Certification Problem

Input (TPDB TRS_Standard/SK90/4.02)

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 2 components.

• The 1^st component contains the pair

  i#(*(x,y))	→	i#(x)	(24)
  i#(*(x,y))	→	i#(y)	(19)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [1]	=	0
  [k(x1, x2)]	=	0
  [*#(x1, x2)]	=	0
  [k#(x1, x2)]	=	0
  [i(x1)]	=	0
  [*(x1, x2)]	=	x1 + x2 + 1
  [i#(x1)]	=	x1 + 0

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

  i#(*(x,y))	→	i#(x)	(24)
  i#(*(x,y))	→	i#(y)	(19)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

  *#(*(i(x),k(y,z)),x)	→	*#(*(i(x),z),x)	(25)
  *#(*(i(x),k(y,z)),x)	→	*#(i(x),y)	(22)
  *#(x,*(y,z))	→	*#(*(x,y),z)	(18)
  *#(*(i(x),k(y,z)),x)	→	*#(i(x),z)	(21)
  *#(x,*(y,z))	→	*#(x,y)	(20)
  *#(*(i(x),k(y,z)),x)	→	*#(*(i(x),y),x)	(16)

  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(1)	=	0		status(1)	=	[]		list-extension(1)	=	Lex
  prec(k)	=	0		status(k)	=	[2, 1]		list-extension(k)	=	Lex
  prec(*#)	=	1		status(*#)	=	[2, 1]		list-extension(*#)	=	Lex
  prec(k#)	=	0		status(k#)	=	[2, 1]		list-extension(k#)	=	Lex
  prec(i)	=	2		status(i)	=	[1]		list-extension(i)	=	Lex
  prec(*)	=	1		status(*)	=	[2, 1]		list-extension(*)	=	Lex
  prec(i#)	=	0		status(i#)	=	[]		list-extension(i#)	=	Lex

  and the following Max-polynomial interpretation

  [1]	=	0
  [k(x1, x2)]	=	max(x1 + 0, x2 + 0, 0)
  [*#(x1, x2)]	=	max(x1 + 0, x2 + 0, 0)
  [k#(x1, x2)]	=	max(x1 + 0, x2 + 0, 0)
  [i(x1)]	=	x1 + 0
  [*(x1, x2)]	=	max(x1 + 0, x2 + 0, 0)
  [i#(x1)]	=	0

  together with the usable rules

  *(x,i(x))	→	1	(4)
  k(*(x,i(y)),*(y,i(x)))	→	1	(15)
  *(*(x,i(y)),y)	→	x	(8)
  *(x,1)	→	x	(1)
  *(i(x),x)	→	1	(3)
  *(x,*(y,z))	→	*(*(x,y),z)	(5)
  i(*(x,y))	→	*(i(y),i(x))	(10)
  *(*(x,y),i(y))	→	x	(7)
  *(*(i(x),k(y,z)),x)	→	k(*(*(i(x),y),x),*(*(i(x),z),x))	(14)
  k(x,x)	→	1	(12)
  k(x,1)	→	1	(11)
  i(i(x))	→	x	(9)
  *(k(x,y),k(y,x))	→	1	(13)
  i(1)	→	1	(6)
  *(1,y)	→	y	(2)

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

  *#(*(i(x),k(y,z)),x)	→	*#(i(x),y)	(22)
  *#(x,*(y,z))	→	*#(*(x,y),z)	(18)
  *#(*(i(x),k(y,z)),x)	→	*#(i(x),z)	(21)
  *#(x,*(y,z))	→	*#(x,y)	(20)
  *#(*(i(x),k(y,z)),x)	→	*#(*(i(x),y),x)	(16)

  could be deleted.

  1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    *#(*(i(x),k(y,z)),x)

    →

    *#(*(i(x),z),x)

    (25)

    1.1.2.1.1 Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [1]	=	1
    [k(x1, x2)]	=	x2 + 2438
    [*#(x1, x2)]	=	x1 + 0
    [k#(x1, x2)]	=	0
    [i(x1)]	=	x1 + 0
    [*(x1, x2)]	=	x1 + x2 + 20653
    [i#(x1)]	=	0

    together with the usable rules

    *(x,i(x))	→	1	(4)
    k(*(x,i(y)),*(y,i(x)))	→	1	(15)
    *(*(x,i(y)),y)	→	x	(8)
    *(x,1)	→	x	(1)
    *(i(x),x)	→	1	(3)
    *(x,*(y,z))	→	*(*(x,y),z)	(5)
    i(*(x,y))	→	*(i(y),i(x))	(10)
    *(*(x,y),i(y))	→	x	(7)
    *(*(i(x),k(y,z)),x)	→	k(*(*(i(x),y),x),*(*(i(x),z),x))	(14)
    k(x,x)	→	1	(12)
    k(x,1)	→	1	(11)
    i(i(x))	→	x	(9)
    *(k(x,y),k(y,x))	→	1	(13)
    i(1)	→	1	(6)
    *(1,y)	→	y	(2)

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

    *#(*(i(x),k(y,z)),x)

    →

    *#(*(i(x),z),x)

    (25)

    could be deleted.

    1.1.2.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.