Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex7_BLR02_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

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

  →

  sel#(N,activate(XS))

  (25)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 1
  [take#(x1, x2)]	=	0
  [activate(x1)]	=	x1 + 3
  [take(x1, x2)]	=	x1 + x2 + 0
  [n__from(x1)]	=	1
  [activate#(x1)]	=	0
  [2nd(x1)]	=	0
  [head#(x1)]	=	0
  [n__s(x1)]	=	2
  [0]	=	23676
  [n__take(x1, x2)]	=	2
  [sel#(x1, x2)]	=	x1 + 0
  [from(x1)]	=	x1 + 2
  [sel(x1, x2)]	=	0
  [s#(x1)]	=	0
  [nil]	=	23677
  [2nd#(x1)]	=	0
  [from#(x1)]	=	0
  [head(x1)]	=	0
  [cons(x1, x2)]	=	x1 + x2 + 2

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

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

  →

  sel#(N,activate(XS))

  (25)

  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__take(X1,X2))	→	take#(activate(X1),activate(X2))	(24)
  activate#(n__take(X1,X2))	→	activate#(X2)	(23)
  activate#(n__from(X))	→	activate#(X)	(22)
  take#(s(N),cons(X,XS))	→	activate#(XS)	(21)
  activate#(n__s(X))	→	activate#(X)	(20)
  activate#(n__take(X1,X2))	→	activate#(X1)	(15)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 0
  [take#(x1, x2)]	=	x2 + 1
  [activate(x1)]	=	x1 + 0
  [take(x1, x2)]	=	x1 + x2 + 20539
  [n__from(x1)]	=	x1 + 282
  [activate#(x1)]	=	x1 + 0
  [2nd(x1)]	=	0
  [head#(x1)]	=	0
  [n__s(x1)]	=	x1 + 0
  [0]	=	23676
  [n__take(x1, x2)]	=	x1 + x2 + 20539
  [sel#(x1, x2)]	=	0
  [from(x1)]	=	x1 + 282
  [sel(x1, x2)]	=	0
  [s#(x1)]	=	0
  [nil]	=	44215
  [2nd#(x1)]	=	0
  [from#(x1)]	=	0
  [head(x1)]	=	0
  [cons(x1, x2)]	=	x2 + 0

  together with the usable rules

  take(0,XS)	→	nil	(4)
  from(X)	→	n__from(X)	(8)
  from(X)	→	cons(X,n__from(n__s(X)))	(1)
  take(s(N),cons(X,XS))	→	cons(X,n__take(N,activate(XS)))	(5)
  take(X1,X2)	→	n__take(X1,X2)	(10)
  activate(X)	→	X	(14)
  activate(n__s(X))	→	s(activate(X))	(12)
  activate(n__from(X))	→	from(activate(X))	(11)
  s(X)	→	n__s(X)	(9)
  activate(n__take(X1,X2))	→	take(activate(X1),activate(X2))	(13)

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

  activate#(n__take(X1,X2))	→	take#(activate(X1),activate(X2))	(24)
  activate#(n__take(X1,X2))	→	activate#(X2)	(23)
  activate#(n__from(X))	→	activate#(X)	(22)
  take#(s(N),cons(X,XS))	→	activate#(XS)	(21)
  activate#(n__take(X1,X2))	→	activate#(X1)	(15)

  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)

    (20)

    1.1.2.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [s(x1)]	=	0
    [take#(x1, x2)]	=	1
    [activate(x1)]	=	1
    [take(x1, x2)]	=	1
    [n__from(x1)]	=	1
    [activate#(x1)]	=	x1 + 0
    [2nd(x1)]	=	0
    [head#(x1)]	=	0
    [n__s(x1)]	=	x1 + 1
    [0]	=	23676
    [n__take(x1, x2)]	=	1
    [sel#(x1, x2)]	=	0
    [from(x1)]	=	x1 + 0
    [sel(x1, x2)]	=	0
    [s#(x1)]	=	0
    [nil]	=	2
    [2nd#(x1)]	=	0
    [from#(x1)]	=	0
    [head(x1)]	=	0
    [cons(x1, x2)]	=	x1 + 0

    together with the usable rules

    from(X)	→	cons(X,n__from(n__s(X)))	(1)
    take(X1,X2)	→	n__take(X1,X2)	(10)

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

    activate#(n__s(X))

    →

    activate#(X)

    (20)

    could be deleted.

    1.1.2.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.