Termination Proof

by AProVE (version ba869e7b28377cd372aedcb96abeb62c4ad6aaa5 rene 20130719 unpublished dirty )

Input

The rewrite relation of the following TRS is considered.

Proof

1 Rule Removal

1.1 Dependency Pair Transformation

1.1.1 Dependency Graph Processor

The dependency pairs are split into 5 components.

• The 1^st component contains the pairs

  Tcpcx#(SRcp(x))	→	Tcpcx#(x)
  Tcpcx#(SRlbeta(x))	→	Tcpcx#(x)
  Tcpcx#(SRllet(x))	→	Tcpcx#(x)
  Tcpcx#(SRlapp(x))	→	Tcpcx#(x)
  Tcpcx#(SRlcase(x))	→	Tcpcx#(x)
  Tcpcx#(SRlseq(x))	→	Tcpcx#(x)
  Tcpcx#(SRcase(x))	→	Tcpcx#(x)
  Tcpcx#(SRseq(x))	→	Tcpcx#(x)
  Tcpcx#(SRcp(x))	→	Tcpcx#(Tcpcx(x))

  1.1.1.1 Innermost Lhss Removal Processor

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

  Tcpcx(SRlbeta(x0))

  Tcpcx(SRcp(x0))

  Tcpcx(SRllet(x0))

  Tcpcx(SRlapp(x0))

  Tcpcx(SRlcase(x0))

  Tcpcx(SRlseq(x0))

  Tcpcx(SRcase(x0))

  Tcpcx(SRseq(x0))

  Tcpcx(Answer)

  Tabs(SRlbeta(x0))

  Tabs(SRcp(x0))

  Tabs(SRllet(x0))

  Tabs(SRlapp(x0))

  Tabs(SRlcase(x0))

  Tabs(SRlseq(x0))

  Tabs(SRcase(x0))

  Tabs(SRseq(x0))

  Tabs(Answer)

  Tcpx(SRlbeta(x0))

  Tcpx(SRcp(x0))

  Tcpx(SRllet(x0))

  Tcpx(SRlapp(x0))

  Tcpx(SRlcase(x0))

  Tcpx(SRlseq(x0))

  Tcpx(SRcase(x0))

  Tcpx(SRseq(x0))

  Tcpx(Answer)

  Txch(SRlbeta(x0))

  Txch(SRcp(x0))

  Txch(SRllet(x0))

  Txch(SRlapp(x0))

  Txch(SRlcase(x0))

  Txch(SRlseq(x0))

  Txch(SRcase(x0))

  Txch(SRseq(x0))

  Txch(Answer)

  Tgc(SRlbeta(x0))

  Tgc(SRcp(x0))

  Tgc(SRllet(x0))

  Tgc(SRlapp(x0))

  Tgc(SRlcase(x0))

  Tgc(SRlseq(x0))

  Tgc(SRcase(x0))

  Tgc(SRseq(x0))

  Tgc(Answer)

  1.1.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [Tcpcx#(x1)]	=	1 · x1
  [SRcp(x1)]	=	1 · x1
  [SRlbeta(x1)]	=	1 + 1 · x1
  [SRllet(x1)]	=	1 + 1 · x1
  [SRlapp(x1)]	=	1 + 1 · x1
  [SRlcase(x1)]	=	1 + 1 · x1
  [SRlseq(x1)]	=	1 + 1 · x1
  [SRcase(x1)]	=	1 + 1 · x1
  [SRseq(x1)]	=	1 + 1 · x1
  [Tcpcx(x1)]	=	1 · x1
  [Answer]	=	1
  [Tabs(x1)]	=	1 · x1
  [Txch(x1)]	=	1 · x1
  [Tcpx(x1)]	=	1 · x1
  [Tgc(x1)]	=	1 · x1

  the pairs

  Tcpcx#(SRcp(x))	→	Tcpcx#(x)
  Tcpcx#(SRcp(x))	→	Tcpcx#(Tcpcx(x))

  remain.

  1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [Tcpcx#(x1)]	=	1 · x1
  [SRcp(x1)]	=	1 + 1 · x1
  [Tcpcx(x1)]	=	1 · x1
  [SRlbeta(x1)]	=	1 + 1 · x1
  [SRllet(x1)]	=	1
  [SRlapp(x1)]	=	1 + 1 · x1
  [SRlcase(x1)]	=	1 · x1
  [SRlseq(x1)]	=	1 · x1
  [SRcase(x1)]	=	0
  [SRseq(x1)]	=	1
  [Answer]	=	1
  [Tabs(x1)]	=	1 + 1 · x1
  [Txch(x1)]	=	1 + 1 · x1
  [Tcpx(x1)]	=	1 · x1
  [Tgc(x1)]	=	1 · x1

  together with the usable rules

  Tcpcx(SRlbeta(x))	→	SRlbeta(Tcpcx(x))
  Tcpcx(SRcp(x))	→	SRcp(Tcpcx(x))
  Tcpcx(SRllet(x))	→	SRllet(Tcpcx(x))
  Tcpcx(SRlapp(x))	→	SRlapp(Tcpcx(x))
  Tcpcx(SRlcase(x))	→	SRlcase(Tcpcx(x))
  Tcpcx(SRlseq(x))	→	SRlseq(Tcpcx(x))
  Tcpcx(SRcase(x))	→	SRcase(Tcpcx(x))
  Tcpcx(SRcase(x))	→	SRcase(x)
  Tcpcx(SRseq(x))	→	SRseq(Tcpcx(x))
  Tcpcx(SRseq(x))	→	SRseq(x)
  Tcpcx(Answer)	→	Answer
  Tcpcx(SRcase(x))	→	SRcase(Tabs(x))
  Tcpcx(SRcase(x))	→	SRcase(Txch(Txch(Tcpx(Tcpx(Tcpcx(x))))))
  Tcpcx(SRcase(x))	→	SRcase(Txch(Txch(Tcpx(Tcpx(Tabs(x))))))
  Tcpcx(SRseq(x))	→	SRseq(Tabs(x))
  Tcpcx(SRcp(x))	→	SRcp(Tcpcx(Tcpcx(x)))
  Tcpcx(SRcp(x))	→	SRcp(Tgc(Tcpx(Tcpx(Tcpx(Tcpx(Tcpcx(Tcpcx(x))))))))
  Tcpx(SRlbeta(x))	→	SRlbeta(Tcpx(x))
  Tcpx(SRcp(x))	→	SRcp(Tcpx(x))
  Tcpx(SRcp(x))	→	SRcp(x)
  Tcpx(SRllet(x))	→	SRllet(Tcpx(x))
  Tcpx(SRlapp(x))	→	SRlapp(Tcpx(x))
  Tcpx(SRlcase(x))	→	SRlcase(Tcpx(x))
  Tcpx(SRlseq(x))	→	SRlseq(Tcpx(x))
  Tcpx(SRcase(x))	→	SRcase(Tcpx(x))
  Tcpx(SRcase(x))	→	SRcase(x)
  Tcpx(SRseq(x))	→	SRseq(Tcpx(x))
  Tcpx(SRseq(x))	→	SRseq(x)
  Tcpx(Answer)	→	Answer
  Tcpx(SRcp(x))	→	SRcp(Tcpx(Tcpx(x)))
  Tgc(SRlbeta(x))	→	SRlbeta(Tgc(x))
  Tgc(SRcp(x))	→	SRcp(Tgc(x))
  Tgc(SRllet(x))	→	SRllet(Tgc(x))
  Tgc(SRlapp(x))	→	SRlapp(Tgc(x))
  Tgc(SRlcase(x))	→	SRlcase(Tgc(x))
  Tgc(SRlseq(x))	→	SRlseq(Tgc(x))
  Tgc(SRcase(x))	→	SRcase(Tgc(x))
  Tgc(SRcase(x))	→	SRcase(x)
  Tgc(SRseq(x))	→	SRseq(Tgc(x))
  Tgc(SRseq(x))	→	SRseq(x)
  Tgc(Answer)	→	Answer

  (w.r.t. the implicit argument filter of the reduction pair), all pairs could be removed.

  1.1.1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pairs

  Tabs#(SRcp(x))	→	Tabs#(x)
  Tabs#(SRlbeta(x))	→	Tabs#(x)
  Tabs#(SRllet(x))	→	Tabs#(x)
  Tabs#(SRlapp(x))	→	Tabs#(x)
  Tabs#(SRlcase(x))	→	Tabs#(x)
  Tabs#(SRlseq(x))	→	Tabs#(x)
  Tabs#(SRcase(x))	→	Tabs#(x)
  Tabs#(SRseq(x))	→	Tabs#(x)

  1.1.1.2 Usable Rules Processor

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

  There are no rules.

  1.1.1.2.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.2.1.1 Size-Change Termination

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

  • Tabs^#(SRcp(x)) → Tabs^#(x):

    1>1

  • Tabs^#(SRlbeta(x)) → Tabs^#(x):

    1>1

  • Tabs^#(SRllet(x)) → Tabs^#(x):

    1>1

  • Tabs^#(SRlapp(x)) → Tabs^#(x):

    1>1

  • Tabs^#(SRlcase(x)) → Tabs^#(x):

    1>1

  • Tabs^#(SRlseq(x)) → Tabs^#(x):

    1>1

  • Tabs^#(SRcase(x)) → Tabs^#(x):

    1>1

  • Tabs^#(SRseq(x)) → Tabs^#(x):

    1>1

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

• The 3^rd component contains the pairs

  Tcpx#(SRcp(x))	→	Tcpx#(x)
  Tcpx#(SRlbeta(x))	→	Tcpx#(x)
  Tcpx#(SRllet(x))	→	Tcpx#(x)
  Tcpx#(SRlapp(x))	→	Tcpx#(x)
  Tcpx#(SRlcase(x))	→	Tcpx#(x)
  Tcpx#(SRlseq(x))	→	Tcpx#(x)
  Tcpx#(SRcase(x))	→	Tcpx#(x)
  Tcpx#(SRseq(x))	→	Tcpx#(x)
  Tcpx#(SRcp(x))	→	Tcpx#(Tcpx(x))

  1.1.1.3 Usable Rules Processor

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

  Tcpx(SRlbeta(x))	→	SRlbeta(Tcpx(x))
  Tcpx(SRcp(x))	→	SRcp(Tcpx(x))
  Tcpx(SRcp(x))	→	SRcp(x)
  Tcpx(SRllet(x))	→	SRllet(Tcpx(x))
  Tcpx(SRlapp(x))	→	SRlapp(Tcpx(x))
  Tcpx(SRlcase(x))	→	SRlcase(Tcpx(x))
  Tcpx(SRlseq(x))	→	SRlseq(Tcpx(x))
  Tcpx(SRcase(x))	→	SRcase(Tcpx(x))
  Tcpx(SRcase(x))	→	SRcase(x)
  Tcpx(SRseq(x))	→	SRseq(Tcpx(x))
  Tcpx(SRseq(x))	→	SRseq(x)
  Tcpx(Answer)	→	Answer
  Tcpx(SRcp(x))	→	SRcp(Tcpx(Tcpx(x)))

  1.1.1.3.1 Innermost Lhss Removal Processor

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

  Tcpx(SRlbeta(x0))

  Tcpx(SRcp(x0))

  Tcpx(SRllet(x0))

  Tcpx(SRlapp(x0))

  Tcpx(SRlcase(x0))

  Tcpx(SRlseq(x0))

  Tcpx(SRcase(x0))

  Tcpx(SRseq(x0))

  Tcpx(Answer)

  1.1.1.3.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [Tcpx#(x1)]	=	1 · x1
  [SRcp(x1)]	=	1 · x1
  [SRlbeta(x1)]	=	1 + 1 · x1
  [SRllet(x1)]	=	1 + 1 · x1
  [SRlapp(x1)]	=	1 + 1 · x1
  [SRlcase(x1)]	=	1 + 1 · x1
  [SRlseq(x1)]	=	1 + 1 · x1
  [SRcase(x1)]	=	1 + 1 · x1
  [SRseq(x1)]	=	1 + 1 · x1
  [Tcpx(x1)]	=	1 · x1
  [Answer]	=	1

  the pairs

  Tcpx#(SRcp(x))	→	Tcpx#(x)
  Tcpx#(SRcp(x))	→	Tcpx#(Tcpx(x))

  remain.

  1.1.1.3.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [Tcpx#(x1)]	=	1 · x1
  [SRcp(x1)]	=	1 + 1 · x1
  [Tcpx(x1)]	=	1 · x1
  [SRlbeta(x1)]	=	1 + 1 · x1
  [SRllet(x1)]	=	1 + 1 · x1
  [SRlapp(x1)]	=	1 + 1 · x1
  [SRlcase(x1)]	=	1 + 1 · x1
  [SRlseq(x1)]	=	1 + 1 · x1
  [SRcase(x1)]	=	1 + 1 · x1
  [SRseq(x1)]	=	1 + 1 · x1
  [Answer]	=	1

  all pairs could be removed.

  1.1.1.3.1.1.1.1 P is empty

  There are no pairs anymore.

• The 4^th component contains the pairs

  Txch#(SRcp(x))	→	Txch#(x)
  Txch#(SRlbeta(x))	→	Txch#(x)
  Txch#(SRllet(x))	→	Txch#(x)
  Txch#(SRlapp(x))	→	Txch#(x)
  Txch#(SRlcase(x))	→	Txch#(x)
  Txch#(SRlseq(x))	→	Txch#(x)
  Txch#(SRcase(x))	→	Txch#(x)
  Txch#(SRseq(x))	→	Txch#(x)

  1.1.1.4 Usable Rules Processor

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

  There are no rules.

  1.1.1.4.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.4.1.1 Size-Change Termination

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

  • Txch^#(SRcp(x)) → Txch^#(x):

    1>1

  • Txch^#(SRlbeta(x)) → Txch^#(x):

    1>1

  • Txch^#(SRllet(x)) → Txch^#(x):

    1>1

  • Txch^#(SRlapp(x)) → Txch^#(x):

    1>1

  • Txch^#(SRlcase(x)) → Txch^#(x):

    1>1

  • Txch^#(SRlseq(x)) → Txch^#(x):

    1>1

  • Txch^#(SRcase(x)) → Txch^#(x):

    1>1

  • Txch^#(SRseq(x)) → Txch^#(x):

    1>1

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

• The 5^th component contains the pairs

  Tgc#(SRcp(x))	→	Tgc#(x)
  Tgc#(SRlbeta(x))	→	Tgc#(x)
  Tgc#(SRllet(x))	→	Tgc#(x)
  Tgc#(SRlapp(x))	→	Tgc#(x)
  Tgc#(SRlcase(x))	→	Tgc#(x)
  Tgc#(SRlseq(x))	→	Tgc#(x)
  Tgc#(SRcase(x))	→	Tgc#(x)
  Tgc#(SRseq(x))	→	Tgc#(x)

  1.1.1.5 Usable Rules Processor

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

  There are no rules.

  1.1.1.5.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.5.1.1 Size-Change Termination

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

  • Tgc^#(SRcp(x)) → Tgc^#(x):

    1>1

  • Tgc^#(SRlbeta(x)) → Tgc^#(x):

    1>1

  • Tgc^#(SRllet(x)) → Tgc^#(x):

    1>1

  • Tgc^#(SRlapp(x)) → Tgc^#(x):

    1>1

  • Tgc^#(SRlcase(x)) → Tgc^#(x):

    1>1

  • Tgc^#(SRlseq(x)) → Tgc^#(x):

    1>1

  • Tgc^#(SRcase(x)) → Tgc^#(x):

    1>1

  • Tgc^#(SRseq(x)) → Tgc^#(x):

    1>1

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