Certification Problem

Input (TPDB TRS_Standard/AProVE_04/IJCAR_26)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 5 components.

• The 1^st component contains the pair

  if#(false,x,y)	→	pr#(x,y)	(45)
  pr#(x,s(s(y)))	→	if#(divides(s(s(y)),x),x,s(y))	(44)

  1.1.1 Size-Change Termination

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

  if#(false,x,y)	→	pr#(x,y)	(45)
  3	≥	2
  2	≥	1
  pr#(x,s(s(y)))	→	if#(divides(s(s(y)),x),x,s(y))	(44)
  2	>	3
  1	≥	2

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

• The 2^nd component contains the pair

  eq#(s(x),s(y))

  →

  eq#(x,y)

  (38)

  1.1.2 Size-Change Termination

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

  eq#(s(x),s(y))	→	eq#(x,y)	(38)
  2	>	2
  1	>	1

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

• The 3^rd component contains the pair

  div#(x,y)	→	quot#(x,y,y)	(33)
  quot#(s(x),s(y),z)	→	quot#(x,y,z)	(34)
  quot#(x,0,s(z))	→	div#(x,s(z))	(35)
  div#(div(x,y),z)	→	div#(x,times(y,z))	(37)

  1.1.3 Subterm Criterion Processor

  We use the projection

  π(quot#)	=	1
  π(div#)	=	1

  and remove the pairs:

  quot#(s(x),s(y),z)	→	quot#(x,y,z)	(34)
  div#(div(x,y),z)	→	div#(x,times(y,z))	(37)

  1.1.3.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the arctic semiring over the integers

  [quot#(x1, x2, x3)]	=	-∞ · x1 + 0 · x2 + 0 · x3 + -∞
  [div#(x1, x2)]	=	-∞ · x1 + 1 · x2 + -16
  [s(x1)]	=	-∞ · x1 + 3
  [0]	=	4

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

  div#(x,y)

  →

  quot#(x,y,y)

  (33)

  could be deleted.

  1.1.3.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 4^th component contains the pair

  times#(s(x),y)

  →

  times#(x,y)

  (31)

  1.1.4 Size-Change Termination

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

  times#(s(x),y)	→	times#(x,y)	(31)
  2	≥	2
  1	>	1

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

• The 5^th component contains the pair

  plus#(s(x),y)	→	plus#(x,y)	(26)
  plus#(s(x),y)	→	plus#(p(s(x)),y)	(28)
  plus#(x,s(y))	→	plus#(x,p(s(y)))	(30)

  1.1.5 Subterm Criterion Processor

  We use the projection to multisets

  π(plus#)	=	{ 1 }
  π(p)	=	{ 1 }

  to remove the pairs:

  plus#(s(x),y)

  →

  plus#(x,y)

  (26)

  1.1.5.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the naturals

  [plus#(x1, x2)]	=	0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 · x1 + 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 · x2 + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  [p(x1)]	=	1 0 1 0 0 0 0 1 0 1 0 1 1 0 0 0 · x1 + 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
  [s(x1)]	=	1 0 0 1 0 1 1 0 1 0 0 0 0 1 1 0 · x1 + 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

  together with the usable rule

  p(s(x))

  →

  x

  (1)

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

  plus#(x,s(y))

  →

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

  (30)

  could be deleted.

  1.1.5.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the naturals

  [plus#(x1, x2)]	=	0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 · x1 + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 · x2 + 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  [p(x1)]	=	0 1 0 0 1 0 0 0 0 0 1 0 1 0 1 1 · x1 + 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0
  [s(x1)]	=	1 1 1 0 1 1 1 0 0 1 1 1 0 1 1 0 · x1 + 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

  together with the usable rule

  p(s(x))

  →

  x

  (1)

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

  plus#(s(x),y)

  →

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

  (28)

  could be deleted.

  1.1.5.1.1.1 P is empty

  There are no pairs anymore.