Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z094)

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

  n#(f(x1))	→	f#(n(x1))	(14)
  f#(x1)	→	c#(x1)	(13)
  c#(f(x1))	→	f#(c(c(x1)))	(12)
  c#(f(x1))	→	c#(x1)	(11)
  n#(s(x1))	→	f#(s(s(x1)))	(8)
  f#(x1)	→	n#(c(c(x1)))	(7)
  n#(f(x1))	→	n#(x1)	(10)
  f#(x1)	→	c#(c(x1))	(9)
  c#(f(x1))	→	c#(c(x1))	(6)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	20977
  [n(x1)]	=	x1 + 0
  [c(x1)]	=	0
  [f(x1)]	=	x1 + 0
  [f#(x1)]	=	0
  [c#(x1)]	=	0
  [n#(x1)]	=	x1 + 0

  together with the usable rules

  n(s(x1))	→	f(s(s(x1)))	(4)
  f(x1)	→	n(c(c(x1)))	(1)
  c(c(x1))	→	c(x1)	(3)
  n(f(x1))	→	f(n(x1))	(5)
  c(f(x1))	→	f(c(c(x1)))	(2)

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

  n#(s(x1))

  →

  f#(s(s(x1)))

  (8)

  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#(x1)	→	c#(x1)	(13)
    f#(x1)	→	c#(c(x1))	(9)
    f#(x1)	→	n#(c(c(x1)))	(7)
    n#(f(x1))	→	n#(x1)	(10)
    n#(f(x1))	→	f#(n(x1))	(14)
    c#(f(x1))	→	c#(x1)	(11)
    c#(f(x1))	→	c#(c(x1))	(6)
    c#(f(x1))	→	f#(c(c(x1)))	(12)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [s(x1)]	=	18458
    [n(x1)]	=	x1 + 25908
    [c(x1)]	=	x1 + 0
    [f(x1)]	=	x1 + 25908
    [f#(x1)]	=	x1 + 1
    [c#(x1)]	=	x1 + 0
    [n#(x1)]	=	x1 + 1

    together with the usable rules

    n(s(x1))	→	f(s(s(x1)))	(4)
    f(x1)	→	n(c(c(x1)))	(1)
    c(c(x1))	→	c(x1)	(3)
    n(f(x1))	→	f(n(x1))	(5)
    c(f(x1))	→	f(c(c(x1)))	(2)

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

    f#(x1)	→	c#(x1)	(13)
    f#(x1)	→	c#(c(x1))	(9)
    n#(f(x1))	→	n#(x1)	(10)
    c#(f(x1))	→	c#(x1)	(11)
    c#(f(x1))	→	c#(c(x1))	(6)
    c#(f(x1))	→	f#(c(c(x1)))	(12)

    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#(x1)	→	n#(c(c(x1)))	(7)
      n#(f(x1))	→	f#(n(x1))	(14)

      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

      [s(x1)]	=	20695 55291
      [n(x1)]	=	1 0 1 1 · x1 + 0 1
      [c(x1)]	=	0 0 0 1 · x1 + 0 0
      [f(x1)]	=	x1 + 0 3
      [f#(x1)]	=	0 1 0 0 · x1 + 1 0
      [c#(x1)]	=	0 0
      [n#(x1)]	=	1 1 0 0 · x1 + 0 0

      together with the usable rules

      n(s(x1))	→	f(s(s(x1)))	(4)
      f(x1)	→	n(c(c(x1)))	(1)
      c(c(x1))	→	c(x1)	(3)
      n(f(x1))	→	f(n(x1))	(5)
      c(f(x1))	→	f(c(c(x1)))	(2)

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

      f#(x1)	→	n#(c(c(x1)))	(7)
      n#(f(x1))	→	f#(n(x1))	(14)

      could be deleted.

      1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.