Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/27131)

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 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  1#(2(1(0(x1))))	→	1#(x1)	(27)
  1#(2(1(0(x1))))	→	1#(2(1(x1)))	(28)
  1#(2(1(0(x1))))	→	1#(1(2(1(x1))))	(29)

  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

  [2(x1)]	=	0 0 0 -3 -3 0 -3 -3 -3 · x1 + -∞ -∞ -∞
  [1(x1)]	=	0 0 0 0 0 0 -3 -3 0 · x1 + -∞ -∞ -∞
  [0(x1)]	=	0 0 3 0 0 3 0 0 3 · x1 + -∞ -∞ -∞
  [1#(x1)]	=	3 4 6 3 4 6 3 4 6 · x1 + -∞ -∞ -∞

  together with the usable rules

  1(2(1(0(x1))))	→	2(1(0(2(1(0(1(1(2(1(x1))))))))))	(14)
  1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))	(15)
  1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))	(16)
  1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))	(17)
  1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))	(18)
  1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))	(19)
  1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))	(20)
  1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))	(21)
  1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))	(22)
  1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))	(23)
  1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))))))))	(24)
  1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))))))))	(25)
  1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))))))))))))))	(26)

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

  1#(2(1(0(x1))))

  →

  1#(2(1(x1)))

  (28)

  could be deleted.

  1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    1#(2(1(0(x1))))	→	1#(x1)	(27)
    1#(2(1(0(x1))))	→	1#(1(2(1(x1))))	(29)

    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

    [2(x1)]	=	0 0 0 -3 -3 0 -3 -3 -3 · x1 + -∞ -∞ -∞
    [1(x1)]	=	0 0 0 -3 0 0 -3 -3 0 · x1 + -∞ -∞ -∞
    [0(x1)]	=	0 0 3 0 0 3 0 0 0 · x1 + -∞ -∞ -∞
    [1#(x1)]	=	21 22 22 21 22 22 21 22 22 · x1 + -∞ -∞ -∞

    together with the usable rules

    1(2(1(0(x1))))	→	2(1(0(2(1(0(1(1(2(1(x1))))))))))	(14)
    1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))	(15)
    1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))	(16)
    1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))	(17)
    1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))	(18)
    1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))	(19)
    1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))	(20)
    1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))	(21)
    1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))	(22)
    1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))	(23)
    1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))))))))	(24)
    1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))))))))	(25)
    1(2(1(0(x1))))	→	2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))))))))))))))	(26)

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

    1#(2(1(0(x1))))

    →

    1#(1(2(1(x1))))

    (29)

    could be deleted.

    1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 1 component.

    • The 1^st component contains the pair

      1#(2(1(0(x1))))

      →

      1#(x1)

      (27)

      1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

      Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

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

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

      1#(2(1(0(x1))))

      →

      1#(x1)

      (27)

      and no rules could be deleted.

      1.1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.