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by AProVE (version ba869e7b28377cd372aedcb96abeb62c4ad6aaa5 rene 20130719 unpublished dirty )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 11 components.

• The 1^st component contains the pairs

  Tucp#(SRlbeta(x))	→	W121#(k,x)
  W121#(s(k),x)	→	W121#(k,x)
  W121#(s(k),x)	→	Tucp#(x)
  Tucp#(SRcp(x))	→	W161#(k,x)
  W161#(s(k),x)	→	W161#(k,x)
  W161#(s(k),x)	→	Tucp#(x)
  Tucp#(SRllet(x))	→	W181#(k,x)
  W181#(s(k),x)	→	W181#(k,x)
  W181#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlapp(x))	→	W186#(k,x)
  W186#(s(k),x)	→	W186#(k,x)
  W186#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlapp(W188(x)))	→	W193#(k,x)
  W193#(s(k),x)	→	W193#(k,x)
  W193#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlapp(x))	→	W199#(k,x)
  W199#(s(k),x)	→	W199#(k,x)
  W199#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlapp(W200(x)))	→	W205#(k,x)
  W205#(s(k),x)	→	W205#(k,x)
  W205#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlcase(x))	→	W255#(k,x)
  W255#(s(k),x)	→	W255#(k,x)
  W255#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlcase(W257(x)))	→	W262#(k,x)
  W262#(s(k),x)	→	W262#(k,x)
  W262#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlcase(x))	→	W268#(k,x)
  W268#(s(k),x)	→	W268#(k,x)
  W268#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlcase(W269(x)))	→	W274#(k,x)
  W274#(s(k),x)	→	W274#(k,x)
  W274#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlseq(x))	→	W324#(k,x)
  W324#(s(k),x)	→	W324#(k,x)
  W324#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlseq(W326(x)))	→	W331#(k,x)
  W331#(s(k),x)	→	W331#(k,x)
  W331#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlseq(x))	→	W337#(k,x)
  W337#(s(k),x)	→	W337#(k,x)
  W337#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlseq(W338(x)))	→	W343#(k,x)
  W343#(s(k),x)	→	W343#(k,x)
  W343#(s(k),x)	→	Tucp#(x)
  Tucp#(SRcase(x))	→	W404#(k,x)
  W404#(s(k),x)	→	W404#(k,x)
  W404#(s(k),x)	→	Tucp#(x)
  Tucp#(SRcase(x))	→	W414#(k,x)
  W414#(s(k),x)	→	W414#(k,x)
  W414#(s(k),x)	→	Tucp#(x)
  Tucp#(SRcase(x))	→	W422#(k,x)
  W422#(s(k),x)	→	W422#(k,x)
  W422#(s(k),x)	→	Tucp#(Tucp(x))
  Tucp#(SRcase(x))	→	W430#(k,x)
  W430#(s(k),x)	→	W430#(k,x)
  W430#(s(k),x)	→	Tucp#(Tucp(x))
  Tucp#(SRseq(x))	→	W509#(k,x)
  W509#(s(k),x)	→	W509#(k,x)
  W509#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlbeta(x))	→	Tucp#(x)
  Tucp#(SRcp(x))	→	Tucp#(x)
  Tucp#(SRllet(x))	→	Tucp#(x)
  Tucp#(SRlapp(x))	→	Tucp#(x)
  Tucp#(SRlapp(W26(x)))	→	Tucp#(x)
  Tucp#(SRlapp(SRllet(x)))	→	Tucp#(x)
  Tucp#(SRlcase(x))	→	Tucp#(x)
  Tucp#(SRlcase(W40(x)))	→	Tucp#(x)
  Tucp#(SRlcase(SRllet(x)))	→	Tucp#(x)
  Tucp#(SRlseq(x))	→	Tucp#(x)
  Tucp#(SRlseq(W54(x)))	→	Tucp#(x)
  Tucp#(SRlseq(SRllet(x)))	→	Tucp#(x)
  Tucp#(SRcase(x))	→	Tucp#(x)
  Tucp#(SRcase(x))	→	Tucp#(Tucp(x))
  Tucp#(SRseq(x))	→	Tucp#(x)
  Tucp#(SRlapp(W239(x)))	→	Tucp#(x)
  Tucp#(SRlcase(W308(x)))	→	Tucp#(x)
  Tucp#(SRlseq(W377(x)))	→	Tucp#(x)
  W430#(s(k),x)	→	Tucp#(x)
  W422#(s(k),x)	→	Tucp#(x)

  1.1.1.1 Usable Rules Processor

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

  Tucp(SRlbeta(x))	→	SRlapp(W109(k,x))
  Tucp(SRlbeta(x))	→	W116(k,x)
  Tucp(SRlbeta(x))	→	W121(k,x)
  Tucp(SRcp(x))	→	W156(k,x)
  Tucp(SRcp(x))	→	W161(k,x)
  Tucp(SRllet(x))	→	W181(k,x)
  Tucp(SRlapp(x))	→	SRlapp(W186(k,x))
  Tucp(SRlapp(W188(x)))	→	SRlapp(W193(k,x))
  Tucp(SRlapp(x))	→	W199(k,x)
  Tucp(SRlapp(W200(x)))	→	W205(k,x)
  Tucp(SRlcase(x))	→	SRlcase(W255(k,x))
  Tucp(SRlcase(W257(x)))	→	SRlcase(W262(k,x))
  Tucp(SRlcase(x))	→	W268(k,x)
  Tucp(SRlcase(W269(x)))	→	W274(k,x)
  Tucp(SRlseq(x))	→	SRlseq(W324(k,x))
  Tucp(SRlseq(W326(x)))	→	SRlseq(W331(k,x))
  Tucp(SRlseq(x))	→	W337(k,x)
  Tucp(SRlseq(W338(x)))	→	W343(k,x)
  Tucp(SRcase(x))	→	SRlcase(W393(k,x))
  Tucp(SRcase(x))	→	W399(k,x)
  Tucp(SRcase(x))	→	W404(k,x)
  Tucp(SRcase(x))	→	SRlcase(W414(k,x))
  Tucp(SRcase(x))	→	SRlcase(W422(k,x))
  Tucp(SRcase(x))	→	W430(k,x)
  Tucp(SRseq(x))	→	SRlseq(W498(k,x))
  Tucp(SRseq(x))	→	W504(k,x)
  Tucp(SRseq(x))	→	W509(k,x)
  Tucp(SRseq(x))	→	SRlseq(W520(k,x))
  Tucp(SRseq(x))	→	W527(k,x)
  Tucp(SRlbeta(x))	→	SRlbeta(Tucp(x))
  Tucp(SRlbeta(x))	→	SRcp(SRlbeta(Tgc(x)))
  Tucp(SRcp(x))	→	SRcp(Tucp(x))
  Tucp(SRcp(x))	→	SRcp(Tgc(x))
  Tucp(SRllet(x))	→	SRllet(Tucp(x))
  Tucp(SRlapp(x))	→	SRlapp(Tucp(x))
  Tucp(SRlapp(W26(x)))	→	SRllet(Tucp(x))
  Tucp(SRlapp(SRllet(x)))	→	SRllet(Tucp(x))
  Tucp(SRlcase(x))	→	SRlcase(Tucp(x))
  Tucp(SRlcase(W40(x)))	→	SRllet(Tucp(x))
  Tucp(SRlcase(SRllet(x)))	→	SRllet(Tucp(x))
  Tucp(SRlseq(x))	→	SRlseq(Tucp(x))
  Tucp(SRlseq(W54(x)))	→	SRllet(Tucp(x))
  Tucp(SRlseq(SRllet(x)))	→	SRllet(Tucp(x))
  Tucp(SRcase(x))	→	SRcase(Tucp(x))
  Tucp(SRcase(x))	→	SRcase(x)
  Tucp(SRcase(x))	→	SRcase(Tgc(x))
  Tucp(SRcase(x))	→	SRcase(Tgc(Tucp(Tucp(x))))
  Tucp(SRseq(x))	→	SRseq(Tucp(x))
  Tucp(SRseq(x))	→	SRseq(x)
  Tucp(SRseq(x))	→	SRseq(Tgc(x))
  Tucp(SRseq(x))	→	SRcp(SRseq(Tgc(x)))
  Tucp(Answer)	→	Answer
  Tucp(Answer)	→	SRcp(Answer)
  Tucp(SRlbeta(x))	→	SRlapp(SRcp(SRlbeta(SRllet(Tgc(x)))))
  Tucp(SRlbeta(x))	→	SRlbeta(SRllet(Tucp(x)))
  Tucp(SRlbeta(x))	→	SRlapp(SRllet(SRcp(SRlbeta(Tgc(x)))))
  Tucp(SRlbeta(x))	→	SRllet(SRcp(SRlbeta(Tgc(x))))
  Tucp(SRlbeta(x))	→	SRllet(SRlbeta(Tucp(x)))
  Tucp(SRcp(x))	→	SRllet(SRcp(Tgc(x)))
  Tucp(SRcp(x))	→	SRllet(SRcp(Tucp(x)))
  Tucp(SRllet(x))	→	SRllet(SRllet(Tucp(x)))
  Tucp(SRlapp(x))	→	SRlapp(SRlapp(SRllet(Tucp(x))))
  Tucp(SRlapp(x))	→	SRlapp(SRllet(Tucp(x)))
  Tucp(SRlapp(x))	→	SRllet(Tucp(x))
  Tucp(SRlapp(x))	→	SRlapp(SRllet(SRlapp(Tucp(x))))
  Tucp(SRlapp(SRllet(x)))	→	SRlapp(SRllet(SRllet(Tucp(x))))
  Tucp(SRlapp(x))	→	SRllet(SRlapp(Tucp(x)))
  Tucp(SRlapp(W239(x)))	→	SRllet(SRllet(Tucp(x)))
  Tucp(SRlapp(SRllet(x)))	→	SRllet(SRllet(Tucp(x)))
  Tucp(SRlcase(x))	→	SRlcase(SRlcase(SRllet(Tucp(x))))
  Tucp(SRlcase(x))	→	SRlcase(SRllet(Tucp(x)))
  Tucp(SRlcase(x))	→	SRllet(Tucp(x))
  Tucp(SRlcase(x))	→	SRlcase(SRllet(SRlcase(Tucp(x))))
  Tucp(SRlcase(SRllet(x)))	→	SRlcase(SRllet(SRllet(Tucp(x))))
  Tucp(SRlcase(x))	→	SRllet(SRlcase(Tucp(x)))
  Tucp(SRlcase(W308(x)))	→	SRllet(SRllet(Tucp(x)))
  Tucp(SRlcase(SRllet(x)))	→	SRllet(SRllet(Tucp(x)))
  Tucp(SRlseq(x))	→	SRlseq(SRlseq(SRllet(Tucp(x))))
  Tucp(SRlseq(x))	→	SRlseq(SRllet(Tucp(x)))
  Tucp(SRlseq(x))	→	SRllet(Tucp(x))
  Tucp(SRlseq(x))	→	SRlseq(SRllet(SRlseq(Tucp(x))))
  Tucp(SRlseq(SRllet(x)))	→	SRlseq(SRllet(SRllet(Tucp(x))))
  Tucp(SRlseq(x))	→	SRllet(SRlseq(Tucp(x)))
  Tucp(SRlseq(W377(x)))	→	SRllet(SRllet(Tucp(x)))
  Tucp(SRlseq(SRllet(x)))	→	SRllet(SRllet(Tucp(x)))
  Tucp(SRcase(x))	→	SRlcase(SRcase(Tgc(x)))
  Tucp(SRcase(x))	→	SRlcase(SRcase(SRllet(Tucp(x))))
  Tucp(SRcase(x))	→	SRlcase(SRcase(SRllet(Tgc(Tucp(Tucp(x))))))
  Tucp(SRcase(x))	→	SRcase(SRllet(Tucp(x)))
  Tucp(SRcase(x))	→	SRcase(SRllet(Tgc(Tucp(Tucp(x)))))
  Tucp(SRcase(x))	→	SRlcase(SRllet(SRcase(Tgc(x))))
  Tucp(SRcase(x))	→	SRllet(SRcase(Tgc(x)))
  Tucp(SRcase(x))	→	SRllet(SRcase(Tucp(x)))
  Tucp(SRcase(x))	→	SRlcase(SRllet(SRcase(Tucp(x))))
  Tucp(SRcase(x))	→	SRlcase(SRllet(SRcase(Tgc(Tucp(Tucp(x))))))
  Tucp(SRcase(x))	→	SRllet(SRcase(Tgc(Tucp(Tucp(x)))))
  Tucp(SRseq(x))	→	SRlseq(SRseq(Tgc(x)))
  Tucp(SRseq(x))	→	SRlseq(SRcp(SRseq(Tgc(x))))
  Tucp(SRseq(x))	→	SRlseq(SRllet(SRseq(Tgc(x))))
  Tucp(SRseq(x))	→	SRllet(SRseq(Tgc(x)))
  Tucp(SRseq(x))	→	SRllet(SRseq(Tucp(x)))
  Tucp(SRseq(x))	→	SRlseq(SRllet(SRcp(SRseq(Tgc(x)))))
  Tucp(SRseq(x))	→	SRllet(SRcp(SRseq(Tgc(x))))
  Tucp(Answer)	→	SRllet(SRcp(Answer))
  Tucp(Answer)	→	SRllet(Answer)
  Tgc(SRlbeta(x))	→	SRlapp(W625(k,x))
  Tgc(SRlbeta(x))	→	W631(k,x)
  Tgc(SRcp(x))	→	W653(k,x)
  Tgc(SRllet(x))	→	W668(k,x)
  Tgc(SRlapp(x))	→	SRlapp(W673(k,x))
  Tgc(SRlapp(x))	→	W679(k,x)
  Tgc(SRlcase(x))	→	SRlcase(W701(k,x))
  Tgc(SRlcase(x))	→	W707(k,x)
  Tgc(SRlseq(x))	→	SRlseq(W729(k,x))
  Tgc(SRlseq(x))	→	W735(k,x)
  Tgc(SRcase(x))	→	SRlcase(W757(k,x))
  Tgc(SRcase(x))	→	W763(k,x)
  Tgc(SRseq(x))	→	SRlseq(W790(k,x))
  Tgc(SRseq(x))	→	W796(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)
  W796(s(k),x)	→	SRlll(W796(k,x))
  W796(s(k),x)	→	SRlll(SRseq(Tgc(x)))
  SRlll(x)	→	SRlseq(x)
  SRlll(x)	→	SRlapp(x)
  SRlll(x)	→	SRllet(x)
  SRlll(x)	→	SRlcase(x)
  W790(s(k),x)	→	SRlll(W790(k,x))
  W790(s(k),x)	→	SRlll(SRseq(Tgc(x)))
  W763(s(k),x)	→	SRlll(W763(k,x))
  W763(s(k),x)	→	SRlll(SRcase(Tgc(x)))
  W757(s(k),x)	→	SRlll(W757(k,x))
  W757(s(k),x)	→	SRlll(SRcase(Tgc(x)))
  W735(s(k),x)	→	SRlll(W735(k,x))
  W735(s(k),x)	→	SRlll(SRlseq(Tgc(x)))
  W729(s(k),x)	→	SRlll(W729(k,x))
  W729(s(k),x)	→	SRlll(SRlseq(Tgc(x)))
  W707(s(k),x)	→	SRlll(W707(k,x))
  W707(s(k),x)	→	SRlll(SRlcase(Tgc(x)))
  W701(s(k),x)	→	SRlll(W701(k,x))
  W701(s(k),x)	→	SRlll(SRlcase(Tgc(x)))
  W679(s(k),x)	→	SRlll(W679(k,x))
  W679(s(k),x)	→	SRlll(SRlapp(Tgc(x)))
  W673(s(k),x)	→	SRlll(W673(k,x))
  W673(s(k),x)	→	SRlll(SRlapp(Tgc(x)))
  W668(s(k),x)	→	SRlll(W668(k,x))
  W668(s(k),x)	→	SRlll(SRllet(Tgc(x)))
  W653(s(k),x)	→	SRlll(W653(k,x))
  W653(s(k),x)	→	SRlll(SRcp(Tgc(x)))
  W631(s(k),x)	→	SRlll(W631(k,x))
  W631(s(k),x)	→	SRlll(SRlbeta(Tgc(x)))
  W625(s(k),x)	→	SRlll(W625(k,x))
  W625(s(k),x)	→	SRlll(SRlbeta(Tgc(x)))
  W527(s(k),x)	→	SRlll(W527(k,x))
  W527(s(k),x)	→	SRlll(SRcp(SRseq(Tgc(x))))
  W520(s(k),x)	→	SRlll(W520(k,x))
  W520(s(k),x)	→	SRlll(SRcp(SRseq(Tgc(x))))
  W509(s(k),x)	→	SRlll(W509(k,x))
  W509(s(k),x)	→	SRlll(SRseq(Tucp(x)))
  W504(s(k),x)	→	SRlll(W504(k,x))
  W504(s(k),x)	→	SRlll(SRseq(Tgc(x)))
  W498(s(k),x)	→	SRlll(W498(k,x))
  W498(s(k),x)	→	SRlll(SRseq(Tgc(x)))
  W430(s(k),x)	→	SRlll(W430(k,x))
  W430(s(k),x)	→	SRlll(SRcase(Tgc(Tucp(Tucp(x)))))
  W422(s(k),x)	→	SRlll(W422(k,x))
  W422(s(k),x)	→	SRlll(SRcase(Tgc(Tucp(Tucp(x)))))
  W414(s(k),x)	→	SRlll(W414(k,x))
  W414(s(k),x)	→	SRlll(SRcase(Tucp(x)))
  W404(s(k),x)	→	SRlll(W404(k,x))
  W404(s(k),x)	→	SRlll(SRcase(Tucp(x)))
  W399(s(k),x)	→	SRlll(W399(k,x))
  W399(s(k),x)	→	SRlll(SRcase(Tgc(x)))
  W393(s(k),x)	→	SRlll(W393(k,x))
  W393(s(k),x)	→	SRlll(SRcase(Tgc(x)))
  W343(s(k),x)	→	SRlll(W343(k,x))
  W343(s(k),x)	→	SRlll(SRllet(Tucp(x)))
  W337(s(k),x)	→	SRlll(W337(k,x))
  W337(s(k),x)	→	SRlll(SRlseq(Tucp(x)))
  W331(s(k),x)	→	SRlll(W331(k,x))
  W331(s(k),x)	→	SRlll(SRllet(Tucp(x)))
  W324(s(k),x)	→	SRlll(W324(k,x))
  W324(s(k),x)	→	SRlll(SRlseq(Tucp(x)))
  W274(s(k),x)	→	SRlll(W274(k,x))
  W274(s(k),x)	→	SRlll(SRllet(Tucp(x)))
  W268(s(k),x)	→	SRlll(W268(k,x))
  W268(s(k),x)	→	SRlll(SRlcase(Tucp(x)))
  W262(s(k),x)	→	SRlll(W262(k,x))
  W262(s(k),x)	→	SRlll(SRllet(Tucp(x)))
  W255(s(k),x)	→	SRlll(W255(k,x))
  W255(s(k),x)	→	SRlll(SRlcase(Tucp(x)))
  W205(s(k),x)	→	SRlll(W205(k,x))
  W205(s(k),x)	→	SRlll(SRllet(Tucp(x)))
  W199(s(k),x)	→	SRlll(W199(k,x))
  W199(s(k),x)	→	SRlll(SRlapp(Tucp(x)))
  W193(s(k),x)	→	SRlll(W193(k,x))
  W193(s(k),x)	→	SRlll(SRllet(Tucp(x)))
  W186(s(k),x)	→	SRlll(W186(k,x))
  W186(s(k),x)	→	SRlll(SRlapp(Tucp(x)))
  W181(s(k),x)	→	SRlll(W181(k,x))
  W181(s(k),x)	→	SRlll(SRllet(Tucp(x)))
  W161(s(k),x)	→	SRlll(W161(k,x))
  W161(s(k),x)	→	SRlll(SRcp(Tucp(x)))
  W156(s(k),x)	→	SRlll(W156(k,x))
  W156(s(k),x)	→	SRlll(SRcp(Tgc(x)))
  W121(s(k),x)	→	SRlll(W121(k,x))
  W121(s(k),x)	→	SRlll(SRlbeta(Tucp(x)))
  W116(s(k),x)	→	SRlll(W116(k,x))
  W116(s(k),x)	→	SRlll(SRcp(SRlbeta(Tgc(x))))
  W109(s(k),x)	→	SRlll(W109(k,x))
  W109(s(k),x)	→	SRlll(SRcp(SRlbeta(Tgc(x))))

  1.1.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [Tucp#(x1)]	=	1 · x1
  [SRlbeta(x1)]	=	1 · x1
  [W121#(x1, x2)]	=	1 · x2
  [s(x1)]	=	1 · x1
  [SRcp(x1)]	=	1 · x1
  [W161#(x1, x2)]	=	1 · x2
  [SRllet(x1)]	=	1 · x1
  [W181#(x1, x2)]	=	1 · x2
  [SRlapp(x1)]	=	1 · x1
  [W186#(x1, x2)]	=	1 · x2
  [W188(x1)]	=	1 · x1
  [W193#(x1, x2)]	=	1 · x2
  [W199#(x1, x2)]	=	1 · x2
  [W200(x1)]	=	1 · x1
  [W205#(x1, x2)]	=	1 · x2
  [SRlcase(x1)]	=	1 · x1
  [W255#(x1, x2)]	=	1 · x2
  [W257(x1)]	=	1 · x1
  [W262#(x1, x2)]	=	1 · x2
  [W268#(x1, x2)]	=	1 · x2
  [W269(x1)]	=	1 · x1
  [W274#(x1, x2)]	=	1 · x2
  [SRlseq(x1)]	=	1 · x1
  [W324#(x1, x2)]	=	1 · x2
  [W326(x1)]	=	1 · x1
  [W331#(x1, x2)]	=	1 · x2
  [W337#(x1, x2)]	=	1 · x2
  [W338(x1)]	=	1 · x1
  [W343#(x1, x2)]	=	1 · x2
  [SRcase(x1)]	=	1 + 1 · x1
  [W404#(x1, x2)]	=	1 · x2
  [W414#(x1, x2)]	=	1 · x2
  [W422#(x1, x2)]	=	1 · x2
  [Tucp(x1)]	=	1 · x1
  [W430#(x1, x2)]	=	1 + 1 · x2
  [SRseq(x1)]	=	1 · x1
  [W509#(x1, x2)]	=	1 · x2
  [W26(x1)]	=	1 · x1
  [W40(x1)]	=	1 · x1
  [W54(x1)]	=	1 · x1
  [W239(x1)]	=	1 · x1
  [W308(x1)]	=	1 · x1
  [W377(x1)]	=	1 · x1
  [W109(x1, x2)]	=	1 · x2
  [W116(x1, x2)]	=	1 · x2
  [W121(x1, x2)]	=	1 · x2
  [W156(x1, x2)]	=	1 · x2
  [W161(x1, x2)]	=	1 · x2
  [W181(x1, x2)]	=	1 · x2
  [W186(x1, x2)]	=	1 · x2
  [W193(x1, x2)]	=	1 · x2
  [W199(x1, x2)]	=	1 · x2
  [W205(x1, x2)]	=	1 · x2
  [W255(x1, x2)]	=	1 · x2
  [W262(x1, x2)]	=	1 · x2
  [W268(x1, x2)]	=	1 · x2
  [W274(x1, x2)]	=	1 · x2
  [W324(x1, x2)]	=	1 · x2
  [W331(x1, x2)]	=	1 · x2
  [W337(x1, x2)]	=	1 · x2
  [W343(x1, x2)]	=	1 · x2
  [W393(x1, x2)]	=	1 + 1 · x2
  [W399(x1, x2)]	=	1 + 1 · x2
  [W404(x1, x2)]	=	1 + 1 · x2
  [W414(x1, x2)]	=	1 + 1 · x2
  [W422(x1, x2)]	=	1 + 1 · x2
  [W430(x1, x2)]	=	1 + 1 · x2
  [W498(x1, x2)]	=	1 · x2
  [W504(x1, x2)]	=	1 · x2
  [W509(x1, x2)]	=	1 · x2
  [W520(x1, x2)]	=	1 · x2
  [W527(x1, x2)]	=	1 · x2
  [Tgc(x1)]	=	1 · x1
  [Answer]	=	0
  [SRlll(x1)]	=	1 · x1
  [W625(x1, x2)]	=	1 · x2
  [W631(x1, x2)]	=	1 · x2
  [W653(x1, x2)]	=	1 · x2
  [W668(x1, x2)]	=	1 · x2
  [W673(x1, x2)]	=	1 · x2
  [W679(x1, x2)]	=	1 · x2
  [W701(x1, x2)]	=	1 · x2
  [W707(x1, x2)]	=	1 · x2
  [W729(x1, x2)]	=	1 · x2
  [W735(x1, x2)]	=	1 · x2
  [W757(x1, x2)]	=	1 + 1 · x2
  [W763(x1, x2)]	=	1 + 1 · x2
  [W790(x1, x2)]	=	1 · x2
  [W796(x1, x2)]	=	1 · x2

  the pairs

  Tucp#(SRlbeta(x))	→	W121#(k,x)
  W121#(s(k),x)	→	W121#(k,x)
  W121#(s(k),x)	→	Tucp#(x)
  Tucp#(SRcp(x))	→	W161#(k,x)
  W161#(s(k),x)	→	W161#(k,x)
  W161#(s(k),x)	→	Tucp#(x)
  Tucp#(SRllet(x))	→	W181#(k,x)
  W181#(s(k),x)	→	W181#(k,x)
  W181#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlapp(x))	→	W186#(k,x)
  W186#(s(k),x)	→	W186#(k,x)
  W186#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlapp(W188(x)))	→	W193#(k,x)
  W193#(s(k),x)	→	W193#(k,x)
  W193#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlapp(x))	→	W199#(k,x)
  W199#(s(k),x)	→	W199#(k,x)
  W199#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlapp(W200(x)))	→	W205#(k,x)
  W205#(s(k),x)	→	W205#(k,x)
  W205#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlcase(x))	→	W255#(k,x)
  W255#(s(k),x)	→	W255#(k,x)
  W255#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlcase(W257(x)))	→	W262#(k,x)
  W262#(s(k),x)	→	W262#(k,x)
  W262#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlcase(x))	→	W268#(k,x)
  W268#(s(k),x)	→	W268#(k,x)
  W268#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlcase(W269(x)))	→	W274#(k,x)
  W274#(s(k),x)	→	W274#(k,x)
  W274#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlseq(x))	→	W324#(k,x)
  W324#(s(k),x)	→	W324#(k,x)
  W324#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlseq(W326(x)))	→	W331#(k,x)
  W331#(s(k),x)	→	W331#(k,x)
  W331#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlseq(x))	→	W337#(k,x)
  W337#(s(k),x)	→	W337#(k,x)
  W337#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlseq(W338(x)))	→	W343#(k,x)
  W343#(s(k),x)	→	W343#(k,x)
  W343#(s(k),x)	→	Tucp#(x)
  W404#(s(k),x)	→	W404#(k,x)
  W404#(s(k),x)	→	Tucp#(x)
  W414#(s(k),x)	→	W414#(k,x)
  W414#(s(k),x)	→	Tucp#(x)
  W422#(s(k),x)	→	W422#(k,x)
  W422#(s(k),x)	→	Tucp#(Tucp(x))
  Tucp#(SRcase(x))	→	W430#(k,x)
  W430#(s(k),x)	→	W430#(k,x)
  Tucp#(SRseq(x))	→	W509#(k,x)
  W509#(s(k),x)	→	W509#(k,x)
  W509#(s(k),x)	→	Tucp#(x)
  Tucp#(SRlbeta(x))	→	Tucp#(x)
  Tucp#(SRcp(x))	→	Tucp#(x)
  Tucp#(SRllet(x))	→	Tucp#(x)
  Tucp#(SRlapp(x))	→	Tucp#(x)
  Tucp#(SRlapp(W26(x)))	→	Tucp#(x)
  Tucp#(SRlapp(SRllet(x)))	→	Tucp#(x)
  Tucp#(SRlcase(x))	→	Tucp#(x)
  Tucp#(SRlcase(W40(x)))	→	Tucp#(x)
  Tucp#(SRlcase(SRllet(x)))	→	Tucp#(x)
  Tucp#(SRlseq(x))	→	Tucp#(x)
  Tucp#(SRlseq(W54(x)))	→	Tucp#(x)
  Tucp#(SRlseq(SRllet(x)))	→	Tucp#(x)
  Tucp#(SRseq(x))	→	Tucp#(x)
  Tucp#(SRlapp(W239(x)))	→	Tucp#(x)
  Tucp#(SRlcase(W308(x)))	→	Tucp#(x)
  Tucp#(SRlseq(W377(x)))	→	Tucp#(x)
  W422#(s(k),x)	→	Tucp#(x)

  remain.

  1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 5 components.

  • The 1^st component contains the pair

    W404#(s(k),x)

    →

    W404#(k,x)

    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.

    • W404^#(s(k),x) → W404^#(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

    W414#(s(k),x)

    →

    W414#(k,x)

    1.1.1.1.1.1.2 Usable Rules Processor

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

    There are no rules.

    1.1.1.1.1.1.2.1 Innermost Lhss Removal Processor

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

    There are no lhss.

    1.1.1.1.1.1.2.1.1 Size-Change Termination

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

    • W414^#(s(k),x) → W414^#(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

    W422#(s(k),x)

    →

    W422#(k,x)

    1.1.1.1.1.1.3 Usable Rules Processor

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

    There are no rules.

    1.1.1.1.1.1.3.1 Innermost Lhss Removal Processor

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

    There are no lhss.

    1.1.1.1.1.1.3.1.1 Size-Change Termination

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

    • W422^#(s(k),x) → W422^#(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

    W121#(s(k),x)	→	W121#(k,x)
    W121#(s(k),x)	→	Tucp#(x)
    Tucp#(SRlbeta(x))	→	W121#(k,x)
    Tucp#(SRcp(x))	→	W161#(k,x)
    W161#(s(k),x)	→	W161#(k,x)
    W161#(s(k),x)	→	Tucp#(x)
    Tucp#(SRllet(x))	→	W181#(k,x)
    W181#(s(k),x)	→	W181#(k,x)
    W181#(s(k),x)	→	Tucp#(x)
    Tucp#(SRlapp(x))	→	W186#(k,x)
    W186#(s(k),x)	→	W186#(k,x)
    W186#(s(k),x)	→	Tucp#(x)
    Tucp#(SRlapp(W188(x)))	→	W193#(k,x)
    W193#(s(k),x)	→	W193#(k,x)
    W193#(s(k),x)	→	Tucp#(x)
    Tucp#(SRlapp(x))	→	W199#(k,x)
    W199#(s(k),x)	→	W199#(k,x)
    W199#(s(k),x)	→	Tucp#(x)
    Tucp#(SRlapp(W200(x)))	→	W205#(k,x)
    W205#(s(k),x)	→	W205#(k,x)
    W205#(s(k),x)	→	Tucp#(x)
    Tucp#(SRlcase(x))	→	W255#(k,x)
    W255#(s(k),x)	→	W255#(k,x)
    W255#(s(k),x)	→	Tucp#(x)
    Tucp#(SRlcase(W257(x)))	→	W262#(k,x)
    W262#(s(k),x)	→	W262#(k,x)
    W262#(s(k),x)	→	Tucp#(x)
    Tucp#(SRlcase(x))	→	W268#(k,x)
    W268#(s(k),x)	→	W268#(k,x)
    W268#(s(k),x)	→	Tucp#(x)
    Tucp#(SRlcase(W269(x)))	→	W274#(k,x)
    W274#(s(k),x)	→	W274#(k,x)
    W274#(s(k),x)	→	Tucp#(x)
    Tucp#(SRlseq(x))	→	W324#(k,x)
    W324#(s(k),x)	→	W324#(k,x)
    W324#(s(k),x)	→	Tucp#(x)
    Tucp#(SRlseq(W326(x)))	→	W331#(k,x)
    W331#(s(k),x)	→	W331#(k,x)
    W331#(s(k),x)	→	Tucp#(x)
    Tucp#(SRlseq(x))	→	W337#(k,x)
    W337#(s(k),x)	→	W337#(k,x)
    W337#(s(k),x)	→	Tucp#(x)
    Tucp#(SRlseq(W338(x)))	→	W343#(k,x)
    W343#(s(k),x)	→	W343#(k,x)
    W343#(s(k),x)	→	Tucp#(x)
    Tucp#(SRseq(x))	→	W509#(k,x)
    W509#(s(k),x)	→	W509#(k,x)
    W509#(s(k),x)	→	Tucp#(x)
    Tucp#(SRlbeta(x))	→	Tucp#(x)
    Tucp#(SRcp(x))	→	Tucp#(x)
    Tucp#(SRllet(x))	→	Tucp#(x)
    Tucp#(SRlapp(x))	→	Tucp#(x)
    Tucp#(SRlapp(W26(x)))	→	Tucp#(x)
    Tucp#(SRlapp(SRllet(x)))	→	Tucp#(x)
    Tucp#(SRlcase(x))	→	Tucp#(x)
    Tucp#(SRlcase(W40(x)))	→	Tucp#(x)
    Tucp#(SRlcase(SRllet(x)))	→	Tucp#(x)
    Tucp#(SRlseq(x))	→	Tucp#(x)
    Tucp#(SRlseq(W54(x)))	→	Tucp#(x)
    Tucp#(SRlseq(SRllet(x)))	→	Tucp#(x)
    Tucp#(SRseq(x))	→	Tucp#(x)
    Tucp#(SRlapp(W239(x)))	→	Tucp#(x)
    Tucp#(SRlcase(W308(x)))	→	Tucp#(x)
    Tucp#(SRlseq(W377(x)))	→	Tucp#(x)

    1.1.1.1.1.1.4 Usable Rules Processor

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

    There are no rules.

    1.1.1.1.1.1.4.1 Innermost Lhss Removal Processor

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

    There are no lhss.

    1.1.1.1.1.1.4.1.1 Reduction Pair Processor

    Using the linear polynomial interpretation over the naturals

    [W121#(x1, x2)]	=	1 + 1 · x2
    [s(x1)]	=	0
    [Tucp#(x1)]	=	1 + 1 · x1
    [SRlbeta(x1)]	=	1 + 1 · x1
    [SRcp(x1)]	=	1 + 1 · x1
    [W161#(x1, x2)]	=	1 + 1 · x2
    [SRllet(x1)]	=	1 + 1 · x1
    [W181#(x1, x2)]	=	1 + 1 · x2
    [SRlapp(x1)]	=	1 + 1 · x1
    [W186#(x1, x2)]	=	1 + 1 · x2
    [W188(x1)]	=	1 · x1
    [W193#(x1, x2)]	=	1 + 1 · x2
    [W199#(x1, x2)]	=	1 + 1 · x2
    [W200(x1)]	=	1 · x1
    [W205#(x1, x2)]	=	1 + 1 · x2
    [SRlcase(x1)]	=	1 + 1 · x1
    [W255#(x1, x2)]	=	1 + 1 · x2
    [W257(x1)]	=	1 · x1
    [W262#(x1, x2)]	=	1 + 1 · x2
    [W268#(x1, x2)]	=	1 + 1 · x2
    [W269(x1)]	=	1 · x1
    [W274#(x1, x2)]	=	1 + 1 · x2
    [SRlseq(x1)]	=	1 + 1 · x1
    [W324#(x1, x2)]	=	1 + 1 · x2
    [W326(x1)]	=	1 · x1
    [W331#(x1, x2)]	=	1 + 1 · x2
    [W337#(x1, x2)]	=	1 + 1 · x2
    [W338(x1)]	=	1 · x1
    [W343#(x1, x2)]	=	1 + 1 · x2
    [SRseq(x1)]	=	1 + 1 · x1
    [W509#(x1, x2)]	=	1 + 1 · x2
    [W26(x1)]	=	1 · x1
    [W40(x1)]	=	1 · x1
    [W54(x1)]	=	1 + 1 · x1
    [W239(x1)]	=	1 + 1 · x1
    [W308(x1)]	=	1 · x1
    [W377(x1)]	=	1 + 1 · x1

    the pairs

    W121#(s(k),x)	→	W121#(k,x)
    W121#(s(k),x)	→	Tucp#(x)
    W161#(s(k),x)	→	W161#(k,x)
    W161#(s(k),x)	→	Tucp#(x)
    W181#(s(k),x)	→	W181#(k,x)
    W181#(s(k),x)	→	Tucp#(x)
    W186#(s(k),x)	→	W186#(k,x)
    W186#(s(k),x)	→	Tucp#(x)
    W193#(s(k),x)	→	W193#(k,x)
    W193#(s(k),x)	→	Tucp#(x)
    W199#(s(k),x)	→	W199#(k,x)
    W199#(s(k),x)	→	Tucp#(x)
    W205#(s(k),x)	→	W205#(k,x)
    W205#(s(k),x)	→	Tucp#(x)
    W255#(s(k),x)	→	W255#(k,x)
    W255#(s(k),x)	→	Tucp#(x)
    W262#(s(k),x)	→	W262#(k,x)
    W262#(s(k),x)	→	Tucp#(x)
    W268#(s(k),x)	→	W268#(k,x)
    W268#(s(k),x)	→	Tucp#(x)
    W274#(s(k),x)	→	W274#(k,x)
    W274#(s(k),x)	→	Tucp#(x)
    W324#(s(k),x)	→	W324#(k,x)
    W324#(s(k),x)	→	Tucp#(x)
    W331#(s(k),x)	→	W331#(k,x)
    W331#(s(k),x)	→	Tucp#(x)
    W337#(s(k),x)	→	W337#(k,x)
    W337#(s(k),x)	→	Tucp#(x)
    W343#(s(k),x)	→	W343#(k,x)
    W343#(s(k),x)	→	Tucp#(x)
    W509#(s(k),x)	→	W509#(k,x)
    W509#(s(k),x)	→	Tucp#(x)

    remain.

    1.1.1.1.1.1.4.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 16 components.

    • The 1^st component contains the pair

      W121#(s(k),x)

      →

      W121#(k,x)

      1.1.1.1.1.1.4.1.1.1.1 Size-Change Termination

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

      • W121^#(s(k),x) → W121^#(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

      W161#(s(k),x)

      →

      W161#(k,x)

      1.1.1.1.1.1.4.1.1.1.2 Size-Change Termination

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

      • W161^#(s(k),x) → W161^#(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

      W181#(s(k),x)

      →

      W181#(k,x)

      1.1.1.1.1.1.4.1.1.1.3 Size-Change Termination

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

      • W181^#(s(k),x) → W181^#(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

      W186#(s(k),x)

      →

      W186#(k,x)

      1.1.1.1.1.1.4.1.1.1.4 Size-Change Termination

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

      • W186^#(s(k),x) → W186^#(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

      W193#(s(k),x)

      →

      W193#(k,x)

      1.1.1.1.1.1.4.1.1.1.5 Size-Change Termination

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

      • W193^#(s(k),x) → W193^#(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

      W199#(s(k),x)

      →

      W199#(k,x)

      1.1.1.1.1.1.4.1.1.1.6 Size-Change Termination

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

      • W199^#(s(k),x) → W199^#(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

      W205#(s(k),x)

      →

      W205#(k,x)

      1.1.1.1.1.1.4.1.1.1.7 Size-Change Termination

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

      • W205^#(s(k),x) → W205^#(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

      W255#(s(k),x)

      →

      W255#(k,x)

      1.1.1.1.1.1.4.1.1.1.8 Size-Change Termination

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

      • W255^#(s(k),x) → W255^#(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

      W262#(s(k),x)

      →

      W262#(k,x)

      1.1.1.1.1.1.4.1.1.1.9 Size-Change Termination

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

      • W262^#(s(k),x) → W262^#(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

      W268#(s(k),x)

      →

      W268#(k,x)

      1.1.1.1.1.1.4.1.1.1.10 Size-Change Termination

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

      • W268^#(s(k),x) → W268^#(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

      W274#(s(k),x)

      →

      W274#(k,x)

      1.1.1.1.1.1.4.1.1.1.11 Size-Change Termination

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

      • W274^#(s(k),x) → W274^#(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

      W324#(s(k),x)

      →

      W324#(k,x)

      1.1.1.1.1.1.4.1.1.1.12 Size-Change Termination

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

      • W324^#(s(k),x) → W324^#(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

      W331#(s(k),x)

      →

      W331#(k,x)

      1.1.1.1.1.1.4.1.1.1.13 Size-Change Termination

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

      • W331^#(s(k),x) → W331^#(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

      W337#(s(k),x)

      →

      W337#(k,x)

      1.1.1.1.1.1.4.1.1.1.14 Size-Change Termination

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

      • W337^#(s(k),x) → W337^#(k,x):

        1>1, 2>=2

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

    • The 15^th component contains the pair

      W343#(s(k),x)

      →

      W343#(k,x)

      1.1.1.1.1.1.4.1.1.1.15 Size-Change Termination

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

      • W343^#(s(k),x) → W343^#(k,x):

        1>1, 2>=2

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

    • The 16^th component contains the pair

      W509#(s(k),x)

      →

      W509#(k,x)

      1.1.1.1.1.1.4.1.1.1.16 Size-Change Termination

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

      • W509^#(s(k),x) → W509^#(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

    W430#(s(k),x)

    →

    W430#(k,x)

    1.1.1.1.1.1.5 Usable Rules Processor

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

    There are no rules.

    1.1.1.1.1.1.5.1 Innermost Lhss Removal Processor

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

    There are no lhss.

    1.1.1.1.1.1.5.1.1 Size-Change Termination

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

    • W430^#(s(k),x) → W430^#(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

  W109#(s(k),x)

  →

  W109#(k,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.

  • W109^#(s(k),x) → W109^#(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

  W116#(s(k),x)

  →

  W116#(k,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 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [W116#(x1, x2)]	=	3 · x1 + 1 · x2
  [s(x1)]	=	1 + 3 · x1

  all pairs and rules could be removed.

  1.1.1.3.1.1.1 P is empty

  There are no pairs anymore.

• The 4^th component contains the pair

  W156#(s(k),x)

  →

  W156#(k,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.

  • W156^#(s(k),x) → W156^#(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

  W393#(s(k),x)

  →

  W393#(k,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.

  • W393^#(s(k),x) → W393^#(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

  W399#(s(k),x)

  →

  W399#(k,x)

  1.1.1.6 Usable Rules Processor

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

  There are no rules.

  1.1.1.6.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.6.1.1 Size-Change Termination

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

  • W399^#(s(k),x) → W399^#(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

  W498#(s(k),x)

  →

  W498#(k,x)

  1.1.1.7 Usable Rules Processor

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

  There are no rules.

  1.1.1.7.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.7.1.1 Size-Change Termination

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

  • W498^#(s(k),x) → W498^#(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

  W504#(s(k),x)

  →

  W504#(k,x)

  1.1.1.8 Usable Rules Processor

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

  There are no rules.

  1.1.1.8.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.8.1.1 Size-Change Termination

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

  • W504^#(s(k),x) → W504^#(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

  W520#(s(k),x)

  →

  W520#(k,x)

  1.1.1.9 Usable Rules Processor

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

  There are no rules.

  1.1.1.9.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.9.1.1 Size-Change Termination

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

  • W520^#(s(k),x) → W520^#(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

  W527#(s(k),x)

  →

  W527#(k,x)

  1.1.1.10 Usable Rules Processor

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

  There are no rules.

  1.1.1.10.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.10.1.1 Size-Change Termination

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

  • W527^#(s(k),x) → W527^#(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 pairs

  W625#(s(k),x)	→	W625#(k,x)
  W625#(s(k),x)	→	Tgc#(x)
  Tgc#(SRlbeta(x))	→	W625#(k,x)
  Tgc#(SRlbeta(x))	→	W631#(k,x)
  W631#(s(k),x)	→	W631#(k,x)
  W631#(s(k),x)	→	Tgc#(x)
  Tgc#(SRcp(x))	→	W653#(k,x)
  W653#(s(k),x)	→	W653#(k,x)
  W653#(s(k),x)	→	Tgc#(x)
  Tgc#(SRllet(x))	→	W668#(k,x)
  W668#(s(k),x)	→	W668#(k,x)
  W668#(s(k),x)	→	Tgc#(x)
  Tgc#(SRlapp(x))	→	W673#(k,x)
  W673#(s(k),x)	→	W673#(k,x)
  W673#(s(k),x)	→	Tgc#(x)
  Tgc#(SRlapp(x))	→	W679#(k,x)
  W679#(s(k),x)	→	W679#(k,x)
  W679#(s(k),x)	→	Tgc#(x)
  Tgc#(SRlcase(x))	→	W701#(k,x)
  W701#(s(k),x)	→	W701#(k,x)
  W701#(s(k),x)	→	Tgc#(x)
  Tgc#(SRlcase(x))	→	W707#(k,x)
  W707#(s(k),x)	→	W707#(k,x)
  W707#(s(k),x)	→	Tgc#(x)
  Tgc#(SRlseq(x))	→	W729#(k,x)
  W729#(s(k),x)	→	W729#(k,x)
  W729#(s(k),x)	→	Tgc#(x)
  Tgc#(SRlseq(x))	→	W735#(k,x)
  W735#(s(k),x)	→	W735#(k,x)
  W735#(s(k),x)	→	Tgc#(x)
  Tgc#(SRcase(x))	→	W757#(k,x)
  W757#(s(k),x)	→	W757#(k,x)
  W757#(s(k),x)	→	Tgc#(x)
  Tgc#(SRcase(x))	→	W763#(k,x)
  W763#(s(k),x)	→	W763#(k,x)
  W763#(s(k),x)	→	Tgc#(x)
  Tgc#(SRseq(x))	→	W790#(k,x)
  W790#(s(k),x)	→	W790#(k,x)
  W790#(s(k),x)	→	Tgc#(x)
  Tgc#(SRseq(x))	→	W796#(k,x)
  W796#(s(k),x)	→	W796#(k,x)
  W796#(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)

  1.1.1.11 Usable Rules Processor

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

  There are no rules.

  1.1.1.11.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.11.1.1 Size-Change Termination

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

  • W625^#(s(k),x) → W625^#(k,x):

    1>1, 2>=2

  • W625^#(s(k),x) → Tgc^#(x):

    2>=1

  • Tgc^#(SRlbeta(x)) → W625^#(k,x):

    1>2

  • W631^#(s(k),x) → Tgc^#(x):

    2>=1

  • W631^#(s(k),x) → W631^#(k,x):

    1>1, 2>=2

  • Tgc^#(SRlbeta(x)) → W631^#(k,x):

    1>2

  • W653^#(s(k),x) → Tgc^#(x):

    2>=1

  • W653^#(s(k),x) → W653^#(k,x):

    1>1, 2>=2

  • Tgc^#(SRcp(x)) → W653^#(k,x):

    1>2

  • W668^#(s(k),x) → Tgc^#(x):

    2>=1

  • W668^#(s(k),x) → W668^#(k,x):

    1>1, 2>=2

  • Tgc^#(SRllet(x)) → W668^#(k,x):

    1>2

  • W673^#(s(k),x) → Tgc^#(x):

    2>=1

  • W673^#(s(k),x) → W673^#(k,x):

    1>1, 2>=2

  • Tgc^#(SRlapp(x)) → W673^#(k,x):

    1>2

  • W679^#(s(k),x) → Tgc^#(x):

    2>=1

  • W679^#(s(k),x) → W679^#(k,x):

    1>1, 2>=2

  • Tgc^#(SRlapp(x)) → W679^#(k,x):

    1>2

  • W701^#(s(k),x) → Tgc^#(x):

    2>=1

  • W701^#(s(k),x) → W701^#(k,x):

    1>1, 2>=2

  • Tgc^#(SRlcase(x)) → W701^#(k,x):

    1>2

  • W707^#(s(k),x) → Tgc^#(x):

    2>=1

  • W707^#(s(k),x) → W707^#(k,x):

    1>1, 2>=2

  • Tgc^#(SRlcase(x)) → W707^#(k,x):

    1>2

  • W729^#(s(k),x) → Tgc^#(x):

    2>=1

  • W729^#(s(k),x) → W729^#(k,x):

    1>1, 2>=2

  • Tgc^#(SRlseq(x)) → W729^#(k,x):

    1>2

  • W735^#(s(k),x) → Tgc^#(x):

    2>=1

  • W735^#(s(k),x) → W735^#(k,x):

    1>1, 2>=2

  • Tgc^#(SRlseq(x)) → W735^#(k,x):

    1>2

  • W757^#(s(k),x) → Tgc^#(x):

    2>=1

  • W757^#(s(k),x) → W757^#(k,x):

    1>1, 2>=2

  • Tgc^#(SRcase(x)) → W757^#(k,x):

    1>2

  • W763^#(s(k),x) → Tgc^#(x):

    2>=1

  • W763^#(s(k),x) → W763^#(k,x):

    1>1, 2>=2

  • Tgc^#(SRcase(x)) → W763^#(k,x):

    1>2

  • W790^#(s(k),x) → Tgc^#(x):

    2>=1

  • W796^#(s(k),x) → Tgc^#(x):

    2>=1

  • W790^#(s(k),x) → W790^#(k,x):

    1>1, 2>=2

  • W796^#(s(k),x) → W796^#(k,x):

    1>1, 2>=2

  • Tgc^#(SRseq(x)) → W790^#(k,x):

    1>2

  • Tgc^#(SRseq(x)) → W796^#(k,x):

    1>2

  • Tgc^#(SRlbeta(x)) → Tgc^#(x):

    1>1

  • Tgc^#(SRcp(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.