Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z100)

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 3 components.

• The 1^st component contains the pair

  l2#(a(a(x1)))

  →

  l2#(x1)

  (18)

  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

  [a(x1)]	=	x1 + 1
  [l2#(x1)]	=	x1 + 0

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

  l2#(a(a(x1)))

  →

  l2#(x1)

  (18)

  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

  l1#(a(x1))

  →

  l1#(x1)

  (21)

  1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules

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

  [a(x1)]	=	x1 + 1
  [l1#(x1)]	=	x1 + 0

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

  l1#(a(x1))

  →

  l1#(x1)

  (21)

  and no rules could be deleted.

  1.1.1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 3^rd component contains the pair

  b#(r1(x1))	→	b#(l1(x1))	(26)
  b#(r2(x1))	→	b#(a(l2(x1)))	(28)

  1.1.1.1.3 Reduction Pair Processor with Usable Rules

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

  [l2(x1)]	=	17 -∞ 16 1 · x1 + 17 4
  [l1(x1)]	=	17 15 15 17 · x1 + 0 17
  [r1(x1)]	=	17 15 15 17 · x1 + 30 30
  [r2(x1)]	=	17 3 3 17 · x1 + 25 25
  [b(x1)]	=	14 14 13 13 · x1 + 14 12
  [a(x1)]	=	-∞ 0 0 -∞ · x1 + 5 -∞
  [b#(x1)]	=	12 -∞ -∞ -∞ · x1 + -∞ -∞

  together with the usable rules

  a(r1(x1))	→	r1(a(a(a(x1))))	(10)
  a(r2(x1))	→	r2(a(a(a(x1))))	(11)
  l1(a(x1))	→	a(a(a(l1(x1))))	(12)
  l2(a(a(x1)))	→	a(a(l2(x1)))	(13)
  b(r1(x1))	→	b(l1(x1))	(14)
  b(r2(x1))	→	b(a(l2(x1)))	(15)
  l1(b(x1))	→	r2(b(x1))	(16)
  l2(b(x1))	→	r1(b(x1))	(17)
  a(a(x1))	→	x1	(9)

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

  b#(r2(x1))

  →

  b#(a(l2(x1)))

  (28)

  could be deleted.

  1.1.1.1.3.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    b#(r1(x1))

    →

    b#(l1(x1))

    (26)

    1.1.1.1.3.1.1 Reduction Pair Processor with Usable Rules

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

    [l2(x1)]	=	0 0 0 0 · x1 + -∞ -∞
    [l1(x1)]	=	0 0 -2 -2 · x1 + -∞ -∞
    [r1(x1)]	=	0 0 0 0 · x1 + -∞ -∞
    [r2(x1)]	=	0 0 -2 0 · x1 + -∞ -∞
    [b(x1)]	=	0 0 -2 -2 · x1 + -∞ -∞
    [a(x1)]	=	0 0 -2 0 · x1 + -∞ -∞
    [b#(x1)]	=	1 2 1 2 · x1 + -∞ -∞

    together with the usable rules

    a(r1(x1))	→	r1(a(a(a(x1))))	(10)
    a(r2(x1))	→	r2(a(a(a(x1))))	(11)
    l1(a(x1))	→	a(a(a(l1(x1))))	(12)
    l2(a(a(x1)))	→	a(a(l2(x1)))	(13)
    b(r1(x1))	→	b(l1(x1))	(14)
    b(r2(x1))	→	b(a(l2(x1)))	(15)
    l1(b(x1))	→	r2(b(x1))	(16)
    l2(b(x1))	→	r1(b(x1))	(17)
    a(a(x1))	→	x1	(9)

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

    b#(r1(x1))

    →

    b#(l1(x1))

    (26)

    could be deleted.

    1.1.1.1.3.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.