Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex14_AEGL02_GM)

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

  mark#(from(X))	→	a__from#(mark(X))	(23)
  mark#(cons(X1,X2))	→	mark#(X1)	(22)
  mark#(from(X))	→	mark#(X)	(19)
  a__from#(X)	→	mark#(X)	(18)
  mark#(s(X))	→	mark#(X)	(15)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [a__length1#(x1)]	=	0
  [s(x1)]	=	x1 + 0
  [a__from#(x1)]	=	x1 + 1
  [a__from(x1)]	=	x1 + 3
  [length1(x1)]	=	1
  [a__length1(x1)]	=	1
  [mark#(x1)]	=	x1 + 0
  [0]	=	1
  [from(x1)]	=	x1 + 3
  [nil]	=	44083
  [mark(x1)]	=	x1 + 1
  [a__length#(x1)]	=	0
  [cons(x1, x2)]	=	x1 + 2
  [length(x1)]	=	1
  [a__length(x1)]	=	1

  together with the usable rules

  a__length1(X)	→	a__length(X)	(4)
  mark(cons(X1,X2))	→	cons(mark(X1),X2)	(8)
  a__from(X)	→	cons(mark(X),from(s(X)))	(1)
  a__length(cons(X,Y))	→	s(a__length1(Y))	(3)
  mark(from(X))	→	a__from(mark(X))	(5)
  mark(nil)	→	nil	(10)
  mark(length1(X))	→	a__length1(X)	(7)
  a__length1(X)	→	length1(X)	(14)
  a__from(X)	→	from(X)	(12)
  mark(0)	→	0	(11)
  mark(s(X))	→	s(mark(X))	(9)
  a__length(X)	→	length(X)	(13)
  mark(length(X))	→	a__length(X)	(6)
  a__length(nil)	→	0	(2)

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

  mark#(from(X))	→	a__from#(mark(X))	(23)
  mark#(cons(X1,X2))	→	mark#(X1)	(22)
  mark#(from(X))	→	mark#(X)	(19)
  a__from#(X)	→	mark#(X)	(18)

  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

    mark#(s(X))

    →

    mark#(X)

    (15)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [a__length1#(x1)]	=	0
    [s(x1)]	=	x1 + 2
    [a__from#(x1)]	=	1
    [a__from(x1)]	=	44603
    [length1(x1)]	=	1
    [a__length1(x1)]	=	1
    [mark#(x1)]	=	x1 + 0
    [0]	=	3
    [from(x1)]	=	14681
    [nil]	=	44083
    [mark(x1)]	=	x1 + 29922
    [a__length#(x1)]	=	0
    [cons(x1, x2)]	=	x1 + 26288
    [length(x1)]	=	3000
    [a__length(x1)]	=	2

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

    mark#(s(X))

    →

    mark#(X)

    (15)

    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

  a__length1#(X)	→	a__length#(X)	(17)
  a__length#(cons(X,Y))	→	a__length1#(Y)	(16)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [a__length1#(x1)]	=	x1 + 1
  [s(x1)]	=	x1 + 9727
  [a__from#(x1)]	=	1
  [a__from(x1)]	=	33603
  [length1(x1)]	=	1
  [a__length1(x1)]	=	1
  [mark#(x1)]	=	0
  [0]	=	9728
  [from(x1)]	=	33602
  [nil]	=	28882
  [mark(x1)]	=	x1 + 1
  [a__length#(x1)]	=	x1 + 0
  [cons(x1, x2)]	=	x1 + x2 + 2242
  [length(x1)]	=	9728
  [a__length(x1)]	=	9727

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

  a__length1#(X)	→	a__length#(X)	(17)
  a__length#(cons(X,Y))	→	a__length1#(Y)	(16)

  could be deleted.

  1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.