Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/aprove2)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

1.1 Rule Removal

1.1.1 Dependency Pair Transformation

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

• The 1^st component contains the pair

  s#(s(f(f(x1))))	→	s#(s(x1))	(22)
  s#(s(f(f(x1))))	→	s#(s(s(x1)))	(21)
  s#(s(f(f(x1))))	→	s#(x1)	(23)

  1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [f(x1)]	=	1 · x1
  [h(x1)]	=	1 · x1
  [g(x1)]	=	1 · x1
  [s#(x1)]	=	1 · x1

  together with the usable rules

  s(s(s(f(x1))))	→	h(s(f(h(x1))))	(10)
  s(s(f(f(x1))))	→	f(f(s(s(s(x1)))))	(12)
  h(g(x1))	→	s(f(g(x1)))	(9)
  h(f(x1))	→	h(s(f(h(x1))))	(11)
  h(x1)	→	x1	(4)

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

  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

  [s#(x1)]	=	0 -∞ -∞ + -∞ 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [s(x1)]	=	0 0 0 + 0 -∞ 0 -∞ 0 -∞ -∞ 0 -∞ · x1
  [f(x1)]	=	0 -∞ 1 + 0 0 0 -∞ 0 0 -∞ 1 -∞ · x1
  [h(x1)]	=	1 0 0 + 0 0 0 -∞ 0 -∞ -∞ 0 0 · x1
  [g(x1)]	=	-∞ -∞ -∞ + 1 1 0 0 0 -∞ 0 0 -∞ · x1

  the pairs

  s#(s(f(f(x1))))	→	s#(s(x1))	(22)
  s#(s(f(f(x1))))	→	s#(s(s(x1)))	(21)

  could be deleted.

  1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [f(x1)]	=	1 · x1
  [s#(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.

  s#(s(f(f(x1))))	→	s#(x1)	(23)
  1	>	1

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

• The 2^nd component contains the pair

  h#(f(x1))

  →

  h#(x1)

  (20)

  1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

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

  1.1.1.1.2.1 Size-Change Termination

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

  h#(f(x1))	→	h#(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)

  (24)

  1.1.1.1.3 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.

  1.1.1.1.3.1 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)	(24)
  1	>	1

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