Certification Problem

Input (TPDB TRS_Standard/AProVE_04/IJCAR_18)

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 5 components.

• The 1^st component contains the pair

  pr#(x,s(s(y)))	→	if#(divides(s(s(y)),x),x,s(y))	(36)
  if#(false,x,y)	→	pr#(x,y)	(38)

  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)	(38)
  2	≥	1
  3	≥	2
  pr#(x,s(s(y)))	→	if#(divides(s(s(y)),x),x,s(y))	(36)
  1	≥	2
  2	>	3

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

• The 2^nd component contains the pair

  div#(x,y)	→	quot#(x,y,y)	(26)
  quot#(s(x),s(y),z)	→	quot#(x,y,z)	(27)
  quot#(x,0,s(z))	→	div#(x,s(z))	(28)
  div#(div(x,y),z)	→	div#(x,times(y,z))	(29)

  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(div)

  =

  1

  weight(div)

  =

  1

  in combination with the following argument filter

  π(div#)	=	1
  π(quot#)	=	1
  π(s)	=	1
  π(div)	=	[1]

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

  div#(div(x,y),z)

  →

  div#(x,times(y,z))

  (29)

  could be deleted.

  1.1.2.1 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(div#)	=	1
  π(quot#)	=	1
  π(s)	=	[1]

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

  quot#(s(x),s(y),z)

  →

  quot#(x,y,z)

  (27)

  could be deleted.

  1.1.2.1.1 Reduction Pair Processor with Usable Rules

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

  prec(0)	=	1		weight(0)	=	2
  prec(s)	=	0		weight(s)	=	1

  in combination with the following argument filter

  π(div#)	=	2
  π(quot#)	=	2
  π(0)	=	[]
  π(s)	=	[]

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

  quot#(x,0,s(z))

  →

  div#(x,s(z))

  (28)

  could be deleted.

  1.1.2.1.1.1 Size-Change Termination

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

  div#(x,y)	→	quot#(x,y,y)	(26)
  1	≥	1
  2	≥	2
  2	≥	3

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

• The 3^rd component contains the pair

  times#(s(x),y)

  →

  times#(x,y)

  (25)

  1.1.3 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [times#(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.

  1.1.3.1 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)	(25)
  1	>	1
  2	≥	2

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

• The 4^th component contains the pair

  plus#(s(x),y)

  →

  plus#(x,y)

  (23)

  1.1.4 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [plus#(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.

  plus#(s(x),y)	→	plus#(x,y)	(23)
  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

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

  →

  eq#(x,y)

  (31)

  1.1.5 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [eq#(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.

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

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