Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex3_12_Luc96a_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#(sel(X1,X2))	→	a__sel#(mark(X1),mark(X2))	(22)
  mark#(from(X))	→	mark#(X)	(15)
  a__sel#(s(X),cons(Y,Z))	→	mark#(X)	(21)
  a__sel#(0,cons(X,Y))	→	mark#(X)	(20)
  mark#(s(X))	→	mark#(X)	(19)
  mark#(from(X))	→	a__from#(mark(X))	(14)
  a__sel#(s(X),cons(Y,Z))	→	mark#(Z)	(13)
  mark#(sel(X1,X2))	→	mark#(X2)	(12)
  mark#(sel(X1,X2))	→	mark#(X1)	(18)
  a__from#(X)	→	mark#(X)	(17)
  a__sel#(s(X),cons(Y,Z))	→	a__sel#(mark(X),mark(Z))	(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 + 35231
  [a__from(x1)]	=	x1 + 20539
  [mark#(x1)]	=	x1 + 35230
  [0]	=	1
  [from(x1)]	=	x1 + 20539
  [sel(x1, x2)]	=	max(x1 + 4902, x2 + 284, 0)
  [a__sel#(x1, x2)]	=	max(x1 + 40131, x2 + 35231, 0)
  [mark(x1)]	=	x1 + 0
  [a__sel(x1, x2)]	=	max(x1 + 4902, x2 + 284, 0)
  [cons(x1, x2)]	=	max(x1 + 4618, 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__sel(s(X),cons(Y,Z))	→	a__sel(mark(X),mark(Z))	(3)
  mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(5)
  a__sel(X1,X2)	→	sel(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__sel(0,cons(X,Y))	→	mark(X)	(2)

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

  mark#(sel(X1,X2))	→	a__sel#(mark(X1),mark(X2))	(22)
  mark#(from(X))	→	mark#(X)	(15)
  a__sel#(s(X),cons(Y,Z))	→	mark#(X)	(21)
  a__sel#(0,cons(X,Y))	→	mark#(X)	(20)
  mark#(from(X))	→	a__from#(mark(X))	(14)
  a__sel#(s(X),cons(Y,Z))	→	mark#(Z)	(13)
  mark#(sel(X1,X2))	→	mark#(X2)	(12)
  mark#(sel(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__sel#(s(X),cons(Y,Z))

    →

    a__sel#(mark(X),mark(Z))

    (16)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the argument filter

    π(a__sel#)	=	1
    π(mark)	=	1

    in combination with the following Weighted Path Order with the following precedence and status

    prec(s)	=	2		status(s)	=	[1]		list-extension(s)	=	Lex
    prec(a__from#)	=	0		status(a__from#)	=	[]		list-extension(a__from#)	=	Lex
    prec(a__from)	=	1		status(a__from)	=	[]		list-extension(a__from)	=	Lex
    prec(mark#)	=	0		status(mark#)	=	[]		list-extension(mark#)	=	Lex
    prec(0)	=	0		status(0)	=	[]		list-extension(0)	=	Lex
    prec(from)	=	1		status(from)	=	[]		list-extension(from)	=	Lex
    prec(sel)	=	3		status(sel)	=	[1, 2]		list-extension(sel)	=	Lex
    prec(a__sel)	=	3		status(a__sel)	=	[1, 2]		list-extension(a__sel)	=	Lex
    prec(cons)	=	0		status(cons)	=	[]		list-extension(cons)	=	Lex

    and the following Max-polynomial interpretation

    [s(x1)]	=	x1 + 0
    [a__from#(x1)]	=	1
    [a__from(x1)]	=	x1 + 8858
    [mark#(x1)]	=	1
    [0]	=	1
    [from(x1)]	=	x1 + 8858
    [sel(x1, x2)]	=	x1 + x2 + 8856
    [a__sel(x1, x2)]	=	x1 + x2 + 8856
    [cons(x1, x2)]	=	max(x1 + 8857, 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__sel(s(X),cons(Y,Z))	→	a__sel(mark(X),mark(Z))	(3)
    mark(sel(X1,X2))	→	a__sel(mark(X1),mark(X2))	(5)
    a__sel(X1,X2)	→	sel(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__sel(0,cons(X,Y))	→	mark(X)	(2)

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

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

    →

    a__sel#(mark(X),mark(Z))

    (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 + 1
    [a__from#(x1)]	=	2
    [a__from(x1)]	=	2
    [mark#(x1)]	=	x1 + 2
    [0]	=	2
    [from(x1)]	=	3
    [sel(x1, x2)]	=	x1 + 3
    [a__sel#(x1, x2)]	=	0
    [mark(x1)]	=	1
    [a__sel(x1, x2)]	=	2
    [cons(x1, x2)]	=	3

    having no usable rules (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.