Certification Problem

Input (TPDB SRS_Standard/Trafo_06/dup10)

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

  B#(B(B(B(x1))))	→	b#(x1)	(10)
  b#(b(b(b(x1))))	→	B#(x1)	(26)
  B#(B(B(B(x1))))	→	b#(b(x1))	(11)
  b#(b(b(b(x1))))	→	B#(B(x1))	(27)

  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

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

  together with the usable rules

  b(b(b(b(x1))))	→	B(B(x1))	(3)
  B(B(B(B(x1))))	→	b(b(x1))	(4)
  b(b(B(B(x1))))	→	x1	(6)
  B(B(b(b(x1))))	→	x1	(7)

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

  B#(B(B(B(x1))))	→	b#(x1)	(10)
  b#(b(b(b(x1))))	→	B#(x1)	(26)
  B#(B(B(B(x1))))	→	b#(b(x1))	(11)
  b#(b(b(b(x1))))	→	B#(B(x1))	(27)

  and the rules

  b(b(b(b(x1))))	→	B(B(x1))	(3)
  B(B(B(B(x1))))	→	b(b(x1))	(4)
  b(b(B(B(x1))))	→	x1	(6)
  B(B(b(b(x1))))	→	x1	(7)

  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

  c#(c(B(B(c(c(b(b(c(c(x1))))))))))

  →

  c#(c(b(b(c(c(B(B(c(c(b(b(x1))))))))))))

  (19)

  1.1.1.1.2 Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the naturals

  [B(x1)]	=	0 1 0 1 0 1 1 0 1 · x1 + 0 0 1
  [c(x1)]	=	0 0 0 0 0 1 0 0 1 · x1 + 0 0 0
  [b(x1)]	=	1 0 1 1 0 0 0 1 0 · x1 + 1 0 0
  [c#(x1)]	=	3 1 0 0 0 0 0 0 0 · x1 + 0 0 0

  together with the usable rules

  b(b(b(b(x1))))	→	B(B(x1))	(3)
  B(B(B(B(x1))))	→	b(b(x1))	(4)
  c(c(B(B(c(c(b(b(c(c(x1))))))))))	→	B(B(c(c(b(b(c(c(B(B(c(c(b(b(x1))))))))))))))	(5)
  b(b(B(B(x1))))	→	x1	(6)
  B(B(b(b(x1))))	→	x1	(7)

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

  c#(c(B(B(c(c(b(b(c(c(x1))))))))))

  →

  c#(c(b(b(c(c(B(B(c(c(b(b(x1))))))))))))

  (19)

  could be deleted.

  1.1.1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.