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 AProVE @ termCOMP 2023)

1 String Reversal

1.1 Dependency Pair Transformation

1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  b#(a(d(x1)))	→	c#(d(d(b(x1))))	(15)
  c#(a(x1))	→	b#(b(x1))	(13)
  b#(a(d(x1)))	→	b#(x1)	(18)
  c#(a(x1))	→	b#(x1)	(14)

  1.1.1.1 Reduction Pair Processor

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

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

  the pair

  c#(a(x1))

  →

  b#(b(x1))

  (13)

  could be deleted.

  1.1.1.1.1 Reduction Pair Processor

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

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

  the pair

  c#(a(x1))

  →

  b#(x1)

  (14)

  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

    b#(a(d(x1)))

    →

    b#(x1)

    (18)

    1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

    Using the linear polynomial interpretation over the naturals

    [a(x1)]	=	1 · x1
    [d(x1)]	=	1 · x1
    [b#(x1)]	=	1 · x1

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

    1.1.1.1.1.1.1.1 Size-Change Termination

    Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

    b#(a(d(x1)))	→	b#(x1)	(18)
    1	>	1

    As there is no critical graph in the transitive closure, there are no infinite chains.

• The 2^nd component contains the pair

  d#(x1)	→	a#(x1)	(19)
  a#(b(x1))	→	d#(a(a(x1)))	(10)
  a#(b(x1))	→	a#(a(x1))	(11)
  a#(b(x1))	→	a#(x1)	(12)

  1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [d(x1)]	=	1 · x1
  [a(x1)]	=	1 · x1
  [b(x1)]	=	1 · x1
  [c(x1)]	=	1 · x1
  [a#(x1)]	=	1 · x1
  [d#(x1)]	=	1 · x1

  together with the usable rules

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

  (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

  1.1.1.2.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

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

  the pairs

  d#(x1)	→	a#(x1)	(19)
  a#(b(x1))	→	d#(a(a(x1)))	(10)
  a#(b(x1))	→	a#(a(x1))	(11)
  a#(b(x1))	→	a#(x1)	(12)

  could be deleted.

  1.1.1.2.1.1 P is empty

  There are no pairs anymore.