Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z023)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by matchbox @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  b#(c(a(x1)))	→	b#(x1)	(4)
  b#(c(a(x1)))	→	a#(b(x1))	(5)
  a#(a(b(x1)))	→	b#(a(b(x1)))	(10)
  b#(c(a(x1)))	→	a#(a(b(x1)))	(6)
  b#(a(x1))	→	b#(x1)	(7)
  b#(a(x1))	→	b#(b(x1))	(8)
  b#(a(x1))	→	a#(b(b(x1)))	(9)

  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

  [c(x1)]	=	0 0 0 0 · x1 + -∞ -∞
  [b(x1)]	=	0 0 -2 0 · x1 + -∞ -∞
  [a(x1)]	=	0 2 -2 0 · x1 + -∞ -∞
  [b#(x1)]	=	9 10 9 10 · x1 + -∞ -∞
  [a#(x1)]	=	9 9 9 9 · x1 + -∞ -∞

  together with the usable rules

  a(a(b(x1)))	→	b(a(b(x1)))	(1)
  b(a(x1))	→	a(b(b(x1)))	(2)
  b(c(a(x1)))	→	c(c(a(a(b(x1)))))	(3)

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

  b#(c(a(x1)))	→	b#(x1)	(4)
  b#(c(a(x1)))	→	a#(b(x1))	(5)
  b#(c(a(x1)))	→	a#(a(b(x1)))	(6)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    a#(a(b(x1)))	→	b#(a(b(x1)))	(10)
    b#(a(x1))	→	b#(x1)	(7)
    b#(a(x1))	→	b#(b(x1))	(8)
    b#(a(x1))	→	a#(b(b(x1)))	(9)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the matrix interpretations of dimension 1 with strict dimension 1 over the naturals

    [c(x1)]	=	0 · x1 + 1
    [b(x1)]	=	1 · x1 + 0
    [a(x1)]	=	1 · x1 + 1
    [b#(x1)]	=	2 · x1 + 0
    [a#(x1)]	=	2 · x1 + 0

    together with the usable rules

    a(a(b(x1)))	→	b(a(b(x1)))	(1)
    b(a(x1))	→	a(b(b(x1)))	(2)
    b(c(a(x1)))	→	c(c(a(a(b(x1)))))	(3)

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

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

    could be deleted.

    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

    [c(x1)]	=	x1 + 0
    [b(x1)]	=	x1 + 0
    [a(x1)]	=	x1 + 0
    [b#(x1)]	=	x1 + 0
    [a#(x1)]	=	x1 + 1

    together with the usable rules

    a(a(b(x1)))	→	b(a(b(x1)))	(1)
    b(a(x1))	→	a(b(b(x1)))	(2)
    b(c(a(x1)))	→	c(c(a(a(b(x1)))))	(3)

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

    a#(a(b(x1)))

    →

    b#(a(b(x1)))

    (10)

    and no rules could be deleted.

    1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.