Certification Problem

Input (TPDB SRS_Standard/Waldmann_06_SRS/uni-5)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by matchbox @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

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

  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

  [c(x1)]	=	0 1 -∞ -∞ -∞ -∞ 1 0 -∞ · x1 + 9 -∞ -∞
  [b(x1)]	=	-∞ 0 -∞ 0 1 -∞ 6 -∞ 1 · x1 + 0 9 16
  [a(x1)]	=	-∞ 0 0 -∞ 8 -∞ 0 1 -∞ · x1 + 7 -∞ -∞
  [c#(x1)]	=	6 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + -∞ -∞ -∞
  [b#(x1)]	=	0 7 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + -∞ -∞ -∞

  together with the usable rules

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

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

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

  could be deleted.

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

  together with the usable rules

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

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

  c#(c(x1))

  →

  b#(x1)

  (5)

  and no rules could be deleted.

  1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.