

  c#(x1)

  (16)

  and no rules could be deleted.

  1.1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.