Certification Problem

Input (TPDB TRS_Standard/Zantema_05/jw11)

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 1 component.

• The 1^st component contains the pair

  f#(a,f(x,a))	→	f#(a,f(f(a,x),f(a,a)))	(4)
  f#(a,f(x,a))	→	f#(a,x)	(3)
  f#(a,f(x,a))	→	f#(f(a,x),f(a,a))	(2)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [a]	=	2
  [f(x1, x2)]	=	1
  [f#(x1, x2)]	=	x1 + 0

  together with the usable rule

  f(a,f(x,a))

  →

  f(a,f(f(a,x),f(a,a)))

  (1)

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

  f#(a,f(x,a))

  →

  f#(f(a,x),f(a,a))

  (2)

  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#(a,f(x,a))	→	f#(a,x)	(3)
    f#(a,f(x,a))	→	f#(a,f(f(a,x),f(a,a)))	(4)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

    [a]	=	1 0
    [f(x1, x2)]	=	0 0 1 1 · x1 + 0 0 1 0 · x2 + 0 0
    [f#(x1, x2)]	=	0 1 0 1 · x2 + 0 0

    together with the usable rule

    f(a,f(x,a))

    →

    f(a,f(f(a,x),f(a,a)))

    (1)

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

    f#(a,f(x,a))

    →

    f#(a,x)

    (3)

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 1 component.

    • The 1^st component contains the pair

      f#(a,f(x,a))

      →

      f#(a,f(f(a,x),f(a,a)))

      (4)

      1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

      [a]	=	89133 1
      [f(x1, x2)]	=	0 0 1 0 · x1 + 0 0 1 0 · x2 + 44566 0
      [f#(x1, x2)]	=	0 1 0 0 · x2 + 0 0

      together with the usable rule

      f(a,f(x,a))

      →

      f(a,f(f(a,x),f(a,a)))

      (1)

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

      f#(a,f(x,a))

      →

      f#(a,f(f(a,x),f(a,a)))

      (4)

      could be deleted.

      1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.