Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/jambox5)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by matchbox @ termCOMP 2023)

1 String Reversal

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 2 components.

• The 1^st component contains the pair

  d#(x1)	→	a#(x1)	(10)
  a#(b(x1))	→	d#(a(a(x1)))	(17)
  a#(b(x1))	→	a#(x1)	(18)
  a#(b(x1))	→	a#(a(x1))	(19)

  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

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

  together with the usable rules

  a(b(x1))	→	d(a(a(x1)))	(6)
  d(x1)	→	a(x1)	(4)
  a(c(a(b(x1))))	→	x1	(9)

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

  d#(x1)	→	a#(x1)	(10)
  a#(b(x1))	→	d#(a(a(x1)))	(17)
  a#(b(x1))	→	a#(x1)	(18)
  a#(b(x1))	→	a#(a(x1))	(19)

  and the rules

  a(b(x1))	→	d(a(a(x1)))	(6)
  d(x1)	→	a(x1)	(4)
  a(c(a(b(x1))))	→	x1	(9)

  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

  c#(a(x1))	→	b#(x1)	(11)
  b#(a(d(x1)))	→	c#(d(d(b(x1))))	(15)
  c#(a(x1))	→	b#(b(x1))	(12)
  b#(a(d(x1)))	→	b#(x1)	(16)

  1.1.1.1.2 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 -2 · x1 + -∞ -∞
  [b(x1)]	=	0 0 -2 -2 · x1 + -∞ -∞
  [a(x1)]	=	0 0 0 0 · x1 + -∞ -∞
  [c#(x1)]	=	8 10 8 10 · x1 + -∞ -∞
  [b#(x1)]	=	8 10 8 10 · x1 + -∞ -∞

  together with the usable rules

  a(b(x1))	→	d(a(a(x1)))	(6)
  c(a(x1))	→	b(b(x1))	(7)
  b(a(d(x1)))	→	c(d(d(b(x1))))	(8)
  d(x1)	→	a(x1)	(4)
  a(c(a(b(x1))))	→	x1	(9)

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

  c#(a(x1))

  →

  b#(b(x1))

  (12)

  could be deleted.

  1.1.1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    c#(a(x1))	→	b#(x1)	(11)
    b#(a(d(x1)))	→	c#(d(d(b(x1))))	(15)
    b#(a(d(x1)))	→	b#(x1)	(16)

    1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

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

    [d(x1)]	=	-∞ 0 -∞ -∞ -∞ 0 0 2 -∞ · x1 + 0 -∞ 4
    [c(x1)]	=	6 2 4 4 0 2 2 -∞ 0 · x1 + 7 5 -∞
    [b(x1)]	=	2 4 0 0 2 -∞ -∞ 0 -∞ · x1 + 5 0 1
    [a(x1)]	=	-∞ 0 -∞ -∞ -∞ 0 0 -∞ -∞ · x1 + 0 -∞ -∞
    [c#(x1)]	=	9 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + -∞ -∞ -∞
    [b#(x1)]	=	-∞ 7 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 4 -∞ -∞

    together with the usable rules

    a(b(x1))	→	d(a(a(x1)))	(6)
    c(a(x1))	→	b(b(x1))	(7)
    b(a(d(x1)))	→	c(d(d(b(x1))))	(8)
    d(x1)	→	a(x1)	(4)
    a(c(a(b(x1))))	→	x1	(9)

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

    c#(a(x1))	→	b#(x1)	(11)
    b#(a(d(x1)))	→	b#(x1)	(16)

    could be deleted.

    1.1.1.1.2.1.1.1 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 + 0
    [b(x1)]	=	x1 + 0
    [a(x1)]	=	x1 + 0
    [c#(x1)]	=	x1 + 0
    [b#(x1)]	=	x1 + 1

    together with the usable rules

    a(b(x1))	→	d(a(a(x1)))	(6)
    c(a(x1))	→	b(b(x1))	(7)
    b(a(d(x1)))	→	c(d(d(b(x1))))	(8)
    d(x1)	→	a(x1)	(4)
    a(c(a(b(x1))))	→	x1	(9)

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

    b#(a(d(x1)))

    →

    c#(d(d(b(x1))))

    (15)

    and no rules could be deleted.

    1.1.1.1.2.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.