Certification Problem

Input (TPDB SRS_Standard/Secret_06_SRS/aprove06)

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 4 components.

• The 1^st component contains the pair

  p#(p(s(x1)))

  →

  p#(x1)

  (10)

  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

  [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)

  (10)

  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

  tower#(s(x1))

  →

  tower#(p(s(p(s(x1)))))

  (17)

  1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

  [0(x1)]	=	0 0 0 0 · x1 + 2 2
  [s(x1)]	=	3 3 1 0 · x1 + 3 0
  [p(x1)]	=	0 1 2 2 · x1 + 0 2
  [tower#(x1)]	=	2 0 0 0 · x1 + 0 0

  together with the usable rules

  p(p(s(x1)))	→	p(x1)	(7)
  p(s(x1))	→	x1	(8)
  p(0(x1))	→	0(s(s(s(s(s(s(s(s(x1)))))))))	(9)

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

  tower#(s(x1))

  →

  tower#(p(s(p(s(x1)))))

  (17)

  could be deleted.

  1.1.1.1.2.1 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

  together with the usable rules

  p(p(s(x1)))	→	p(x1)	(7)
  p(s(x1))	→	x1	(8)
  p(0(x1))	→	0(s(s(s(s(s(s(s(s(x1)))))))))	(9)

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

  p(p(s(x1)))	→	p(x1)	(7)
  p(s(x1))	→	x1	(8)
  p(0(x1))	→	0(s(s(s(s(s(s(s(s(x1)))))))))	(9)

  could be deleted.

  1.1.1.1.2.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 3^rd component contains the pair

  twoto#(s(x1))

  →

  twoto#(p(s(p(s(x1)))))

  (34)

  1.1.1.1.3 Reduction Pair Processor with Usable Rules

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

  [0(x1)]	=	0 0 0 0 · x1 + 2 2
  [s(x1)]	=	3 3 1 0 · x1 + 3 0
  [p(x1)]	=	0 1 2 2 · x1 + 0 2
  [twoto#(x1)]	=	2 0 0 0 · x1 + 0 0

  together with the usable rules

  p(p(s(x1)))	→	p(x1)	(7)
  p(s(x1))	→	x1	(8)
  p(0(x1))	→	0(s(s(s(s(s(s(s(s(x1)))))))))	(9)

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

  twoto#(s(x1))

  →

  twoto#(p(s(p(s(x1)))))

  (34)

  could be deleted.

  1.1.1.1.3.1 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

  together with the usable rules

  p(p(s(x1)))	→	p(x1)	(7)
  p(s(x1))	→	x1	(8)
  p(0(x1))	→	0(s(s(s(s(s(s(s(s(x1)))))))))	(9)

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

  p(p(s(x1)))	→	p(x1)	(7)
  p(s(x1))	→	x1	(8)
  p(0(x1))	→	0(s(s(s(s(s(s(s(s(x1)))))))))	(9)

  could be deleted.

  1.1.1.1.3.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 4^th component contains the pair

  twice#(s(x1))

  →

  twice#(p(p(p(s(s(s(x1)))))))

  (42)

  1.1.1.1.4 Reduction Pair Processor with Usable Rules

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

  [0(x1)]	=	0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 · x1 + 2 2 2 2
  [s(x1)]	=	1 2 0 0 0 0 1 1 1 1 0 0 0 1 0 0 · x1 + 3 0 0 0
  [p(x1)]	=	0 0 1 0 0 0 0 1 0 1 0 0 2 2 2 2 · x1 + 0 0 0 2
  [twice#(x1)]	=	1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 · x1 + 0 0 0 0

  together with the usable rules

  p(p(s(x1)))	→	p(x1)	(7)
  p(s(x1))	→	x1	(8)
  p(0(x1))	→	0(s(s(s(s(s(s(s(s(x1)))))))))	(9)

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

  twice#(s(x1))

  →

  twice#(p(p(p(s(s(s(x1)))))))

  (42)

  could be deleted.

  1.1.1.1.4.1 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

  together with the usable rules

  p(p(s(x1)))	→	p(x1)	(7)
  p(s(x1))	→	x1	(8)
  p(0(x1))	→	0(s(s(s(s(s(s(s(s(x1)))))))))	(9)

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

  p(p(s(x1)))	→	p(x1)	(7)
  p(s(x1))	→	x1	(8)
  p(0(x1))	→	0(s(s(s(s(s(s(s(s(x1)))))))))	(9)

  could be deleted.

  1.1.1.1.4.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.