Certification Problem

Input (TPDB SRS_Standard/Secret_06_SRS/multum4)

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 1 component.

• The 1^st component contains the pair

  b#(a(b(b(b(b(b(b(a(b(b(a(b(b(x1))))))))))))))	→	b#(b(b(b(b(b(a(b(b(a(b(b(a(b(b(b(b(b(x1))))))))))))))))))	(6)
  b#(a(b(b(b(b(b(b(a(b(b(a(b(b(x1))))))))))))))	→	b#(b(b(b(b(a(b(b(a(b(b(a(b(b(b(b(b(x1)))))))))))))))))	(7)
  b#(a(b(b(b(b(b(b(a(b(b(a(b(b(x1))))))))))))))	→	b#(b(b(b(a(b(b(a(b(b(a(b(b(b(b(b(x1))))))))))))))))	(8)
  b#(a(b(b(b(b(b(b(a(b(b(a(b(b(x1))))))))))))))	→	b#(b(b(a(b(b(a(b(b(a(b(b(b(b(b(x1)))))))))))))))	(9)
  b#(a(b(b(b(b(b(b(a(b(b(a(b(b(x1))))))))))))))	→	b#(b(a(b(b(a(b(b(a(b(b(b(b(b(x1))))))))))))))	(12)
  b#(a(b(b(b(b(b(b(a(b(b(a(b(b(x1))))))))))))))	→	b#(a(b(b(a(b(b(a(b(b(b(b(b(x1)))))))))))))	(15)

  1.1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 2 with strict dimension 1 over the arctic semiring over the naturals

  [b(x1)]	=	0 0 -2 -2 · x1 + -∞ -∞
  [a(x1)]	=	0 0 0 0 · x1 + -∞ -∞
  [b#(x1)]	=	5 7 5 7 · x1 + -∞ -∞

  together with the usable rule

  b(a(b(b(b(b(b(b(a(b(b(a(b(b(x1))))))))))))))

  →

  b(b(b(b(b(b(a(b(b(a(b(b(a(b(b(b(b(b(x1))))))))))))))))))

  (2)

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

  b#(a(b(b(b(b(b(b(a(b(b(a(b(b(x1))))))))))))))	→	b#(b(b(b(b(b(a(b(b(a(b(b(a(b(b(b(b(b(x1))))))))))))))))))	(6)
  b#(a(b(b(b(b(b(b(a(b(b(a(b(b(x1))))))))))))))	→	b#(b(b(b(b(a(b(b(a(b(b(a(b(b(b(b(b(x1)))))))))))))))))	(7)
  b#(a(b(b(b(b(b(b(a(b(b(a(b(b(x1))))))))))))))	→	b#(b(b(b(a(b(b(a(b(b(a(b(b(b(b(b(x1))))))))))))))))	(8)
  b#(a(b(b(b(b(b(b(a(b(b(a(b(b(x1))))))))))))))	→	b#(b(b(a(b(b(a(b(b(a(b(b(b(b(b(x1)))))))))))))))	(9)
  b#(a(b(b(b(b(b(b(a(b(b(a(b(b(x1))))))))))))))	→	b#(b(a(b(b(a(b(b(a(b(b(b(b(b(x1))))))))))))))	(12)

  could be deleted.

  1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    b#(a(b(b(b(b(b(b(a(b(b(a(b(b(x1))))))))))))))

    →

    b#(a(b(b(a(b(b(a(b(b(b(b(b(x1)))))))))))))

    (15)

    1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

    [b(x1)]	=	-∞ -∞ 0 0 -∞ -∞ -∞ 0 -∞ · x1 + 0 -∞ -∞
    [a(x1)]	=	-∞ 5 -∞ -∞ -∞ 6 -∞ 3 -∞ · x1 + 8 8 -∞
    [b#(x1)]	=	2 -∞ 6 -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 15 -∞ -∞

    together with the usable rule

    b(a(b(b(b(b(b(b(a(b(b(a(b(b(x1))))))))))))))

    →

    b(b(b(b(b(b(a(b(b(a(b(b(a(b(b(b(b(b(x1))))))))))))))))))

    (2)

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

    b#(a(b(b(b(b(b(b(a(b(b(a(b(b(x1))))))))))))))

    →

    b#(a(b(b(a(b(b(a(b(b(b(b(b(x1)))))))))))))

    (15)

    could be deleted.

    1.1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.