Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/ExProp7_Luc06_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#(p(X))	→	a__p#(mark(X))	(19)
  mark#(f(X))	→	a__f#(mark(X))	(18)
  a__f#(s(0))	→	a__f#(a__p(s(0)))	(17)
  mark#(cons(X1,X2))	→	mark#(X1)	(16)
  mark#(f(X))	→	mark#(X)	(13)
  a__p#(s(X))	→	mark#(X)	(12)
  mark#(p(X))	→	mark#(X)	(15)
  mark#(s(X))	→	mark#(X)	(14)
  a__f#(s(0))	→	a__p#(s(0))	(11)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 29485
  [a__f(x1)]	=	x1 + 4
  [a__p(x1)]	=	x1 + 0
  [f(x1)]	=	x1 + 4
  [p(x1)]	=	x1 + 0
  [mark#(x1)]	=	x1 + 29484
  [0]	=	1
  [mark(x1)]	=	x1 + 29483
  [a__p#(x1)]	=	x1 + 0
  [cons(x1, x2)]	=	x1 + 2
  [a__f#(x1)]	=	29487

  together with the usable rules

  mark(f(X))	→	a__f(mark(X))	(4)
  mark(s(X))	→	s(mark(X))	(8)
  a__f(0)	→	cons(0,f(s(0)))	(1)
  a__p(s(X))	→	mark(X)	(3)
  mark(p(X))	→	a__p(mark(X))	(5)
  a__p(X)	→	p(X)	(10)
  mark(cons(X1,X2))	→	cons(mark(X1),X2)	(7)
  a__f(X)	→	f(X)	(9)
  mark(0)	→	0	(6)
  a__f(s(0))	→	a__f(a__p(s(0)))	(2)

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

  mark#(p(X))	→	a__p#(mark(X))	(19)
  mark#(f(X))	→	a__f#(mark(X))	(18)
  mark#(cons(X1,X2))	→	mark#(X1)	(16)
  mark#(f(X))	→	mark#(X)	(13)
  a__p#(s(X))	→	mark#(X)	(12)
  mark#(s(X))	→	mark#(X)	(14)
  a__f#(s(0))	→	a__p#(s(0))	(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

    mark#(p(X))

    →

    mark#(X)

    (15)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [s(x1)]	=	x1 + 29480
    [a__f(x1)]	=	4
    [a__p(x1)]	=	x1 + 1
    [f(x1)]	=	4
    [p(x1)]	=	x1 + 1
    [mark#(x1)]	=	x1 + 29484
    [0]	=	3
    [mark(x1)]	=	x1 + 2241
    [a__p#(x1)]	=	x1 + 0
    [cons(x1, x2)]	=	x1 + 1
    [a__f#(x1)]	=	29487

    together with the usable rules

    mark(f(X))	→	a__f(mark(X))	(4)
    mark(s(X))	→	s(mark(X))	(8)
    a__f(0)	→	cons(0,f(s(0)))	(1)
    a__p(s(X))	→	mark(X)	(3)
    mark(p(X))	→	a__p(mark(X))	(5)
    a__p(X)	→	p(X)	(10)
    mark(cons(X1,X2))	→	cons(mark(X1),X2)	(7)
    a__f(X)	→	f(X)	(9)
    mark(0)	→	0	(6)
    a__f(s(0))	→	a__f(a__p(s(0)))	(2)

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

    mark#(p(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__f#(s(0))

    →

    a__f#(a__p(s(0)))

    (17)

    1.1.1.1.2 Reduction Pair Processor with Usable Rules

    Using the argument filter

    π(mark)	=	1
    π(a__p#)	=	1

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

    prec(s)	=	2		status(s)	=	[]		list-extension(s)	=	Lex
    prec(a__f)	=	1		status(a__f)	=	[]		list-extension(a__f)	=	Lex
    prec(a__p)	=	0		status(a__p)	=	[]		list-extension(a__p)	=	Lex
    prec(f)	=	1		status(f)	=	[]		list-extension(f)	=	Lex
    prec(p)	=	0		status(p)	=	[]		list-extension(p)	=	Lex
    prec(mark#)	=	0		status(mark#)	=	[]		list-extension(mark#)	=	Lex
    prec(0)	=	1		status(0)	=	[]		list-extension(0)	=	Lex
    prec(cons)	=	1		status(cons)	=	[]		list-extension(cons)	=	Lex
    prec(a__f#)	=	0		status(a__f#)	=	[1]		list-extension(a__f#)	=	Lex

    and the following Max-polynomial interpretation

    [s(x1)]	=	x1 + 1
    [a__f(x1)]	=	32286
    [a__p(x1)]	=	x1 + 0
    [f(x1)]	=	32286
    [p(x1)]	=	x1 + 0
    [mark#(x1)]	=	1
    [0]	=	32287
    [cons(x1, x2)]	=	32286
    [a__f#(x1)]	=	x1 + 1

    together with the usable rules

    mark(f(X))	→	a__f(mark(X))	(4)
    mark(s(X))	→	s(mark(X))	(8)
    a__f(0)	→	cons(0,f(s(0)))	(1)
    a__p(s(X))	→	mark(X)	(3)
    mark(p(X))	→	a__p(mark(X))	(5)
    a__p(X)	→	p(X)	(10)
    mark(cons(X1,X2))	→	cons(mark(X1),X2)	(7)
    a__f(X)	→	f(X)	(9)
    mark(0)	→	0	(6)
    a__f(s(0))	→	a__f(a__p(s(0)))	(2)

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

    a__f#(s(0))

    →

    a__f#(a__p(s(0)))

    (17)

    could be deleted.

    1.1.1.1.2.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.