Certification Problem

Input (TPDB SRS_Standard/Zantema_06/while2)

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 3 components.

• The 1^st component contains the pair

  c#(d(x1))

  →

  c#(b(b(a(f(x1)))))

  (13)

  1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rules

  f(d(x1))	→	b(a(f(x1)))	(9)
  f(a(x1))	→	a(x1)	(10)
  a(b(x1))	→	b(a(x1))	(7)
  b(a(x1))	→	d(x1)	(6)

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

  1.1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rules

  a(b(x1))	→	b(a(x1))	(7)
  b(a(x1))	→	d(x1)	(6)

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

  c#(d(x1))

  →

  c#(b(b(a(f(x1)))))

  (13)

  could be deleted.

  1.1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  f#(d(x1))

  →

  f#(x1)

  (20)

  1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [d(x1)]	=	1 · x1
  [f#(x1)]	=	1 · x1

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

  1.1.1.2.1 Size-Change Termination

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

  f#(d(x1))	→	f#(x1)	(20)
  1	>	1

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

• The 3^rd component contains the pair

  a#(b(x1))

  →

  a#(x1)

  (12)

  1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

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

  1.1.1.3.1 Size-Change Termination

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

  a#(b(x1))	→	a#(x1)	(12)
  1	>	1

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