Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z123)

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

  f#(f(x1))	→	a#(x1)	(19)
  b#(b(x1))	→	c#(x1)	(12)
  a#(a(x1))	→	b#(b(x1))	(18)
  a#(x1)	→	c#(d(x1))	(17)
  f#(f(x1))	→	f#(a(x1))	(16)
  b#(b(x1))	→	c#(c(x1))	(11)
  a#(a(x1))	→	b#(x1)	(10)
  b#(b(b(x1)))	→	f#(x1)	(9)
  b#(b(b(x1)))	→	a#(f(x1))	(15)
  b#(b(x1))	→	c#(c(c(x1)))	(14)
  a#(a(x1))	→	b#(b(b(x1)))	(13)
  c#(d(d(x1)))	→	f#(x1)	(8)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [a(x1)]	=	x1 + 678006
  [d(x1)]	=	x1 + 193716
  [b(x1)]	=	x1 + 452004
  [c(x1)]	=	x1 + 290574
  [f(x1)]	=	x1 + 678006
  [f#(x1)]	=	x1 + 387431
  [c#(x1)]	=	x1 + 0
  [a#(x1)]	=	x1 + 355147
  [b#(x1)]	=	x1 + 129145

  together with the usable rules

  b(b(x1))	→	c(c(c(x1)))	(4)
  a(a(x1))	→	b(b(b(x1)))	(1)
  b(b(b(x1)))	→	a(f(x1))	(3)
  c(c(x1))	→	d(d(d(x1)))	(5)
  f(f(x1))	→	f(a(x1))	(7)
  c(d(d(x1)))	→	f(x1)	(6)
  a(x1)	→	d(c(d(x1)))	(2)

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

  f#(f(x1))	→	a#(x1)	(19)
  b#(b(x1))	→	c#(x1)	(12)
  a#(a(x1))	→	b#(b(x1))	(18)
  a#(x1)	→	c#(d(x1))	(17)
  b#(b(x1))	→	c#(c(x1))	(11)
  a#(a(x1))	→	b#(x1)	(10)
  b#(b(b(x1)))	→	f#(x1)	(9)
  b#(b(x1))	→	c#(c(c(x1)))	(14)
  c#(d(d(x1)))	→	f#(x1)	(8)

  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#(a(x1))	→	b#(b(b(x1)))	(13)
    b#(b(b(x1)))	→	a#(f(x1))	(15)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [a(x1)]	=	x1 + 14
    [d(x1)]	=	x1 + 4
    [b(x1)]	=	x1 + 9
    [c(x1)]	=	x1 + 6
    [f(x1)]	=	1
    [f#(x1)]	=	x1 + 387431
    [c#(x1)]	=	x1 + 0
    [a#(x1)]	=	x1 + 16
    [b#(x1)]	=	x1 + 0

    together with the usable rules

    b(b(x1))	→	c(c(c(x1)))	(4)
    a(a(x1))	→	b(b(b(x1)))	(1)
    b(b(b(x1)))	→	a(f(x1))	(3)
    c(c(x1))	→	d(d(d(x1)))	(5)
    f(f(x1))	→	f(a(x1))	(7)
    c(d(d(x1)))	→	f(x1)	(6)
    a(x1)	→	d(c(d(x1)))	(2)

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

    a#(a(x1))	→	b#(b(b(x1)))	(13)
    b#(b(b(x1)))	→	a#(f(x1))	(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

    f#(f(x1))

    →

    f#(a(x1))

    (16)

    1.1.1.1.2 Reduction Pair Processor with Usable Rules

    Using the argument filter
    in combination with the following Weighted Path Order with the following precedence and status

    prec(a)	=	0		status(a)	=	[]		list-extension(a)	=	Lex
    prec(d)	=	0		status(d)	=	[]		list-extension(d)	=	Lex
    prec(b)	=	0		status(b)	=	[]		list-extension(b)	=	Lex
    prec(c)	=	1		status(c)	=	[]		list-extension(c)	=	Lex
    prec(f)	=	1		status(f)	=	[]		list-extension(f)	=	Lex
    prec(f#)	=	0		status(f#)	=	[1]		list-extension(f#)	=	Lex
    prec(c#)	=	0		status(c#)	=	[]		list-extension(c#)	=	Lex
    prec(a#)	=	0		status(a#)	=	[]		list-extension(a#)	=	Lex
    prec(b#)	=	0		status(b#)	=	[]		list-extension(b#)	=	Lex

    and the following Max-polynomial interpretation

    [a(x1)]	=	x1 + 159429
    [d(x1)]	=	x1 + 44286
    [b(x1)]	=	x1 + 106286
    [c(x1)]	=	x1 + 70857
    [f(x1)]	=	x1 + 159429
    [f#(x1)]	=	x1 + 1
    [c#(x1)]	=	1
    [a#(x1)]	=	1
    [b#(x1)]	=	1

    together with the usable rules

    b(b(x1))	→	c(c(c(x1)))	(4)
    a(a(x1))	→	b(b(b(x1)))	(1)
    b(b(b(x1)))	→	a(f(x1))	(3)
    c(c(x1))	→	d(d(d(x1)))	(5)
    f(f(x1))	→	f(a(x1))	(7)
    c(d(d(x1)))	→	f(x1)	(6)
    a(x1)	→	d(c(d(x1)))	(2)

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

    f#(f(x1))

    →

    f#(a(x1))

    (16)

    could be deleted.

    1.1.1.1.2.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.