Certification Problem

Input (TPDB TRS_Standard/AProVE_04/fac)

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 4 components.

• The 1^st component contains the pair

  fac#(s(x))

  →

  fac#(p(s(x)))

  (14)

  1.1.1 Switch to Innermost Termination

  The TRS does not have overlaps with the pairs and is locally confluent:

  20

  Hence, it suffices to show innermost termination in the following.

  1.1.1.1 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  p(s(s(x)))	→	s(p(s(x)))	(6)
  p(s(0))	→	0	(7)

  1.1.1.1.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  p(s(s(x0)))

  p(s(0))

  1.1.1.1.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [p(x1)]	=	1 · x1
  [s(x1)]	=	2 · x1
  [0]	=	2
  [fac#(x1)]	=	1 · x1

  the rules

  plus(x,0)	→	x	(1)
  plus(x,s(y))	→	s(plus(x,y))	(2)
  times(0,y)	→	0	(3)
  times(x,0)	→	0	(4)
  times(s(x),y)	→	plus(times(x,y),y)	(5)
  p(s(0))	→	0	(7)
  fac(s(x))	→	times(fac(p(s(x))),s(x))	(8)

  could be deleted.

  1.1.1.1.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [fac#(x1)]	=	x1
  [p(x1)]	=	-2 + x1
  [s(x1)]	=	2 + x1

  the pair

  fac#(s(x))

  →

  fac#(p(s(x)))

  (14)

  could be deleted.

  1.1.1.1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  times#(s(x),y)

  →

  times#(x,y)

  (11)

  1.1.2 Switch to Innermost Termination

  The TRS does not have overlaps with the pairs and is locally confluent:

  20

  Hence, it suffices to show innermost termination in the following.

  1.1.2.1 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  There are no rules.

  1.1.2.1.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  There are no lhss.

  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.

  times#(s(x),y)	→	times#(x,y)	(11)
  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

  plus#(x,s(y))

  →

  plus#(x,y)

  (9)

  1.1.3 Switch to Innermost Termination

  The TRS does not have overlaps with the pairs and is locally confluent:

  20

  Hence, it suffices to show innermost termination in the following.

  1.1.3.1 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  There are no rules.

  1.1.3.1.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  There are no lhss.

  1.1.3.1.1.1 Size-Change Termination

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

  plus#(x,s(y))	→	plus#(x,y)	(9)
  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

  p#(s(s(x)))

  →

  p#(s(x))

  (12)

  1.1.4 Switch to Innermost Termination

  The TRS does not have overlaps with the pairs and is locally confluent:

  20

  Hence, it suffices to show innermost termination in the following.

  1.1.4.1 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  There are no rules.

  1.1.4.1.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  There are no lhss.

  1.1.4.1.1.1 Size-Change Termination

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

  p#(s(s(x)))	→	p#(s(x))	(12)
  1	>	1

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