Certification Problem

Input (TPDB TRS_Standard/AG01/#3.18)

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 3 components.

• The 1^st component contains the pair

  plus#(s(x),y)	→	plus#(x,s(y))	(12)
  plus#(s(x),y)	→	plus#(x,y)	(11)
  plus#(s(x),y)	→	plus#(minus(x,y),double(y))	(13)

  1.1.1 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(plus#)	=	1
  π(s)	=	[1]
  π(minus)	=	1

  together with the usable rules

  minus(x,0)	→	x	(1)
  minus(s(x),s(y))	→	minus(x,y)	(2)

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

  plus#(s(x),y)	→	plus#(x,s(y))	(12)
  plus#(s(x),y)	→	plus#(x,y)	(11)
  plus#(s(x),y)	→	plus#(minus(x,y),double(y))	(13)

  could be deleted.

  1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

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

  →

  minus#(x,y)

  (9)

  1.1.2 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.

  1.1.2.1 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)	(9)
  1	>	1
  2	>	2

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

• The 3^rd component contains the pair

  double#(s(x))

  →

  double#(x)

  (10)

  1.1.3 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [double#(x1)]	=	1 · x1

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

  1.1.3.1 Size-Change Termination

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

  double#(s(x))	→	double#(x)	(10)
  1	>	1

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