Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex26_Luc03b_Z)

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

  activate#(n__dbl(X))	→	dbl#(X)	(37)
  add#(s(X),Y)	→	activate#(X)	(36)
  sqr#(s(X))	→	activate#(X)	(26)
  terms#(N)	→	sqr#(N)	(35)
  sqr#(s(X))	→	activate#(X)	(26)
  activate#(n__first(X1,X2))	→	first#(X1,X2)	(34)
  first#(s(X),cons(Y,Z))	→	activate#(X)	(33)
  sqr#(s(X))	→	sqr#(activate(X))	(25)
  activate#(n__add(X1,X2))	→	add#(X1,X2)	(24)
  sqr#(s(X))	→	dbl#(activate(X))	(23)
  activate#(n__terms(X))	→	terms#(X)	(30)
  dbl#(s(X))	→	activate#(X)	(22)
  first#(s(X),cons(Y,Z))	→	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)]	=	x1 + x2 + 1324
  [recip(x1)]	=	x1 + 0
  [activate(x1)]	=	x1 + 1
  [dbl(x1)]	=	x1 + 2242
  [dbl#(x1)]	=	x1 + 2240
  [terms#(x1)]	=	x1 + 2242
  [activate#(x1)]	=	x1 + 0
  [n__add(x1, x2)]	=	x1 + x2 + 31328
  [n__s(x1)]	=	x1 + 0
  [sqr#(x1)]	=	x1 + 2241
  [n__dbl(x1)]	=	x1 + 2241
  [0]	=	1
  [s#(x1)]	=	0
  [first#(x1, x2)]	=	x1 + x2 + 0
  [nil]	=	0
  [first(x1, x2)]	=	x1 + x2 + 1325
  [n__terms(x1)]	=	x1 + 2243
  [cons(x1, x2)]	=	x2 + 0
  [add#(x1, x2)]	=	x1 + 0
  [add(x1, x2)]	=	x1 + x2 + 31329
  [sqr(x1)]	=	x1 + 42004
  [terms(x1)]	=	x1 + 2244

  together with the usable rules

  activate(n__dbl(X))	→	dbl(X)	(18)
  dbl(0)	→	0	(4)
  activate(n__terms(X))	→	terms(X)	(15)
  first(0,X)	→	nil	(8)
  terms(N)	→	cons(recip(sqr(N)),n__terms(s(N)))	(1)
  activate(n__add(X1,X2))	→	add(X1,X2)	(16)
  activate(n__first(X1,X2))	→	first(X1,X2)	(19)
  activate(n__s(X))	→	s(X)	(17)
  dbl(s(X))	→	s(n__s(n__dbl(activate(X))))	(5)
  terms(X)	→	n__terms(X)	(10)
  add(s(X),Y)	→	s(n__add(activate(X),Y))	(7)
  activate(X)	→	X	(20)
  first(X1,X2)	→	n__first(X1,X2)	(14)
  s(X)	→	n__s(X)	(12)
  add(X1,X2)	→	n__add(X1,X2)	(11)
  first(s(X),cons(Y,Z))	→	cons(Y,n__first(activate(X),activate(Z)))	(9)
  dbl(X)	→	n__dbl(X)	(13)
  add(0,X)	→	X	(6)

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

  activate#(n__dbl(X))	→	dbl#(X)	(37)
  add#(s(X),Y)	→	activate#(X)	(36)
  sqr#(s(X))	→	activate#(X)	(26)
  terms#(N)	→	sqr#(N)	(35)
  sqr#(s(X))	→	activate#(X)	(26)
  activate#(n__first(X1,X2))	→	first#(X1,X2)	(34)
  first#(s(X),cons(Y,Z))	→	activate#(X)	(33)
  activate#(n__add(X1,X2))	→	add#(X1,X2)	(24)
  sqr#(s(X))	→	dbl#(activate(X))	(23)
  activate#(n__terms(X))	→	terms#(X)	(30)
  dbl#(s(X))	→	activate#(X)	(22)
  first#(s(X),cons(Y,Z))	→	activate#(Z)	(21)

  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

    sqr#(s(X))

    →

    sqr#(activate(X))

    (25)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the argument filter

    π(n__first)	=	2
    π(dbl#)	=	1
    π(activate#)	=	1
    π(sqr#)	=	1
    π(s#)	=	1
    π(first#)	=	2
    π(cons)	=	2
    π(add#)	=	2

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

    prec(s)	=	3		status(s)	=	[1]		list-extension(s)	=	Lex
    prec(recip)	=	2		status(recip)	=	[]		list-extension(recip)	=	Lex
    prec(activate)	=	1		status(activate)	=	[1]		list-extension(activate)	=	Lex
    prec(dbl)	=	4		status(dbl)	=	[1]		list-extension(dbl)	=	Lex
    prec(terms#)	=	0		status(terms#)	=	[]		list-extension(terms#)	=	Lex
    prec(n__add)	=	4		status(n__add)	=	[2, 1]		list-extension(n__add)	=	Lex
    prec(n__s)	=	3		status(n__s)	=	[1]		list-extension(n__s)	=	Lex
    prec(n__dbl)	=	4		status(n__dbl)	=	[1]		list-extension(n__dbl)	=	Lex
    prec(0)	=	0		status(0)	=	[]		list-extension(0)	=	Lex
    prec(nil)	=	0		status(nil)	=	[]		list-extension(nil)	=	Lex
    prec(first)	=	1		status(first)	=	[2]		list-extension(first)	=	Lex
    prec(n__terms)	=	2		status(n__terms)	=	[]		list-extension(n__terms)	=	Lex
    prec(add)	=	4		status(add)	=	[2, 1]		list-extension(add)	=	Lex
    prec(sqr)	=	0		status(sqr)	=	[]		list-extension(sqr)	=	Lex
    prec(terms)	=	2		status(terms)	=	[]		list-extension(terms)	=	Lex

    and the following Max-polynomial interpretation

    [s(x1)]	=	x1 + 0
    [recip(x1)]	=	0
    [activate(x1)]	=	x1 + 0
    [dbl(x1)]	=	x1 + 0
    [terms#(x1)]	=	0
    [n__add(x1, x2)]	=	max(x1 + 0, x2 + 0, 0)
    [n__s(x1)]	=	x1 + 0
    [n__dbl(x1)]	=	x1 + 0
    [0]	=	0
    [nil]	=	0
    [first(x1, x2)]	=	max(x2 + 0, 0)
    [n__terms(x1)]	=	0
    [add(x1, x2)]	=	max(x1 + 0, x2 + 0, 0)
    [sqr(x1)]	=	0
    [terms(x1)]	=	0

    together with the usable rules

    activate(n__dbl(X))	→	dbl(X)	(18)
    dbl(0)	→	0	(4)
    activate(n__terms(X))	→	terms(X)	(15)
    first(0,X)	→	nil	(8)
    terms(N)	→	cons(recip(sqr(N)),n__terms(s(N)))	(1)
    activate(n__add(X1,X2))	→	add(X1,X2)	(16)
    activate(n__first(X1,X2))	→	first(X1,X2)	(19)
    activate(n__s(X))	→	s(X)	(17)
    dbl(s(X))	→	s(n__s(n__dbl(activate(X))))	(5)
    terms(X)	→	n__terms(X)	(10)
    add(s(X),Y)	→	s(n__add(activate(X),Y))	(7)
    activate(X)	→	X	(20)
    first(X1,X2)	→	n__first(X1,X2)	(14)
    s(X)	→	n__s(X)	(12)
    add(X1,X2)	→	n__add(X1,X2)	(11)
    first(s(X),cons(Y,Z))	→	cons(Y,n__first(activate(X),activate(Z)))	(9)
    dbl(X)	→	n__dbl(X)	(13)
    add(0,X)	→	X	(6)

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

    sqr#(s(X))

    →

    sqr#(activate(X))

    (25)

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.