Certification Problem

Input (TPDB SRS_Standard/Secret_06_SRS/aprove03)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by matchbox @ termCOMP 2023)

1 Rule Removal

1.1 String Reversal

1.1.1 Dependency Pair Transformation

1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  0#(p(x1))

  →

  0#(x1)

  (18)

  1.1.1.1.1.1 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)

  (18)

  and no rules could be deleted.

  1.1.1.1.1.1.1 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(s(p(p(s(s(p(s(p(x1))))))))))))))	(25)
  s#(sixtimes(x1))	→	s#(s(s(p(p(p(s(p(sixtimes(s(s(s(p(s(p(p(s(s(s(s(s(s(s(p(p(x1)))))))))))))))))))))))))	(37)
  s#(sixtimes(x1))	→	s#(s(p(p(p(s(p(sixtimes(s(s(s(p(s(p(p(s(s(s(s(s(s(s(p(p(x1))))))))))))))))))))))))	(40)
  s#(half(x1))	→	s#(s(p(p(half(s(s(p(p(s(s(p(p(s(p(x1)))))))))))))))	(48)

  1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

  [s(x1)]	=	0 0 0 0 0 0 -3 0 0 · x1 + -∞ -∞ -∞
  [p(x1)]	=	0 0 0 -3 -3 0 -3 -3 -3 · x1 + -∞ -∞ -∞
  [sixtimes(x1)]	=	9 9 9 9 9 9 9 9 9 · x1 + -∞ -∞ -∞
  [half(x1)]	=	9 9 9 9 9 9 9 9 9 · x1 + -∞ -∞ -∞
  [s#(x1)]	=	12 13 15 12 13 15 12 13 15 · x1 + -∞ -∞ -∞

  together with the usable rules

  s(half(x1))	→	s(p(s(s(p(p(half(s(s(p(p(s(s(p(p(s(p(x1)))))))))))))))))	(12)
  s(s(half(x1)))	→	s(p(s(s(p(p(half(s(s(p(p(s(s(p(s(p(x1))))))))))))))))	(13)
  s(sixtimes(x1))	→	s(s(s(p(p(p(s(p(sixtimes(s(s(s(p(s(p(p(s(s(s(s(s(s(s(p(p(x1)))))))))))))))))))))))))	(14)
  s(p(p(x1)))	→	p(x1)	(15)
  s(p(x1))	→	x1	(16)

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

  s#(s(half(x1)))	→	s#(s(p(p(half(s(s(p(p(s(s(p(s(p(x1))))))))))))))	(25)
  s#(sixtimes(x1))	→	s#(s(p(p(p(s(p(sixtimes(s(s(s(p(s(p(p(s(s(s(s(s(s(s(p(p(x1))))))))))))))))))))))))	(40)
  s#(half(x1))	→	s#(s(p(p(half(s(s(p(p(s(s(p(p(s(p(x1)))))))))))))))	(48)

  could be deleted.

  1.1.1.1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    s#(sixtimes(x1))

    →

    s#(s(s(p(p(p(s(p(sixtimes(s(s(s(p(s(p(p(s(s(s(s(s(s(s(p(p(x1)))))))))))))))))))))))))

    (37)

    1.1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

    Using the matrix interpretations of dimension 4 with strict dimension 1 over the arctic semiring over the naturals

    [s(x1)]	=	0 0 0 0 0 0 0 0 -4 0 0 0 -4 -4 0 0 · x1 + -∞ -∞ -∞ -∞
    [p(x1)]	=	0 0 0 0 -4 -4 0 0 -4 -4 -4 0 -4 -4 -4 -4 · x1 + -∞ -∞ -∞ -∞
    [sixtimes(x1)]	=	8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 · x1 + -∞ -∞ -∞ -∞
    [half(x1)]	=	28 28 28 28 28 28 28 28 28 28 28 28 24 24 28 28 · x1 + -∞ -∞ -∞ -∞
    [s#(x1)]	=	1 1 1 3 1 1 1 3 1 1 1 3 1 1 1 3 · x1 + -∞ -∞ -∞ -∞

    together with the usable rules

    s(half(x1))	→	s(p(s(s(p(p(half(s(s(p(p(s(s(p(p(s(p(x1)))))))))))))))))	(12)
    s(s(half(x1)))	→	s(p(s(s(p(p(half(s(s(p(p(s(s(p(s(p(x1))))))))))))))))	(13)
    s(sixtimes(x1))	→	s(s(s(p(p(p(s(p(sixtimes(s(s(s(p(s(p(p(s(s(s(s(s(s(s(p(p(x1)))))))))))))))))))))))))	(14)
    s(p(p(x1)))	→	p(x1)	(15)
    s(p(x1))	→	x1	(16)

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

    s#(sixtimes(x1))

    →

    s#(s(s(p(p(p(s(p(sixtimes(s(s(s(p(s(p(p(s(s(s(s(s(s(s(p(p(x1)))))))))))))))))))))))))

    (37)

    could be deleted.

    1.1.1.1.1.2.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.