Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex4_7_56_Bor03_FR)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  after#(s(N),cons(X,XS))

  →

  after#(N,activate(XS))

  (11)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 1
  [activate(x1)]	=	8856
  [n__from(x1)]	=	8858
  [activate#(x1)]	=	0
  [after(x1, x2)]	=	0
  [n__s(x1)]	=	2
  [0]	=	0
  [from(x1)]	=	8857
  [s#(x1)]	=	0
  [after#(x1, x2)]	=	x1 + 0
  [from#(x1)]	=	0
  [cons(x1, x2)]	=	x1 + 8858

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

  after#(s(N),cons(X,XS))

  →

  after#(N,activate(XS))

  (11)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

  activate#(n__s(X))	→	activate#(X)	(14)
  activate#(n__from(X))	→	activate#(X)	(9)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 1
  [activate(x1)]	=	8856
  [n__from(x1)]	=	x1 + 8858
  [activate#(x1)]	=	x1 + 0
  [after(x1, x2)]	=	0
  [n__s(x1)]	=	x1 + 0
  [0]	=	0
  [from(x1)]	=	8857
  [s#(x1)]	=	0
  [after#(x1, x2)]	=	0
  [from#(x1)]	=	0
  [cons(x1, x2)]	=	x1 + 8858

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

  activate#(n__from(X))

  →

  activate#(X)

  (9)

  could be deleted.

  1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    activate#(n__s(X))

    →

    activate#(X)

    (14)

    1.1.2.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [s(x1)]	=	x1 + 1
    [activate(x1)]	=	8856
    [n__from(x1)]	=	x1 + 8858
    [activate#(x1)]	=	x1 + 0
    [after(x1, x2)]	=	0
    [n__s(x1)]	=	x1 + 1
    [0]	=	0
    [from(x1)]	=	8857
    [s#(x1)]	=	0
    [after#(x1, x2)]	=	0
    [from#(x1)]	=	0
    [cons(x1, x2)]	=	x1 + 8858

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

    activate#(n__s(X))

    →

    activate#(X)

    (14)

    could be deleted.

    1.1.2.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.