Certification Problem

Input (TPDB TRS_Standard/AProVE_07/kabasci04)

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

  gcd#(s(x),s(y))

  →

  gcd#(minus(max(x,y),min(x,y)),s(min(x,y)))

  (16)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [gcd#(x1, x2)]	=	-2 + 2 · x1 + x2
  [minus(x1, x2)]	=	x1
  [max(x1, x2)]	=	x1 + x2
  [0]	=	0
  [s(x1)]	=	1 + 2 · x1
  [min(x1, x2)]	=	x1
  [any(x1)]	=	0

  together with the usable rules

  max(x,0)	→	x	(4)
  max(0,y)	→	y	(5)
  max(s(x),s(y))	→	s(max(x,y))	(6)
  min(x,0)	→	0	(1)
  min(0,y)	→	0	(2)
  min(s(x),s(y))	→	s(min(x,y))	(3)
  minus(x,0)	→	x	(7)
  minus(s(x),s(y))	→	s(minus(x,any(y)))	(8)

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

  gcd#(s(x),s(y))

  →

  gcd#(minus(max(x,y),min(x,y)),s(min(x,y)))

  (16)

  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,any(y))

  (14)

  1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

  together with the usable rules

  any(s(x))	→	s(s(any(x)))	(10)
  any(x)	→	x	(11)

  (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,any(y))	(14)
  1	>	1

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

• The 3^rd component contains the pair

  min#(s(x),s(y))

  →

  min#(x,y)

  (12)

  1.1.3 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [min#(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.

  min#(s(x),s(y))	→	min#(x,y)	(12)
  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

  max#(s(x),s(y))

  →

  max#(x,y)

  (13)

  1.1.4 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [max#(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.

  max#(s(x),s(y))	→	max#(x,y)	(13)
  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

  any#(s(x))

  →

  any#(x)

  (20)

  1.1.5 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

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

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

  1.1.5.1 Size-Change Termination

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

  any#(s(x))	→	any#(x)	(20)
  1	>	1

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