Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/jambox1)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

1.1 Closure Under Flat Contexts

{c(☐), a(☐), b(☐)}

1.1.1 Semantic Labeling

Root-labeling is applied.

1.1.1.1 Rule Removal

1.1.1.1.1 Dependency Pair Transformation

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  cb#(bc(ca(ab(x1))))	→	cb#(x1)	(37)
  cb#(bc(ca(ab(x1))))	→	cb#(bc(cb(x1)))	(35)
  cb#(bc(ca(aa(x1))))	→	cb#(bc(ca(x1)))	(39)

  1.1.1.1.1.1.1 Reduction Pair Processor

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

  [cb#(x1)]	=	0 -∞ -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [bc(x1)]	=	0 0 0 + -∞ -∞ 0 0 0 0 -∞ 0 0 · x1
  [ca(x1)]	=	0 1 0 + -∞ 0 0 0 -∞ -∞ -∞ 0 0 · x1
  [ab(x1)]	=	0 0 0 + 0 0 1 0 -∞ 0 -∞ -∞ -∞ · x1
  [cb(x1)]	=	1 0 0 + 1 -∞ 0 -∞ -∞ 0 -∞ -∞ 0 · x1
  [aa(x1)]	=	0 1 0 + -∞ 0 0 0 0 1 0 1 0 · x1
  [cc(x1)]	=	0 1 0 + 0 1 1 -∞ 0 0 0 0 0 · x1
  [ac(x1)]	=	-∞ 0 -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [ba(x1)]	=	0 0 -∞ + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [bb(x1)]	=	0 0 -∞ + -∞ -∞ -∞ 0 -∞ -∞ -∞ -∞ -∞ · x1

  the pair

  cb#(bc(ca(aa(x1))))

  →

  cb#(bc(ca(x1)))

  (39)

  could be deleted.

  1.1.1.1.1.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [cb#(x1)]	=	1 · x1
  [bc(x1)]	=	1 + 1 · x1
  [ca(x1)]	=	1 · x1
  [ab(x1)]	=	1 · x1
  [cb(x1)]	=	0
  [cc(x1)]	=	1 · x1
  [aa(x1)]	=	0
  [ac(x1)]	=	1 · x1
  [ba(x1)]	=	0
  [bb(x1)]	=	1 · x1

  the pair

  cb#(bc(ca(ab(x1))))

  →

  cb#(x1)

  (37)

  could be deleted.

  1.1.1.1.1.1.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [cb#(x1)]	=	1 · x1
  [bc(x1)]	=	1 · x1
  [ca(x1)]	=	1 · x1
  [ab(x1)]	=	1
  [cb(x1)]	=	0
  [cc(x1)]	=	1 · x1
  [aa(x1)]	=	0
  [ac(x1)]	=	1 · x1
  [ba(x1)]	=	0
  [bb(x1)]	=	1 · x1

  the pair

  cb#(bc(ca(ab(x1))))

  →

  cb#(bc(cb(x1)))

  (35)

  could be deleted.

  1.1.1.1.1.1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  bc#(cc(x1))

  →

  bc#(x1)

  (42)

  1.1.1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [cc(x1)]	=	1 · x1
  [bc#(x1)]	=	1 · x1

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

  1.1.1.1.1.1.2.1 Size-Change Termination

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

  bc#(cc(x1))	→	bc#(x1)	(42)
  1	>	1

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