Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/27280)

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)	(35)
  1#(2(1(0(x1))))	→	1#(2(1(x1)))	(36)
  1#(2(1(0(x1))))	→	1#(1(2(1(x1))))	(37)

  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)]	=	5 7 8 5 7 8 5 7 8 · x1 + -∞ -∞ -∞

  together with the usable rules

  1(2(1(0(x1))))	→	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(1(1(2(1(x1)))))))))))))	(19)
  1(2(1(0(x1))))	→	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(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(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(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(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(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(1(1(2(1(x1))))))))))))))))))))))))))))))))))	(26)
  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)))))))))))))))))))))))))))))))))))))	(27)
  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))))))))))))))))))))))))))))))))))))))))	(28)
  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)))))))))))))))))))))))))))))))))))))))))))	(29)
  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))))))))))))))))))))))))))))))))))))))))))))))	(30)
  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(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))	(31)
  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(2(1(0(2(1(0(1(1(2(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(32)
  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(2(1(0(2(1(0(2(1(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(33)
  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(2(1(0(2(1(0(2(1(0(...display limit reached...)))))))))))))))))))))))))))))))))))))))))))))))))))	(34)

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

  1#(2(1(0(x1))))	→	1#(2(1(x1)))	(36)
  1#(2(1(0(x1))))	→	1#(1(2(1(x1))))	(37)

  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)

    (35)

    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)

    (35)

    and no rules could be deleted.

    1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.