Certification Problem

Input (TPDB TRS_Standard/AProVE_07/wiehe12)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 8 components.

• The 1^st component contains the pair

  g#(s(x),s(y))	→	g#(minus(m(x,y),n(x,y)),n(s(x),s(y)))	(42)
  g#(s(x),s(y))	→	g#(k(minus(m(x,y),n(x,y)),s(s(0))),k(n(s(x),s(y)),s(s(0))))	(35)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [g#(x1, x2)]	=	-2 + 2 · x1 + x2
  [minus(x1, x2)]	=	x1
  [m(x1, x2)]	=	x1 + x2
  [0]	=	0
  [s(x1)]	=	1 + 2 · x1
  [n(x1, x2)]	=	x1
  [k(x1, x2)]	=	x1

  together with the usable rules

  m(0,y)	→	y	(5)
  m(x,0)	→	x	(6)
  m(s(x),s(y))	→	s(m(x,y))	(7)
  n(0,y)	→	0	(2)
  n(x,0)	→	0	(3)
  n(s(x),s(y))	→	s(n(x,y))	(4)
  minus(x,0)	→	x	(15)
  minus(s(x),s(y))	→	minus(x,y)	(16)
  k(0,s(y))	→	0	(8)
  k(s(x),s(y))	→	s(k(minus(x,y),s(y)))	(9)

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

  g#(s(x),s(y))	→	g#(minus(m(x,y),n(x,y)),n(s(x),s(y)))	(42)
  g#(s(x),s(y))	→	g#(k(minus(m(x,y),n(x,y)),s(s(0))),k(n(s(x),s(y)),s(s(0))))	(35)

  could be deleted.

  1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  k#(s(x),s(y))

  →

  k#(minus(x,y),s(y))

  (45)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions

  prec(s)

  =

  0

  weight(s)

  =

  1

  in combination with the following argument filter

  π(k#)	=	1
  π(s)	=	[1]
  π(minus)	=	1

  together with the usable rules

  minus(x,0)	→	x	(15)
  minus(s(x),s(y))	→	minus(x,y)	(16)

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

  k#(s(x),s(y))

  →

  k#(minus(x,y),s(y))

  (45)

  could be deleted.

  1.1.2.1 P is empty

  There are no pairs anymore.

• The 3^rd component contains the pair

  p#(s(x),x)	→	p#(if(gt(x,x),id(x),id(x)),s(x))	(55)
  p#(s(x),s(y))	→	p#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))	(48)
  p#(id(x),s(y))	→	p#(x,if(gt(s(y),y),y,s(y)))	(59)

  1.1.3 Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

  [p#(x1, x2)]	=	0 + 1 · x1 + 0 · x2
  [s(x1)]	=	0 + 4 · x1
  [if(x1, x2, x3)]	=	-∞ + 0 · x1 + 0 · x2 + 0 · x3
  [gt(x1, x2)]	=	-∞ + 0 · x1 + 0 · x2
  [id(x1)]	=	-∞ + 0 · x1
  [not(x1)]	=	1 + 4 · x1
  [0]	=	1
  [true]	=	0
  [false]	=	0

  together with the usable rules

  gt(s(x),0)	→	true	(27)
  gt(0,y)	→	false	(28)
  gt(s(x),s(y))	→	gt(x,y)	(29)
  id(x)	→	x	(17)
  if(true,x,y)	→	x	(18)
  if(false,x,y)	→	y	(19)
  not(x)	→	if(x,false,true)	(20)

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

  p#(s(x),x)

  →

  p#(if(gt(x,x),id(x),id(x)),s(x))

  (55)

  could be deleted.

  1.1.3.1 Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

  [p#(x1, x2)]	=	0 + 1 · x1 + 0 · x2
  [s(x1)]	=	0 + 2 · x1
  [if(x1, x2, x3)]	=	-∞ + -∞ · x1 + 0 · x2 + 0 · x3
  [gt(x1, x2)]	=	-∞ + 0 · x1 + 1 · x2
  [not(x1)]	=	4 + 2 · x1
  [id(x1)]	=	-∞ + 1 · x1
  [0]	=	0
  [true]	=	0
  [false]	=	0

  together with the usable rules

  if(true,x,y)	→	x	(18)
  if(false,x,y)	→	y	(19)
  id(x)	→	x	(17)

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

  p#(s(x),s(y))

  →

  p#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))

  (48)

  could be deleted.

  1.1.3.1.1 Size-Change Termination

  Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

  p#(id(x),s(y))	→	p#(x,if(gt(s(y),y),y,s(y)))	(59)
  1	>	1

  As there is no critical graph in the transitive closure, there are no infinite chains.

• The 4^th component contains the pair

  n#(s(x),s(y))

  →

  n#(x,y)

  (43)

  1.1.4 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [n#(x1, x2)]	=	1 · x1 + 1 · x2

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

  1.1.4.1 Size-Change Termination

  Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

  n#(s(x),s(y))	→	n#(x,y)	(43)
  1	>	1
  2	>	2

  As there is no critical graph in the transitive closure, there are no infinite chains.

• The 5^th component contains the pair

  m#(s(x),s(y))

  →

  m#(x,y)

  (44)

  1.1.5 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [m#(x1, x2)]	=	1 · x1 + 1 · x2

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

  1.1.5.1 Size-Change Termination

  Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

  m#(s(x),s(y))	→	m#(x,y)	(44)
  1	>	1
  2	>	2

  As there is no critical graph in the transitive closure, there are no infinite chains.

• The 6^th component contains the pair

  minus#(s(x),s(y))

  →

  minus#(x,y)

  (62)

  1.1.6 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [minus#(x1, x2)]	=	1 · x1 + 1 · x2

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

  1.1.6.1 Size-Change Termination

  Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

  minus#(s(x),s(y))	→	minus#(x,y)	(62)
  1	>	1
  2	>	2

  As there is no critical graph in the transitive closure, there are no infinite chains.

• The 7^th component contains the pair

  h#(s(x))	→	f#(x)	(65)
  f#(s(x))	→	h#(x)	(64)

  1.1.7 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [f#(x1)]	=	1 · x1
  [h#(x1)]	=	1 · x1

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

  1.1.7.1 Size-Change Termination

  Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

  f#(s(x))	→	h#(x)	(64)
  1	>	1
  h#(s(x))	→	f#(x)	(65)
  1	>	1

  As there is no critical graph in the transitive closure, there are no infinite chains.

• The 8^th component contains the pair

  gt#(s(x),s(y))

  →

  gt#(x,y)

  (66)

  1.1.8 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [gt#(x1, x2)]	=	1 · x1 + 1 · x2

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

  1.1.8.1 Size-Change Termination

  Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

  gt#(s(x),s(y))	→	gt#(x,y)	(66)
  1	>	1
  2	>	2

  As there is no critical graph in the transitive closure, there are no infinite chains.