Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex1_Luc02b_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(X),cons(Y,Z))

  →

  sel#(X,activate(Z))

  (21)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 1
  [n__first(x1, x2)]	=	2
  [activate(x1)]	=	1
  [n__from(x1)]	=	3
  [activate#(x1)]	=	0
  [n__s(x1)]	=	x1 + 2
  [0]	=	0
  [sel#(x1, x2)]	=	x1 + 0
  [from(x1)]	=	2
  [sel(x1, x2)]	=	0
  [s#(x1)]	=	0
  [first#(x1, x2)]	=	0
  [nil]	=	2
  [first(x1, x2)]	=	x1 + 1
  [from#(x1)]	=	0
  [cons(x1, x2)]	=	x1 + 3

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

  sel#(s(X),cons(Y,Z))

  →

  sel#(X,activate(Z))

  (21)

  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__first(X1,X2))	→	activate#(X1)	(22)
  activate#(n__first(X1,X2))	→	first#(activate(X1),activate(X2))	(19)
  activate#(n__first(X1,X2))	→	activate#(X2)	(18)
  first#(s(X),cons(Y,Z))	→	activate#(Z)	(17)
  activate#(n__s(X))	→	activate#(X)	(16)
  activate#(n__from(X))	→	activate#(X)	(13)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 0
  [n__first(x1, x2)]	=	x1 + x2 + 1144
  [activate(x1)]	=	x1 + 0
  [n__from(x1)]	=	x1 + 28101
  [activate#(x1)]	=	x1 + 0
  [n__s(x1)]	=	x1 + 0
  [0]	=	0
  [sel#(x1, x2)]	=	0
  [from(x1)]	=	x1 + 28101
  [sel(x1, x2)]	=	0
  [s#(x1)]	=	0
  [first#(x1, x2)]	=	x2 + 1143
  [nil]	=	0
  [first(x1, x2)]	=	x1 + x2 + 1144
  [from#(x1)]	=	0
  [cons(x1, x2)]	=	x2 + 0

  together with the usable rules

  first(X1,X2)	→	n__first(X1,X2)	(8)
  from(X)	→	cons(X,n__from(n__s(X)))	(1)
  first(s(X),cons(Y,Z))	→	cons(Y,n__first(X,activate(Z)))	(3)
  activate(n__s(X))	→	s(activate(X))	(10)
  s(X)	→	n__s(X)	(7)
  activate(X)	→	X	(12)
  activate(n__first(X1,X2))	→	first(activate(X1),activate(X2))	(11)
  activate(n__from(X))	→	from(activate(X))	(9)
  from(X)	→	n__from(X)	(6)
  first(0,Z)	→	nil	(2)

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

  activate#(n__first(X1,X2))	→	activate#(X1)	(22)
  activate#(n__first(X1,X2))	→	first#(activate(X1),activate(X2))	(19)
  activate#(n__first(X1,X2))	→	activate#(X2)	(18)
  first#(s(X),cons(Y,Z))	→	activate#(Z)	(17)
  activate#(n__from(X))	→	activate#(X)	(13)

  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)

    (16)

    1.1.2.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [s(x1)]	=	x1 + 8946
    [n__first(x1, x2)]	=	x2 + 1144
    [activate(x1)]	=	x1 + 0
    [n__from(x1)]	=	1
    [activate#(x1)]	=	x1 + 0
    [n__s(x1)]	=	x1 + 8946
    [0]	=	0
    [sel#(x1, x2)]	=	0
    [from(x1)]	=	1
    [sel(x1, x2)]	=	0
    [s#(x1)]	=	0
    [first#(x1, x2)]	=	x2 + 1143
    [nil]	=	0
    [first(x1, x2)]	=	x2 + 1144
    [from#(x1)]	=	0
    [cons(x1, x2)]	=	x2 + 0

    together with the usable rules

    first(X1,X2)	→	n__first(X1,X2)	(8)
    from(X)	→	cons(X,n__from(n__s(X)))	(1)
    first(s(X),cons(Y,Z))	→	cons(Y,n__first(X,activate(Z)))	(3)
    activate(n__s(X))	→	s(activate(X))	(10)
    s(X)	→	n__s(X)	(7)
    activate(X)	→	X	(12)
    activate(n__first(X1,X2))	→	first(activate(X1),activate(X2))	(11)
    activate(n__from(X))	→	from(activate(X))	(9)
    from(X)	→	n__from(X)	(6)
    first(0,Z)	→	nil	(2)

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

    activate#(n__s(X))

    →

    activate#(X)

    (16)

    could be deleted.

    1.1.2.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.