Certification Problem

Input (TPDB TRS_Standard/SK90/4.30)

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

  f#(.(.(x,y),z))	→	f#(.(x,.(y,z)))	(10)
  f#(.(nil,y))	→	f#(y)	(9)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [f(x1)]	=	0
  [nil]	=	1
  [f#(x1)]	=	x1 + 0
  [g#(x1)]	=	0
  [.(x1, x2)]	=	x1 + x2 + 28958
  [g(x1)]	=	0

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

  f#(.(nil,y))

  →

  f#(y)

  (9)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    f#(.(.(x,y),z))

    →

    f#(.(x,.(y,z)))

    (10)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [f(x1)]	=	0
    [nil]	=	1
    [f#(x1)]	=	x1 + 0
    [g#(x1)]	=	0
    [.(x1, x2)]	=	x1 + 32286
    [g(x1)]	=	0

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

    f#(.(.(x,y),z))

    →

    f#(.(x,.(y,z)))

    (10)

    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

  g#(.(x,nil))	→	g#(x)	(8)
  g#(.(x,.(y,z)))	→	g#(.(.(x,y),z))	(7)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [f(x1)]	=	0
  [nil]	=	1
  [f#(x1)]	=	0
  [g#(x1)]	=	x1 + 0
  [.(x1, x2)]	=	x1 + x2 + 1424
  [g(x1)]	=	0

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

  g#(.(x,nil))

  →

  g#(x)

  (8)

  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

    g#(.(x,.(y,z)))

    →

    g#(.(.(x,y),z))

    (7)

    1.1.2.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [f(x1)]	=	0
    [nil]	=	1
    [f#(x1)]	=	0
    [g#(x1)]	=	x1 + 0
    [.(x1, x2)]	=	x2 + 4421
    [g(x1)]	=	0

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

    g#(.(x,.(y,z)))

    →

    g#(.(.(x,y),z))

    (7)

    could be deleted.

    1.1.2.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.