Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex4_7_56_Bor03_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 1 component.

• The 1^st component contains the pair

  mark#(after(X1,X2))	→	a__after#(mark(X1),mark(X2))	(22)
  mark#(from(X))	→	mark#(X)	(15)
  a__after#(s(N),cons(X,XS))	→	mark#(N)	(21)
  a__after#(0,XS)	→	mark#(XS)	(20)
  mark#(s(X))	→	mark#(X)	(19)
  mark#(from(X))	→	a__from#(mark(X))	(14)
  a__after#(s(N),cons(X,XS))	→	mark#(XS)	(13)
  mark#(after(X1,X2))	→	mark#(X2)	(12)
  mark#(after(X1,X2))	→	mark#(X1)	(18)
  a__from#(X)	→	mark#(X)	(17)
  a__after#(s(N),cons(X,XS))	→	a__after#(mark(N),mark(XS))	(16)
  mark#(cons(X1,X2))	→	mark#(X1)	(11)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 0
  [a__from#(x1)]	=	x1 + 5854
  [a__from(x1)]	=	x1 + 41065
  [after(x1, x2)]	=	max(x1 + 71680, x2 + 30614, 0)
  [a__after#(x1, x2)]	=	max(x1 + 46918, x2 + 5854, 0)
  [a__after(x1, x2)]	=	max(x1 + 71680, x2 + 30614, 0)
  [mark#(x1)]	=	x1 + 5853
  [0]	=	0
  [from(x1)]	=	x1 + 41065
  [mark(x1)]	=	x1 + 0
  [cons(x1, x2)]	=	max(x1 + 41065, x2 + 0, 0)

  together with the usable rules

  mark(from(X))	→	a__from(mark(X))	(4)
  mark(0)	→	0	(8)
  a__from(X)	→	cons(mark(X),from(s(X)))	(1)
  a__after(s(N),cons(X,XS))	→	a__after(mark(N),mark(XS))	(3)
  mark(after(X1,X2))	→	a__after(mark(X1),mark(X2))	(5)
  a__after(X1,X2)	→	after(X1,X2)	(10)
  mark(s(X))	→	s(mark(X))	(7)
  a__from(X)	→	from(X)	(9)
  mark(cons(X1,X2))	→	cons(mark(X1),X2)	(6)
  a__after(0,XS)	→	mark(XS)	(2)

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

  mark#(after(X1,X2))	→	a__after#(mark(X1),mark(X2))	(22)
  mark#(from(X))	→	mark#(X)	(15)
  a__after#(s(N),cons(X,XS))	→	mark#(N)	(21)
  a__after#(0,XS)	→	mark#(XS)	(20)
  mark#(from(X))	→	a__from#(mark(X))	(14)
  a__after#(s(N),cons(X,XS))	→	mark#(XS)	(13)
  mark#(after(X1,X2))	→	mark#(X2)	(12)
  mark#(after(X1,X2))	→	mark#(X1)	(18)
  a__from#(X)	→	mark#(X)	(17)
  mark#(cons(X1,X2))	→	mark#(X1)	(11)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 2 components.

  • The 1^st component contains the pair

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

    →

    a__after#(mark(N),mark(XS))

    (16)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [s(x1)]	=	x1 + 9727
    [a__from#(x1)]	=	0
    [a__from(x1)]	=	44245
    [after(x1, x2)]	=	x2 + 9726
    [a__after#(x1, x2)]	=	x1 + 0
    [a__after(x1, x2)]	=	x2 + 9726
    [mark#(x1)]	=	0
    [0]	=	31328
    [from(x1)]	=	34519
    [mark(x1)]	=	x1 + 9726
    [cons(x1, x2)]	=	x2 + 9726

    together with the usable rules

    mark(from(X))	→	a__from(mark(X))	(4)
    mark(0)	→	0	(8)
    a__from(X)	→	cons(mark(X),from(s(X)))	(1)
    a__after(s(N),cons(X,XS))	→	a__after(mark(N),mark(XS))	(3)
    mark(after(X1,X2))	→	a__after(mark(X1),mark(X2))	(5)
    a__after(X1,X2)	→	after(X1,X2)	(10)
    mark(s(X))	→	s(mark(X))	(7)
    a__from(X)	→	from(X)	(9)
    mark(cons(X1,X2))	→	cons(mark(X1),X2)	(6)
    a__after(0,XS)	→	mark(XS)	(2)

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

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

    →

    a__after#(mark(N),mark(XS))

    (16)

    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

    mark#(s(X))

    →

    mark#(X)

    (19)

    1.1.1.1.2 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [s(x1)]	=	x1 + 9727
    [a__from#(x1)]	=	0
    [a__from(x1)]	=	35496
    [after(x1, x2)]	=	x2 + 1
    [a__after#(x1, x2)]	=	x1 + 0
    [a__after(x1, x2)]	=	x2 + 1
    [mark#(x1)]	=	x1 + 0
    [0]	=	31328
    [from(x1)]	=	35495
    [mark(x1)]	=	x1 + 1
    [cons(x1, x2)]	=	x2 + 1

    together with the usable rules

    mark(from(X))	→	a__from(mark(X))	(4)
    mark(0)	→	0	(8)
    a__from(X)	→	cons(mark(X),from(s(X)))	(1)
    a__after(s(N),cons(X,XS))	→	a__after(mark(N),mark(XS))	(3)
    mark(after(X1,X2))	→	a__after(mark(X1),mark(X2))	(5)
    a__after(X1,X2)	→	after(X1,X2)	(10)
    mark(s(X))	→	s(mark(X))	(7)
    a__from(X)	→	from(X)	(9)
    mark(cons(X1,X2))	→	cons(mark(X1),X2)	(6)
    a__after(0,XS)	→	mark(XS)	(2)

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

    mark#(s(X))

    →

    mark#(X)

    (19)

    could be deleted.

    1.1.1.1.2.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.