Certification Problem

Input (TPDB SRS_Standard/Secret_06_SRS/aprove08)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by matchbox @ termCOMP 2023)

1 Rule Removal

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 2 components.

• The 1^st component contains the pair

  j#(s(x1))	→	i#(p(s(p(s(x1)))))	(8)
  i#(s(x1))	→	j#(p(s(p(s(p(p(p(p(s(s(s(s(x1)))))))))))))	(13)

  1.1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 4 with strict dimension 1 over the arctic semiring over the naturals

  [0(x1)]	=	-∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 0 1 -∞ 1
  [s(x1)]	=	1 3 4 4 0 -∞ 0 -∞ -∞ 0 1 -∞ -∞ -∞ 0 0 · x1 + 4 3 -∞ 0
  [p(x1)]	=	-∞ 0 -∞ 0 -∞ -∞ 0 0 -∞ -∞ -∞ 0 -∞ -∞ -∞ 0 · x1 + 0 0 -∞ -∞
  [j#(x1)]	=	1 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 1 -∞ -∞ -∞
  [i#(x1)]	=	1 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 2 -∞ -∞ -∞

  together with the usable rules

  p(p(s(x1)))	→	p(x1)	(5)
  p(s(x1))	→	x1	(6)
  p(0(x1))	→	0(s(s(s(s(s(s(s(s(x1)))))))))	(7)

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

  j#(s(x1))

  →

  i#(p(s(p(s(x1)))))

  (8)

  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

  [0(x1)]	=	x1 + 0
  [s(x1)]	=	x1 + 0
  [p(x1)]	=	x1 + 1
  [j#(x1)]	=	x1 + 0
  [i#(x1)]	=	x1 + 7

  together with the usable rules

  p(p(s(x1)))	→	p(x1)	(5)
  p(s(x1))	→	x1	(6)
  p(0(x1))	→	0(s(s(s(s(s(s(s(s(x1)))))))))	(7)

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

  i#(s(x1))

  →

  j#(p(s(p(s(p(p(p(p(s(s(s(s(x1)))))))))))))

  (13)

  and the rules

  p(p(s(x1)))	→	p(x1)	(5)
  p(s(x1))	→	x1	(6)
  p(0(x1))	→	0(s(s(s(s(s(s(s(s(x1)))))))))	(7)

  could be deleted.

  1.1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

  p#(p(s(x1)))

  →

  p#(x1)

  (22)

  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

  [s(x1)]	=	x1 + 1
  [p(x1)]	=	x1 + 1
  [p#(x1)]	=	x1 + 0

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

  p#(p(s(x1)))

  →

  p#(x1)

  (22)

  and no rules could be deleted.

  1.1.1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.