Certification Problem

Input (TPDB TRS_Standard/Strategy_removed_AG01/#4.29)

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 4 components.

• The 1^st component contains the pair

  if_times#(false,s(x),y)	→	times#(x,y)	(21)
  times#(s(x),y)	→	if_times#(even(s(x)),s(x),y)	(20)
  if_times#(true,s(x),y)	→	times#(half(s(x)),y)	(14)
  if_times#(true,s(x),y)	→	times#(half(s(x)),y)	(14)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 8946
  [if_times#(x1, x2, x3)]	=	x2 + 0
  [even(x1)]	=	1
  [plus#(x1, x2)]	=	0
  [false]	=	2
  [half#(x1)]	=	0
  [true]	=	2
  [half(x1)]	=	x1 + 0
  [even#(x1)]	=	0
  [times#(x1, x2)]	=	x1 + 0
  [0]	=	36459
  [times(x1, x2)]	=	0
  [if_times(x1, x2, x3)]	=	0
  [plus(x1, x2)]	=	0

  together with the usable rules

  half(0)	→	0	(4)
  half(s(s(x)))	→	s(half(x))	(5)

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

  if_times#(false,s(x),y)

  →

  times#(x,y)

  (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

    if_times#(true,s(x),y)	→	times#(half(s(x)),y)	(14)
    if_times#(true,s(x),y)	→	times#(half(s(x)),y)	(14)
    times#(s(x),y)	→	if_times#(even(s(x)),s(x),y)	(20)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the argument filter

    π(if_times#)

    =

    2

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

    prec(s)	=	1		status(s)	=	[1]		list-extension(s)	=	Lex
    prec(even)	=	2		status(even)	=	[]		list-extension(even)	=	Lex
    prec(plus#)	=	0		status(plus#)	=	[1, 2]		list-extension(plus#)	=	Lex
    prec(false)	=	0		status(false)	=	[]		list-extension(false)	=	Lex
    prec(half#)	=	0		status(half#)	=	[]		list-extension(half#)	=	Lex
    prec(true)	=	2		status(true)	=	[]		list-extension(true)	=	Lex
    prec(half)	=	1		status(half)	=	[]		list-extension(half)	=	Lex
    prec(even#)	=	0		status(even#)	=	[]		list-extension(even#)	=	Lex
    prec(times#)	=	0		status(times#)	=	[1]		list-extension(times#)	=	Lex
    prec(0)	=	1		status(0)	=	[]		list-extension(0)	=	Lex
    prec(times)	=	0		status(times)	=	[1, 2]		list-extension(times)	=	Lex
    prec(if_times)	=	0		status(if_times)	=	[3, 2, 1]		list-extension(if_times)	=	Lex
    prec(plus)	=	0		status(plus)	=	[2, 1]		list-extension(plus)	=	Lex

    and the following Max-polynomial interpretation

    [s(x1)]	=	x1 + 1
    [even(x1)]	=	3
    [plus#(x1, x2)]	=	max(x1 + 1, x2 + 1, 0)
    [false]	=	0
    [half#(x1)]	=	1
    [true]	=	2
    [half(x1)]	=	x1 + 0
    [even#(x1)]	=	1
    [times#(x1, x2)]	=	max(x1 + 0, 0)
    [0]	=	1
    [times(x1, x2)]	=	max(x1 + 1, x2 + 1, 0)
    [if_times(x1, x2, x3)]	=	x1 + x2 + x3 + 1
    [plus(x1, x2)]	=	x1 + x2 + 1

    together with the usable rules

    half(0)	→	0	(4)
    half(s(s(x)))	→	s(half(x))	(5)

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

    if_times#(true,s(x),y)	→	times#(half(s(x)),y)	(14)
    if_times#(true,s(x),y)	→	times#(half(s(x)),y)	(14)
    times#(s(x),y)	→	if_times#(even(s(x)),s(x),y)	(20)

    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

  plus#(s(x),y)

  →

  plus#(x,y)

  (18)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 1
  [if_times#(x1, x2, x3)]	=	0
  [even(x1)]	=	1
  [plus#(x1, x2)]	=	x1 + 0
  [false]	=	2
  [half#(x1)]	=	0
  [true]	=	2
  [half(x1)]	=	x1 + 61160
  [even#(x1)]	=	0
  [times#(x1, x2)]	=	1
  [0]	=	11966
  [times(x1, x2)]	=	0
  [if_times(x1, x2, x3)]	=	0
  [plus(x1, x2)]	=	0

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

  plus#(s(x),y)

  →

  plus#(x,y)

  (18)

  could be deleted.

  1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 3^rd component contains the pair

  half#(s(s(x)))

  →

  half#(x)

  (15)

  1.1.3 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 1
  [if_times#(x1, x2, x3)]	=	0
  [even(x1)]	=	1
  [plus#(x1, x2)]	=	0
  [false]	=	2
  [half#(x1)]	=	x1 + 0
  [true]	=	2
  [half(x1)]	=	x1 + 61160
  [even#(x1)]	=	0
  [times#(x1, x2)]	=	1
  [0]	=	3037
  [times(x1, x2)]	=	0
  [if_times(x1, x2, x3)]	=	0
  [plus(x1, x2)]	=	0

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

  half#(s(s(x)))

  →

  half#(x)

  (15)

  could be deleted.

  1.1.3.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 4^th component contains the pair

  even#(s(s(x)))

  →

  even#(x)

  (16)

  1.1.4 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [s(x1)]	=	x1 + 1
  [if_times#(x1, x2, x3)]	=	0
  [even(x1)]	=	1
  [plus#(x1, x2)]	=	0
  [false]	=	2
  [half#(x1)]	=	0
  [true]	=	2
  [half(x1)]	=	x1 + 19428
  [even#(x1)]	=	x1 + 0
  [times#(x1, x2)]	=	1
  [0]	=	1
  [times(x1, x2)]	=	0
  [if_times(x1, x2, x3)]	=	0
  [plus(x1, x2)]	=	0

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

  even#(s(s(x)))

  →

  even#(x)

  (16)

  could be deleted.

  1.1.4.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.