Termination Proof

by AProVE (version ba869e7b28377cd372aedcb96abeb62c4ad6aaa5 rene 20130719 unpublished dirty )

Input

The rewrite relation of the following TRS is considered.

Proof

1 Removal of non-applicable rules

There are no rules.

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

  1.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [Tcpcx#(x1)]	=	1 · x1
  [SRlapp(x1)]	=	1 · x1
  [SRlbeta(x1)]	=	1 + 1 · x1
  [SRlcase(x1)]	=	1 · x1
  [SRlseq(x1)]	=	1 · x1
  [SRcase(x1)]	=	1 · x1
  [SRseq(x1)]	=	1 + 1 · x1
  [SRllet(x1)]	=	1 · x1
  [SRcp(x1)]	=	1 · x1
  [Tcpx(x1)]	=	1 · x1
  [Txch(x1)]	=	1 · x1
  [Tcpcx(x1)]	=	1 · x1
  [Tgc(x1)]	=	1 · x1
  [Answer]	=	1
  [Tabs(x1)]	=	1 · x1
  [W273(x1, x2)]	=	1 + 1 · x2
  [W279(x1, x2)]	=	1 + 1 · x2
  [W301(x1, x2)]	=	1 · x2
  [W316(x1, x2)]	=	1 · x2
  [W321(x1, x2)]	=	1 · x2
  [W327(x1, x2)]	=	1 · x2
  [W349(x1, x2)]	=	1 · x2
  [W355(x1, x2)]	=	1 · x2
  [W377(x1, x2)]	=	1 · x2
  [W383(x1, x2)]	=	1 · x2
  [W405(x1, x2)]	=	1 · x2
  [W411(x1, x2)]	=	1 · x2
  [W438(x1, x2)]	=	1 + 1 · x2
  [W444(x1, x2)]	=	1 + 1 · x2
  [s(x1)]	=	1 + 1 · x1
  [SRlll(x1)]	=	1 · x1

  the pairs

  Tcpcx#(SRlapp(x))	→	Tcpcx#(x)
  Tcpcx#(SRlcase(x))	→	Tcpcx#(x)
  Tcpcx#(SRlseq(x))	→	Tcpcx#(x)
  Tcpcx#(SRcase(x))	→	Tcpcx#(x)
  Tcpcx#(SRllet(x))	→	Tcpcx#(x)
  Tcpcx#(SRcp(x))	→	Tcpcx#(x)
  Tcpcx#(SRcase(x))	→	Tcpcx#(Tcpx(Tcpx(Txch(Txch(x)))))
  Tcpcx#(SRcp(x))	→	Tcpcx#(Tcpcx(x))
  Tcpcx#(SRcp(x))	→	Tcpcx#(Tcpcx(Tcpx(Tcpx(Tcpx(Tcpx(Tgc(x)))))))
  Tcpcx#(SRcp(x))	→	Tcpcx#(Tcpx(Tcpx(Tcpx(Tcpx(Tgc(x))))))

  remain.

  1.1.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [Tcpcx#(x1)]	=	1 · x1
  [SRlapp(x1)]	=	1 · x1
  [SRlcase(x1)]	=	1 · x1
  [SRlseq(x1)]	=	1 · x1
  [SRcase(x1)]	=	1 + 1 · x1
  [SRllet(x1)]	=	1 · x1
  [SRcp(x1)]	=	1 · x1
  [Tcpx(x1)]	=	1 · x1
  [Txch(x1)]	=	1 · x1
  [Tcpcx(x1)]	=	1 · x1
  [Tgc(x1)]	=	1 · x1
  [SRlbeta(x1)]	=	1 + 1 · x1
  [SRseq(x1)]	=	1 + 1 · x1
  [Answer]	=	1
  [Tabs(x1)]	=	1 · x1
  [W273(x1, x2)]	=	1 + 1 · x2
  [W279(x1, x2)]	=	1 + 1 · x2
  [W301(x1, x2)]	=	1 · x2
  [W316(x1, x2)]	=	1 · x2
  [W321(x1, x2)]	=	1 · x2
  [W327(x1, x2)]	=	1 · x2
  [W349(x1, x2)]	=	1 · x2
  [W355(x1, x2)]	=	1 · x2
  [W377(x1, x2)]	=	1 · x2
  [W383(x1, x2)]	=	1 · x2
  [W405(x1, x2)]	=	1 + 1 · x2
  [W411(x1, x2)]	=	1 + 1 · x2
  [W438(x1, x2)]	=	1 + 1 · x2
  [W444(x1, x2)]	=	1 + 1 · x2
  [s(x1)]	=	1 + 1 · x1
  [SRlll(x1)]	=	1 · x1

  the pairs

  Tcpcx#(SRlapp(x))	→	Tcpcx#(x)
  Tcpcx#(SRlcase(x))	→	Tcpcx#(x)
  Tcpcx#(SRlseq(x))	→	Tcpcx#(x)
  Tcpcx#(SRllet(x))	→	Tcpcx#(x)
  Tcpcx#(SRcp(x))	→	Tcpcx#(x)
  Tcpcx#(SRcp(x))	→	Tcpcx#(Tcpcx(x))
  Tcpcx#(SRcp(x))	→	Tcpcx#(Tcpcx(Tcpx(Tcpx(Tcpx(Tcpx(Tgc(x)))))))
  Tcpcx#(SRcp(x))	→	Tcpcx#(Tcpx(Tcpx(Tcpx(Tcpx(Tgc(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
  [SRlapp(x1)]	=	1 · x1
  [SRlcase(x1)]	=	1 · x1
  [SRlseq(x1)]	=	1 · x1
  [SRllet(x1)]	=	1 · x1
  [SRcp(x1)]	=	1 + 1 · x1
  [Tcpcx(x1)]	=	1 · x1
  [Tcpx(x1)]	=	1 · x1
  [Tgc(x1)]	=	1 · x1
  [SRlbeta(x1)]	=	1
  [SRcase(x1)]	=	0
  [SRseq(x1)]	=	0
  [Answer]	=	1
  [Tabs(x1)]	=	1 + 1 · x1
  [Txch(x1)]	=	1 + 1 · x1
  [W273(x1, x2)]	=	1
  [W279(x1, x2)]	=	1
  [W301(x1, x2)]	=	1 + 1 · x2
  [W316(x1, x2)]	=	1 · x2
  [W321(x1, x2)]	=	1 · x2
  [W327(x1, x2)]	=	1 · x2
  [W349(x1, x2)]	=	1 · x2
  [W355(x1, x2)]	=	1 · x2
  [W377(x1, x2)]	=	1 · x2
  [W383(x1, x2)]	=	1 · x2
  [W405(x1, x2)]	=	0
  [W411(x1, x2)]	=	0
  [W438(x1, x2)]	=	0
  [W444(x1, x2)]	=	0
  [s(x1)]	=	1 + 1 · x1
  [SRlll(x1)]	=	1 · x1

  together with the usable rules

  Tcpcx(SRlbeta(x))	→	SRlbeta(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(SRllet(x))	→	SRllet(Tcpcx(x))
  Tcpcx(SRcp(x))	→	SRcp(Tcpcx(x))
  Tcpcx(SRcase(x))	→	SRcase(Tabs(x))
  Tcpcx(SRcase(x))	→	SRcase(Tcpcx(Tcpx(Tcpx(Txch(Txch(x))))))
  Tcpcx(SRcase(x))	→	SRcase(Tabs(Tcpx(Tcpx(Txch(Txch(x))))))
  Tcpcx(SRseq(x))	→	SRseq(Tabs(x))
  Tcpcx(SRcp(x))	→	SRcp(Tcpcx(Tcpcx(x)))
  Tcpcx(SRcp(x))	→	SRcp(Tcpcx(Tcpcx(Tcpx(Tcpx(Tcpx(Tcpx(Tgc(x))))))))
  Tgc(SRlbeta(x))	→	SRlapp(W273(k,x))
  Tgc(SRlbeta(x))	→	W279(k,x)
  Tgc(SRcp(x))	→	W301(k,x)
  Tgc(SRllet(x))	→	W316(k,x)
  Tgc(SRlapp(x))	→	SRlapp(W321(k,x))
  Tgc(SRlapp(x))	→	W327(k,x)
  Tgc(SRlcase(x))	→	SRlcase(W349(k,x))
  Tgc(SRlcase(x))	→	W355(k,x)
  Tgc(SRlseq(x))	→	SRlseq(W377(k,x))
  Tgc(SRlseq(x))	→	W383(k,x)
  Tgc(SRcase(x))	→	SRlcase(W405(k,x))
  Tgc(SRcase(x))	→	W411(k,x)
  Tgc(SRseq(x))	→	SRlseq(W438(k,x))
  Tgc(SRseq(x))	→	W444(k,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
  Tgc(SRlbeta(x))	→	SRlapp(SRlbeta(SRllet(Tgc(x))))
  Tgc(SRlbeta(x))	→	SRlapp(SRllet(SRlbeta(Tgc(x))))
  Tgc(SRlbeta(x))	→	SRllet(SRlbeta(Tgc(x)))
  Tgc(SRcp(x))	→	SRllet(SRcp(Tgc(x)))
  Tgc(SRllet(x))	→	SRllet(SRllet(Tgc(x)))
  Tgc(SRlapp(x))	→	SRlapp(SRlapp(SRllet(Tgc(x))))
  Tgc(SRlapp(x))	→	SRlapp(SRllet(SRlapp(Tgc(x))))
  Tgc(SRlapp(x))	→	SRllet(SRlapp(Tgc(x)))
  Tgc(SRlcase(x))	→	SRlcase(SRlcase(SRllet(Tgc(x))))
  Tgc(SRlcase(x))	→	SRlcase(SRllet(SRlcase(Tgc(x))))
  Tgc(SRlcase(x))	→	SRllet(SRlcase(Tgc(x)))
  Tgc(SRlseq(x))	→	SRlseq(SRlseq(SRllet(Tgc(x))))
  Tgc(SRlseq(x))	→	SRlseq(SRllet(SRlseq(Tgc(x))))
  Tgc(SRlseq(x))	→	SRllet(SRlseq(Tgc(x)))
  Tgc(SRcase(x))	→	SRlcase(SRcase(Tgc(x)))
  Tgc(SRcase(x))	→	SRlcase(SRcase(SRllet(Tgc(x))))
  Tgc(SRcase(x))	→	SRlcase(SRllet(SRcase(Tgc(x))))
  Tgc(SRcase(x))	→	SRllet(SRcase(Tgc(x)))
  Tgc(SRseq(x))	→	SRlseq(SRseq(Tgc(x)))
  Tgc(SRseq(x))	→	SRlseq(SRllet(SRseq(Tgc(x))))
  Tgc(SRseq(x))	→	SRllet(SRseq(Tgc(x)))
  Tgc(Answer)	→	SRllet(Answer)
  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)))
  W273(s(k),x)	→	SRlll(W273(k,x))
  W273(s(k),x)	→	SRlll(SRlbeta(Tgc(x)))
  W279(s(k),x)	→	SRlll(W279(k,x))
  W279(s(k),x)	→	SRlll(SRlbeta(Tgc(x)))
  W301(s(k),x)	→	SRlll(W301(k,x))
  W301(s(k),x)	→	SRlll(SRcp(Tgc(x)))
  W316(s(k),x)	→	SRlll(W316(k,x))
  W316(s(k),x)	→	SRlll(SRllet(Tgc(x)))
  W321(s(k),x)	→	SRlll(W321(k,x))
  W321(s(k),x)	→	SRlll(SRlapp(Tgc(x)))
  W327(s(k),x)	→	SRlll(W327(k,x))
  W327(s(k),x)	→	SRlll(SRlapp(Tgc(x)))
  W349(s(k),x)	→	SRlll(W349(k,x))
  W349(s(k),x)	→	SRlll(SRlcase(Tgc(x)))
  W355(s(k),x)	→	SRlll(W355(k,x))
  W355(s(k),x)	→	SRlll(SRlcase(Tgc(x)))
  W377(s(k),x)	→	SRlll(W377(k,x))
  W377(s(k),x)	→	SRlll(SRlseq(Tgc(x)))
  W383(s(k),x)	→	SRlll(W383(k,x))
  W383(s(k),x)	→	SRlll(SRlseq(Tgc(x)))
  W405(s(k),x)	→	SRlll(W405(k,x))
  W405(s(k),x)	→	SRlll(SRcase(Tgc(x)))
  W411(s(k),x)	→	SRlll(W411(k,x))
  W411(s(k),x)	→	SRlll(SRcase(Tgc(x)))
  W438(s(k),x)	→	SRlll(W438(k,x))
  W438(s(k),x)	→	SRlll(SRseq(Tgc(x)))
  W444(s(k),x)	→	SRlll(W444(k,x))
  W444(s(k),x)	→	SRlll(SRseq(Tgc(x)))
  SRlll(x)	→	SRlseq(x)
  SRlll(x)	→	SRlapp(x)
  SRlll(x)	→	SRllet(x)
  SRlll(x)	→	SRlcase(x)

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

  Tcpcx#(SRlapp(x))	→	Tcpcx#(x)
  Tcpcx#(SRlcase(x))	→	Tcpcx#(x)
  Tcpcx#(SRlseq(x))	→	Tcpcx#(x)
  Tcpcx#(SRllet(x))	→	Tcpcx#(x)

  remain.

  1.1.1.1.1.1.1 Usable Rules Processor

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

  There are no rules.

  1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.1.1.1.1.1.1 Size-Change Termination

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

  • Tcpcx^#(SRlapp(x)) → Tcpcx^#(x):

    1>1

  • Tcpcx^#(SRlcase(x)) → Tcpcx^#(x):

    1>1

  • Tcpcx^#(SRlseq(x)) → Tcpcx^#(x):

    1>1

  • Tcpcx^#(SRllet(x)) → Tcpcx^#(x):

    1>1

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

• 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#(SRcase(x))	→	Tabs#(Tcpx(Tcpx(Txch(Txch(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.

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

  1.1.1.2.1 Innermost Lhss Removal Processor

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

  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)

  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)

  1.1.1.2.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [Txch(x1)]	=	1 · x1
  [SRlbeta(x1)]	=	1 + 2 · x1
  [SRcp(x1)]	=	1 + 1 · x1
  [SRllet(x1)]	=	1 + 2 · x1
  [SRlapp(x1)]	=	1 · x1
  [SRlcase(x1)]	=	2 · x1
  [SRlseq(x1)]	=	2 · x1
  [SRcase(x1)]	=	1 + 1 · x1
  [SRseq(x1)]	=	2 · x1
  [Answer]	=	0
  [Tcpx(x1)]	=	1 · x1
  [Tabs#(x1)]	=	3 · x1

  the pairs

  Tabs#(SRlapp(x))	→	Tabs#(x)
  Tabs#(SRlcase(x))	→	Tabs#(x)
  Tabs#(SRlseq(x))	→	Tabs#(x)
  Tabs#(SRseq(x))	→	Tabs#(x)

  and the rules

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

  remain.

  1.1.1.2.1.1.1 Usable Rules Processor

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

  There are no rules.

  1.1.1.2.1.1.1.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.2.1.1.1.1.1 Size-Change Termination

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

  • Tabs^#(SRlapp(x)) → Tabs^#(x):

    1>1

  • Tabs^#(SRlcase(x)) → Tabs^#(x):

    1>1

  • Tabs^#(SRlseq(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

  W273#(s(k),x)	→	W273#(k,x)
  W273#(s(k),x)	→	Tgc#(x)
  Tgc#(SRlbeta(x))	→	W273#(k,x)
  Tgc#(SRlbeta(x))	→	W279#(k,x)
  W279#(s(k),x)	→	W279#(k,x)
  W279#(s(k),x)	→	Tgc#(x)
  Tgc#(SRcp(x))	→	W301#(k,x)
  W301#(s(k),x)	→	W301#(k,x)
  W301#(s(k),x)	→	Tgc#(x)
  Tgc#(SRllet(x))	→	W316#(k,x)
  W316#(s(k),x)	→	W316#(k,x)
  W316#(s(k),x)	→	Tgc#(x)
  Tgc#(SRlapp(x))	→	W321#(k,x)
  W321#(s(k),x)	→	W321#(k,x)
  W321#(s(k),x)	→	Tgc#(x)
  Tgc#(SRlapp(x))	→	W327#(k,x)
  W327#(s(k),x)	→	W327#(k,x)
  W327#(s(k),x)	→	Tgc#(x)
  Tgc#(SRlcase(x))	→	W349#(k,x)
  W349#(s(k),x)	→	W349#(k,x)
  W349#(s(k),x)	→	Tgc#(x)
  Tgc#(SRlcase(x))	→	W355#(k,x)
  W355#(s(k),x)	→	W355#(k,x)
  W355#(s(k),x)	→	Tgc#(x)
  Tgc#(SRlseq(x))	→	W377#(k,x)
  W377#(s(k),x)	→	W377#(k,x)
  W377#(s(k),x)	→	Tgc#(x)
  Tgc#(SRlseq(x))	→	W383#(k,x)
  W383#(s(k),x)	→	W383#(k,x)
  W383#(s(k),x)	→	Tgc#(x)
  Tgc#(SRcase(x))	→	W405#(k,x)
  W405#(s(k),x)	→	W405#(k,x)
  W405#(s(k),x)	→	Tgc#(x)
  Tgc#(SRcase(x))	→	W411#(k,x)
  W411#(s(k),x)	→	W411#(k,x)
  W411#(s(k),x)	→	Tgc#(x)
  Tgc#(SRseq(x))	→	W438#(k,x)
  W438#(s(k),x)	→	W438#(k,x)
  W438#(s(k),x)	→	Tgc#(x)
  Tgc#(SRseq(x))	→	W444#(k,x)
  W444#(s(k),x)	→	W444#(k,x)
  W444#(s(k),x)	→	Tgc#(x)
  Tgc#(SRlbeta(x))	→	Tgc#(x)
  Tgc#(SRcp(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)

  1.1.1.3 Usable Rules Processor

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

  There are no rules.

  1.1.1.3.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.3.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [W273#(x1, x2)]	=	1 + 1 · x2
  [s(x1)]	=	0
  [Tgc#(x1)]	=	1 · x1
  [SRlbeta(x1)]	=	1 + 1 · x1
  [W279#(x1, x2)]	=	1 + 1 · x2
  [SRcp(x1)]	=	1 + 1 · x1
  [W301#(x1, x2)]	=	1 + 1 · x2
  [SRllet(x1)]	=	1 + 1 · x1
  [W316#(x1, x2)]	=	1 + 1 · x2
  [SRlapp(x1)]	=	1 + 1 · x1
  [W321#(x1, x2)]	=	1 + 1 · x2
  [W327#(x1, x2)]	=	1 + 1 · x2
  [SRlcase(x1)]	=	1 + 1 · x1
  [W349#(x1, x2)]	=	1 + 1 · x2
  [W355#(x1, x2)]	=	1 + 1 · x2
  [SRlseq(x1)]	=	1 + 1 · x1
  [W377#(x1, x2)]	=	1 + 1 · x2
  [W383#(x1, x2)]	=	1 + 1 · x2
  [SRcase(x1)]	=	1 + 1 · x1
  [W405#(x1, x2)]	=	1 + 1 · x2
  [W411#(x1, x2)]	=	1 + 1 · x2
  [SRseq(x1)]	=	1 + 1 · x1
  [W438#(x1, x2)]	=	1 + 1 · x2
  [W444#(x1, x2)]	=	1 + 1 · x2

  the pairs

  W273#(s(k),x)	→	W273#(k,x)
  Tgc#(SRlbeta(x))	→	W273#(k,x)
  Tgc#(SRlbeta(x))	→	W279#(k,x)
  W279#(s(k),x)	→	W279#(k,x)
  Tgc#(SRcp(x))	→	W301#(k,x)
  W301#(s(k),x)	→	W301#(k,x)
  Tgc#(SRllet(x))	→	W316#(k,x)
  W316#(s(k),x)	→	W316#(k,x)
  Tgc#(SRlapp(x))	→	W321#(k,x)
  W321#(s(k),x)	→	W321#(k,x)
  Tgc#(SRlapp(x))	→	W327#(k,x)
  W327#(s(k),x)	→	W327#(k,x)
  Tgc#(SRlcase(x))	→	W349#(k,x)
  W349#(s(k),x)	→	W349#(k,x)
  Tgc#(SRlcase(x))	→	W355#(k,x)
  W355#(s(k),x)	→	W355#(k,x)
  Tgc#(SRlseq(x))	→	W377#(k,x)
  W377#(s(k),x)	→	W377#(k,x)
  Tgc#(SRlseq(x))	→	W383#(k,x)
  W383#(s(k),x)	→	W383#(k,x)
  Tgc#(SRcase(x))	→	W405#(k,x)
  W405#(s(k),x)	→	W405#(k,x)
  Tgc#(SRcase(x))	→	W411#(k,x)
  W411#(s(k),x)	→	W411#(k,x)
  Tgc#(SRseq(x))	→	W438#(k,x)
  W438#(s(k),x)	→	W438#(k,x)
  Tgc#(SRseq(x))	→	W444#(k,x)
  W444#(s(k),x)	→	W444#(k,x)

  remain.

  1.1.1.3.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 14 components.

  • The 1^st component contains the pair

    W273#(s(k),x)

    →

    W273#(k,x)

    1.1.1.3.1.1.1.1 Size-Change Termination

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

    • W273^#(s(k),x) → W273^#(k,x):

      1>1, 2>=2

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

  • The 2^nd component contains the pair

    W279#(s(k),x)

    →

    W279#(k,x)

    1.1.1.3.1.1.1.2 Size-Change Termination

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

    • W279^#(s(k),x) → W279^#(k,x):

      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

    W301#(s(k),x)

    →

    W301#(k,x)

    1.1.1.3.1.1.1.3 Size-Change Termination

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

    • W301^#(s(k),x) → W301^#(k,x):

      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

    W316#(s(k),x)

    →

    W316#(k,x)

    1.1.1.3.1.1.1.4 Size-Change Termination

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

    • W316^#(s(k),x) → W316^#(k,x):

      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

    W321#(s(k),x)

    →

    W321#(k,x)

    1.1.1.3.1.1.1.5 Size-Change Termination

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

    • W321^#(s(k),x) → W321^#(k,x):

      1>1, 2>=2

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

  • The 6^th component contains the pair

    W327#(s(k),x)

    →

    W327#(k,x)

    1.1.1.3.1.1.1.6 Size-Change Termination

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

    • W327^#(s(k),x) → W327^#(k,x):

      1>1, 2>=2

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

  • The 7^th component contains the pair

    W349#(s(k),x)

    →

    W349#(k,x)

    1.1.1.3.1.1.1.7 Size-Change Termination

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

    • W349^#(s(k),x) → W349^#(k,x):

      1>1, 2>=2

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

  • The 8^th component contains the pair

    W355#(s(k),x)

    →

    W355#(k,x)

    1.1.1.3.1.1.1.8 Size-Change Termination

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

    • W355^#(s(k),x) → W355^#(k,x):

      1>1, 2>=2

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

  • The 9^th component contains the pair

    W377#(s(k),x)

    →

    W377#(k,x)

    1.1.1.3.1.1.1.9 Size-Change Termination

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

    • W377^#(s(k),x) → W377^#(k,x):

      1>1, 2>=2

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

  • The 10^th component contains the pair

    W383#(s(k),x)

    →

    W383#(k,x)

    1.1.1.3.1.1.1.10 Size-Change Termination

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

    • W383^#(s(k),x) → W383^#(k,x):

      1>1, 2>=2

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

  • The 11^th component contains the pair

    W405#(s(k),x)

    →

    W405#(k,x)

    1.1.1.3.1.1.1.11 Size-Change Termination

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

    • W405^#(s(k),x) → W405^#(k,x):

      1>1, 2>=2

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

  • The 12^th component contains the pair

    W411#(s(k),x)

    →

    W411#(k,x)

    1.1.1.3.1.1.1.12 Size-Change Termination

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

    • W411^#(s(k),x) → W411^#(k,x):

      1>1, 2>=2

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

  • The 13^th component contains the pair

    W438#(s(k),x)

    →

    W438#(k,x)

    1.1.1.3.1.1.1.13 Size-Change Termination

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

    • W438^#(s(k),x) → W438^#(k,x):

      1>1, 2>=2

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

  • The 14^th component contains the pair

    W444#(s(k),x)

    →

    W444#(k,x)

    1.1.1.3.1.1.1.14 Size-Change Termination

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

    • W444^#(s(k),x) → W444^#(k,x):

      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 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

  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))

  1.1.1.5 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)))

  1.1.1.5.1 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)

  1.1.1.5.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the naturals

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

  the pairs

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

  remain.

  1.1.1.5.1.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [Tcpx(x1)]	=	1 · x1
  [SRlbeta(x1)]	=	2 · x1
  [SRcp(x1)]	=	3 + 1 · x1
  [SRllet(x1)]	=	2 · x1
  [SRlapp(x1)]	=	2 · x1
  [SRlcase(x1)]	=	2 · x1
  [SRlseq(x1)]	=	2 · x1
  [SRcase(x1)]	=	2 · x1
  [SRseq(x1)]	=	2 · x1
  [Answer]	=	0
  [Tcpx#(x1)]	=	1 · x1

  the pairs

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

  and the rules

  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)))

  remain.

  1.1.1.5.1.1.1.1 Usable Rules Processor

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

  There are no rules.

  1.1.1.5.1.1.1.1.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.5.1.1.1.1.1.1 Size-Change Termination

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

  • Tcpx^#(SRlbeta(x)) → Tcpx^#(x):

    1>1

  • Tcpx^#(SRllet(x)) → Tcpx^#(x):

    1>1

  • Tcpx^#(SRlapp(x)) → Tcpx^#(x):

    1>1

  • Tcpx^#(SRlseq(x)) → Tcpx^#(x):

    1>1

  • Tcpx^#(SRcase(x)) → Tcpx^#(x):

    1>1

  • Tcpx^#(SRseq(x)) → Tcpx^#(x):

    1>1

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