Certification Problem

Input (TPDB SRS_Standard/Secret_06_SRS/aprove01)

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

  0#(p(x1))

  →

  0#(x1)

  (27)

  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

  [p(x1)]	=	x1 + 1
  [0#(x1)]	=	x1 + 0

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

  0#(p(x1))

  →

  0#(x1)

  (27)

  and no rules 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

  s#(s(half(x1)))	→	s#(s(p(p(half(s(x1))))))	(50)
  s#(f(x1))	→	s#(s(p(g(s(p(x1))))))	(46)

  1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

  [j(x1)]	=	0 0 0 0 0 0 -3 -3 -3 · x1 + -∞ -∞ -∞
  [g(x1)]	=	0 0 0 0 0 0 -3 -3 -3 · x1 + -∞ -∞ -∞
  [f(x1)]	=	0 0 0 0 0 0 0 0 0 · x1 + -∞ -∞ -∞
  [s(x1)]	=	0 0 0 0 0 0 -3 0 0 · x1 + -∞ -∞ -∞
  [p(x1)]	=	0 0 0 -3 -3 0 -3 -3 -3 · x1 + -∞ -∞ -∞
  [half(x1)]	=	0 0 0 0 0 0 -3 0 0 · x1 + -∞ -∞ -∞
  [s#(x1)]	=	2 2 3 2 2 3 2 2 3 · x1 + -∞ -∞ -∞

  together with the usable rules

  s(p(x1))	→	x1	(11)
  s(p(p(x1)))	→	p(x1)	(12)
  s(f(x1))	→	s(s(p(g(s(p(x1))))))	(13)
  s(g(x1))	→	s(p(s(p(s(j(s(s(s(p(p(x1)))))))))))	(14)
  s(j(x1))	→	s(p(p(s(p(f(s(p(s(s(p(x1)))))))))))	(15)
  s(s(half(x1)))	→	s(s(p(p(half(s(x1))))))	(17)

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

  s#(s(half(x1)))	→	s#(s(p(p(half(s(x1))))))	(50)
  s#(f(x1))	→	s#(s(p(g(s(p(x1))))))	(46)

  could be deleted.

  1.1.1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.