107.67/111.38	YES
107.67/111.38	
107.67/111.38	Problem 1: 
107.67/111.38	
107.67/111.38	(VAR v_NonEmpty:S M:S N:S V1:S V2:S X:S X1:S X2:S X3:S)
107.67/111.38	(RULES
107.67/111.38	a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.38	a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.38	a__U12(tt) -> tt
107.67/111.38	a__U12(X:S) -> U12(X:S)
107.67/111.38	a__U21(tt) -> tt
107.67/111.38	a__U21(X:S) -> U21(X:S)
107.67/111.38	a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.38	a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.38	a__U32(tt) -> tt
107.67/111.38	a__U32(X:S) -> U32(X:S)
107.67/111.38	a__U41(tt,N:S) -> mark(N:S)
107.67/111.38	a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.38	a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.38	a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.38	a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.38	a__U61(tt) -> 0
107.67/111.38	a__U61(X:S) -> U61(X:S)
107.67/111.38	a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.38	a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.38	a__isNat(0) -> tt
107.67/111.38	a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.38	a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.38	a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.38	a__isNat(X:S) -> isNat(X:S)
107.67/111.38	a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.38	a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.38	a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.38	a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.38	mark(0) -> 0
107.67/111.38	mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.38	mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.38	mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.38	mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.38	mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.38	mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.38	mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.38	mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.38	mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.38	mark(s(X:S)) -> s(mark(X:S))
107.67/111.38	mark(tt) -> tt
107.67/111.38	mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.38	) 
107.67/111.38	(STRATEGY INNERMOST)
107.67/111.38	
107.67/111.38	Problem 1: 
107.67/111.38	
107.67/111.38	Dependency Pairs Processor:
107.67/111.38	-> Pairs:
107.67/111.38	 A__U11(tt,V2:S) -> A__U12(a__isNat(V2:S))
107.67/111.38	 A__U11(tt,V2:S) -> A__ISNAT(V2:S)
107.67/111.38	 A__U31(tt,V2:S) -> A__U32(a__isNat(V2:S))
107.67/111.38	 A__U31(tt,V2:S) -> A__ISNAT(V2:S)
107.67/111.38	 A__U41(tt,N:S) -> MARK(N:S)
107.67/111.38	 A__U51(tt,M:S,N:S) -> A__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 A__U51(tt,M:S,N:S) -> A__ISNAT(N:S)
107.67/111.38	 A__U52(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S))
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__U71(tt,M:S,N:S) -> A__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 A__U71(tt,M:S,N:S) -> A__ISNAT(N:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
107.67/111.38	 A__ISNAT(s(V1:S)) -> A__U21(a__isNat(V1:S))
107.67/111.38	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
107.67/111.38	 A__ISNAT(x(V1:S,V2:S)) -> A__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 A__ISNAT(x(V1:S,V2:S)) -> A__ISNAT(V1:S)
107.67/111.38	 A__PLUS(N:S,0) -> A__U41(a__isNat(N:S),N:S)
107.67/111.38	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
107.67/111.38	 A__PLUS(N:S,s(M:S)) -> A__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
107.67/111.38	 A__X(N:S,0) -> A__U61(a__isNat(N:S))
107.67/111.38	 A__X(N:S,0) -> A__ISNAT(N:S)
107.67/111.38	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 A__X(N:S,s(M:S)) -> A__ISNAT(M:S)
107.67/111.38	 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S)
107.67/111.38	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U12(X:S)) -> A__U12(mark(X:S))
107.67/111.38	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U21(X:S)) -> A__U21(mark(X:S))
107.67/111.38	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U31(X1:S,X2:S)) -> A__U31(mark(X1:S),X2:S)
107.67/111.38	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U32(X:S)) -> A__U32(mark(X:S))
107.67/111.38	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U41(X1:S,X2:S)) -> A__U41(mark(X1:S),X2:S)
107.67/111.38	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> A__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> A__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U61(X:S)) -> A__U61(mark(X:S))
107.67/111.38	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	-> Rules:
107.67/111.38	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.38	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.38	 a__U12(tt) -> tt
107.67/111.38	 a__U12(X:S) -> U12(X:S)
107.67/111.38	 a__U21(tt) -> tt
107.67/111.38	 a__U21(X:S) -> U21(X:S)
107.67/111.38	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.38	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.38	 a__U32(tt) -> tt
107.67/111.38	 a__U32(X:S) -> U32(X:S)
107.67/111.38	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.38	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.38	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.38	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.38	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.38	 a__U61(tt) -> 0
107.67/111.38	 a__U61(X:S) -> U61(X:S)
107.67/111.38	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.38	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.38	 a__isNat(0) -> tt
107.67/111.38	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.38	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(X:S) -> isNat(X:S)
107.67/111.38	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.38	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.38	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.38	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.38	 mark(0) -> 0
107.67/111.38	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.38	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.38	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.38	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.38	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.38	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.38	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.38	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.38	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.38	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.38	 mark(tt) -> tt
107.67/111.38	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.38	
107.67/111.38	Problem 1: 
107.67/111.38	
107.67/111.38	SCC Processor:
107.67/111.38	-> Pairs:
107.67/111.38	 A__U11(tt,V2:S) -> A__U12(a__isNat(V2:S))
107.67/111.38	 A__U11(tt,V2:S) -> A__ISNAT(V2:S)
107.67/111.38	 A__U31(tt,V2:S) -> A__U32(a__isNat(V2:S))
107.67/111.38	 A__U31(tt,V2:S) -> A__ISNAT(V2:S)
107.67/111.38	 A__U41(tt,N:S) -> MARK(N:S)
107.67/111.38	 A__U51(tt,M:S,N:S) -> A__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 A__U51(tt,M:S,N:S) -> A__ISNAT(N:S)
107.67/111.38	 A__U52(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S))
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__U71(tt,M:S,N:S) -> A__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 A__U71(tt,M:S,N:S) -> A__ISNAT(N:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
107.67/111.38	 A__ISNAT(s(V1:S)) -> A__U21(a__isNat(V1:S))
107.67/111.38	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
107.67/111.38	 A__ISNAT(x(V1:S,V2:S)) -> A__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 A__ISNAT(x(V1:S,V2:S)) -> A__ISNAT(V1:S)
107.67/111.38	 A__PLUS(N:S,0) -> A__U41(a__isNat(N:S),N:S)
107.67/111.38	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
107.67/111.38	 A__PLUS(N:S,s(M:S)) -> A__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
107.67/111.38	 A__X(N:S,0) -> A__U61(a__isNat(N:S))
107.67/111.38	 A__X(N:S,0) -> A__ISNAT(N:S)
107.67/111.38	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 A__X(N:S,s(M:S)) -> A__ISNAT(M:S)
107.67/111.38	 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S)
107.67/111.38	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U12(X:S)) -> A__U12(mark(X:S))
107.67/111.38	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U21(X:S)) -> A__U21(mark(X:S))
107.67/111.38	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U31(X1:S,X2:S)) -> A__U31(mark(X1:S),X2:S)
107.67/111.38	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U32(X:S)) -> A__U32(mark(X:S))
107.67/111.38	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U41(X1:S,X2:S)) -> A__U41(mark(X1:S),X2:S)
107.67/111.38	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> A__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> A__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U61(X:S)) -> A__U61(mark(X:S))
107.67/111.38	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	-> Rules:
107.67/111.38	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.38	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.38	 a__U12(tt) -> tt
107.67/111.38	 a__U12(X:S) -> U12(X:S)
107.67/111.38	 a__U21(tt) -> tt
107.67/111.38	 a__U21(X:S) -> U21(X:S)
107.67/111.38	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.38	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.38	 a__U32(tt) -> tt
107.67/111.38	 a__U32(X:S) -> U32(X:S)
107.67/111.38	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.38	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.38	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.38	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.38	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.38	 a__U61(tt) -> 0
107.67/111.38	 a__U61(X:S) -> U61(X:S)
107.67/111.38	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.38	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.38	 a__isNat(0) -> tt
107.67/111.38	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.38	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(X:S) -> isNat(X:S)
107.67/111.38	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.38	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.38	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.38	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.38	 mark(0) -> 0
107.67/111.38	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.38	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.38	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.38	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.38	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.38	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.38	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.38	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.38	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.38	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.38	 mark(tt) -> tt
107.67/111.38	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.38	->Strongly Connected Components:
107.67/111.38	->->Cycle:
107.67/111.38	->->-> Pairs:
107.67/111.38	 A__U11(tt,V2:S) -> A__ISNAT(V2:S)
107.67/111.38	 A__U31(tt,V2:S) -> A__ISNAT(V2:S)
107.67/111.38	 A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
107.67/111.38	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
107.67/111.38	 A__ISNAT(x(V1:S,V2:S)) -> A__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 A__ISNAT(x(V1:S,V2:S)) -> A__ISNAT(V1:S)
107.67/111.38	->->-> Rules:
107.67/111.38	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.38	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.38	 a__U12(tt) -> tt
107.67/111.38	 a__U12(X:S) -> U12(X:S)
107.67/111.38	 a__U21(tt) -> tt
107.67/111.38	 a__U21(X:S) -> U21(X:S)
107.67/111.38	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.38	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.38	 a__U32(tt) -> tt
107.67/111.38	 a__U32(X:S) -> U32(X:S)
107.67/111.38	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.38	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.38	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.38	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.38	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.38	 a__U61(tt) -> 0
107.67/111.38	 a__U61(X:S) -> U61(X:S)
107.67/111.38	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.38	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.38	 a__isNat(0) -> tt
107.67/111.38	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.38	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(X:S) -> isNat(X:S)
107.67/111.38	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.38	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.38	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.38	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.38	 mark(0) -> 0
107.67/111.38	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.38	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.38	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.38	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.38	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.38	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.38	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.38	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.38	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.38	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.38	 mark(tt) -> tt
107.67/111.38	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.38	->->Cycle:
107.67/111.38	->->-> Pairs:
107.67/111.38	 A__U41(tt,N:S) -> MARK(N:S)
107.67/111.38	 A__U51(tt,M:S,N:S) -> A__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 A__U52(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S))
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__U71(tt,M:S,N:S) -> A__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__PLUS(N:S,0) -> A__U41(a__isNat(N:S),N:S)
107.67/111.38	 A__PLUS(N:S,s(M:S)) -> A__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U41(X1:S,X2:S)) -> A__U41(mark(X1:S),X2:S)
107.67/111.38	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> A__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> A__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	->->-> Rules:
107.67/111.38	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.38	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.38	 a__U12(tt) -> tt
107.67/111.38	 a__U12(X:S) -> U12(X:S)
107.67/111.38	 a__U21(tt) -> tt
107.67/111.38	 a__U21(X:S) -> U21(X:S)
107.67/111.38	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.38	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.38	 a__U32(tt) -> tt
107.67/111.38	 a__U32(X:S) -> U32(X:S)
107.67/111.38	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.38	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.38	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.38	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.38	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.38	 a__U61(tt) -> 0
107.67/111.38	 a__U61(X:S) -> U61(X:S)
107.67/111.38	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.38	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.38	 a__isNat(0) -> tt
107.67/111.38	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.38	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(X:S) -> isNat(X:S)
107.67/111.38	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.38	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.38	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.38	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.38	 mark(0) -> 0
107.67/111.38	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.38	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.38	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.38	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.38	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.38	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.38	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.38	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.38	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.38	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.38	 mark(tt) -> tt
107.67/111.38	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.38	
107.67/111.38	
107.67/111.38	The problem is decomposed in 2 subproblems.
107.67/111.38	
107.67/111.38	Problem 1.1: 
107.67/111.38	
107.67/111.38	Subterm Processor:
107.67/111.38	-> Pairs:
107.67/111.38	 A__U11(tt,V2:S) -> A__ISNAT(V2:S)
107.67/111.38	 A__U31(tt,V2:S) -> A__ISNAT(V2:S)
107.67/111.38	 A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
107.67/111.38	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
107.67/111.38	 A__ISNAT(x(V1:S,V2:S)) -> A__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 A__ISNAT(x(V1:S,V2:S)) -> A__ISNAT(V1:S)
107.67/111.38	-> Rules:
107.67/111.38	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.38	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.38	 a__U12(tt) -> tt
107.67/111.38	 a__U12(X:S) -> U12(X:S)
107.67/111.38	 a__U21(tt) -> tt
107.67/111.38	 a__U21(X:S) -> U21(X:S)
107.67/111.38	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.38	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.38	 a__U32(tt) -> tt
107.67/111.38	 a__U32(X:S) -> U32(X:S)
107.67/111.38	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.38	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.38	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.38	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.38	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.38	 a__U61(tt) -> 0
107.67/111.38	 a__U61(X:S) -> U61(X:S)
107.67/111.38	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.38	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.38	 a__isNat(0) -> tt
107.67/111.38	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.38	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(X:S) -> isNat(X:S)
107.67/111.38	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.38	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.38	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.38	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.38	 mark(0) -> 0
107.67/111.38	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.38	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.38	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.38	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.38	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.38	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.38	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.38	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.38	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.38	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.38	 mark(tt) -> tt
107.67/111.38	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.38	->Projection:
107.67/111.38	 pi(A__U11) = 2
107.67/111.38	 pi(A__U31) = 2
107.67/111.38	 pi(A__ISNAT) = 1
107.67/111.38	
107.67/111.38	Problem 1.1: 
107.67/111.38	
107.67/111.38	SCC Processor:
107.67/111.38	-> Pairs:
107.67/111.38	 A__U11(tt,V2:S) -> A__ISNAT(V2:S)
107.67/111.38	 A__U31(tt,V2:S) -> A__ISNAT(V2:S)
107.67/111.38	-> Rules:
107.67/111.38	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.38	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.38	 a__U12(tt) -> tt
107.67/111.38	 a__U12(X:S) -> U12(X:S)
107.67/111.38	 a__U21(tt) -> tt
107.67/111.38	 a__U21(X:S) -> U21(X:S)
107.67/111.38	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.38	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.38	 a__U32(tt) -> tt
107.67/111.38	 a__U32(X:S) -> U32(X:S)
107.67/111.38	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.38	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.38	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.38	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.38	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.38	 a__U61(tt) -> 0
107.67/111.38	 a__U61(X:S) -> U61(X:S)
107.67/111.38	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.38	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.38	 a__isNat(0) -> tt
107.67/111.38	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.38	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(X:S) -> isNat(X:S)
107.67/111.38	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.38	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.38	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.38	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.38	 mark(0) -> 0
107.67/111.38	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.38	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.38	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.38	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.38	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.38	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.38	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.38	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.38	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.38	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.38	 mark(tt) -> tt
107.67/111.38	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.38	->Strongly Connected Components:
107.67/111.38	 There is no strongly connected component
107.67/111.38	
107.67/111.38	The problem is finite.
107.67/111.38	
107.67/111.38	Problem 1.2: 
107.67/111.38	
107.67/111.38	Reduction Pairs Processor:
107.67/111.38	-> Pairs:
107.67/111.38	 A__U41(tt,N:S) -> MARK(N:S)
107.67/111.38	 A__U51(tt,M:S,N:S) -> A__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 A__U52(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S))
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__U71(tt,M:S,N:S) -> A__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__PLUS(N:S,0) -> A__U41(a__isNat(N:S),N:S)
107.67/111.38	 A__PLUS(N:S,s(M:S)) -> A__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U41(X1:S,X2:S)) -> A__U41(mark(X1:S),X2:S)
107.67/111.38	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> A__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> A__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	-> Rules:
107.67/111.38	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.38	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.38	 a__U12(tt) -> tt
107.67/111.38	 a__U12(X:S) -> U12(X:S)
107.67/111.38	 a__U21(tt) -> tt
107.67/111.38	 a__U21(X:S) -> U21(X:S)
107.67/111.38	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.38	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.38	 a__U32(tt) -> tt
107.67/111.38	 a__U32(X:S) -> U32(X:S)
107.67/111.38	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.38	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.38	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.38	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.38	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.38	 a__U61(tt) -> 0
107.67/111.38	 a__U61(X:S) -> U61(X:S)
107.67/111.38	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.38	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.38	 a__isNat(0) -> tt
107.67/111.38	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.38	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(X:S) -> isNat(X:S)
107.67/111.38	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.38	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.38	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.38	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.38	 mark(0) -> 0
107.67/111.38	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.38	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.38	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.38	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.38	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.38	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.38	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.38	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.38	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.38	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.38	 mark(tt) -> tt
107.67/111.38	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.38	-> Usable rules:
107.67/111.38	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.38	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.38	 a__U12(tt) -> tt
107.67/111.38	 a__U12(X:S) -> U12(X:S)
107.67/111.38	 a__U21(tt) -> tt
107.67/111.38	 a__U21(X:S) -> U21(X:S)
107.67/111.38	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.38	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.38	 a__U32(tt) -> tt
107.67/111.38	 a__U32(X:S) -> U32(X:S)
107.67/111.38	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.38	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.38	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.38	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.38	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.38	 a__U61(tt) -> 0
107.67/111.38	 a__U61(X:S) -> U61(X:S)
107.67/111.38	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.38	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.38	 a__isNat(0) -> tt
107.67/111.38	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.38	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(X:S) -> isNat(X:S)
107.67/111.38	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.38	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.38	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.38	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.38	 mark(0) -> 0
107.67/111.38	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.38	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.38	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.38	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.38	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.38	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.38	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.38	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.38	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.38	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.38	 mark(tt) -> tt
107.67/111.38	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.38	->Interpretation type:
107.67/111.38	 Simple mixed
107.67/111.38	->Coefficients:
107.67/111.38	 Natural Numbers
107.67/111.38	->Dimension:
107.67/111.38	 1
107.67/111.38	->Bound:
107.67/111.38	 2
107.67/111.38	->Interpretation:
107.67/111.38	 
107.67/111.38	[a__U11](X1,X2) = 2.X1.X2 + 2.X1
107.67/111.38	[a__U12](X) = 2.X.X + 2.X
107.67/111.38	[a__U21](X) = 2.X
107.67/111.38	[a__U31](X1,X2) = 2.X1.X2 + 2.X1
107.67/111.38	[a__U32](X) = 2.X
107.67/111.38	[a__U41](X1,X2) = 2.X1.X2 + X1 + X2 + 2
107.67/111.38	[a__U51](X1,X2,X3) = 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2
107.67/111.38	[a__U52](X1,X2,X3) = 2.X1.X2.X3 + X1.X2 + X1 + X2 + X3 + 2
107.67/111.38	[a__U61](X) = 2.X.X + 2.X + 2
107.67/111.38	[a__U71](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X3 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2
107.67/111.38	[a__U72](X1,X2,X3) = 2.X1.X2.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1
107.67/111.38	[a__isNat](X) = 0
107.67/111.38	[a__plus](X1,X2) = X1 + X2 + 1
107.67/111.38	[a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2
107.67/111.38	[mark](X) = X
107.67/111.38	[0] = 2
107.67/111.38	[U11](X1,X2) = 2.X1.X2 + 2.X1
107.67/111.38	[U12](X) = 2.X.X + 2.X
107.67/111.38	[U21](X) = 2.X
107.67/111.38	[U31](X1,X2) = 2.X1.X2 + 2.X1
107.67/111.38	[U32](X) = 2.X
107.67/111.38	[U41](X1,X2) = 2.X1.X2 + X1 + X2 + 2
107.67/111.38	[U51](X1,X2,X3) = 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2
107.67/111.38	[U52](X1,X2,X3) = 2.X1.X2.X3 + X1.X2 + X1 + X2 + X3 + 2
107.67/111.38	[U61](X) = 2.X.X + 2.X + 2
107.67/111.38	[U71](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X3 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2
107.67/111.38	[U72](X1,X2,X3) = 2.X1.X2.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1
107.67/111.38	[fSNonEmpty] = 0
107.67/111.38	[isNat](X) = 0
107.67/111.38	[plus](X1,X2) = X1 + X2 + 1
107.67/111.38	[s](X) = X + 1
107.67/111.38	[tt] = 0
107.67/111.38	[x](X1,X2) = 2.X1.X2 + X1 + 2.X2
107.67/111.38	[A__U11](X1,X2) = 0
107.67/111.38	[A__U12](X) = 0
107.67/111.38	[A__U21](X) = 0
107.67/111.38	[A__U31](X1,X2) = 0
107.67/111.38	[A__U32](X) = 0
107.67/111.38	[A__U41](X1,X2) = X2 + 2
107.67/111.38	[A__U51](X1,X2,X3) = X1.X2 + 2.X1.X3 + X1 + X2 + X3 + 2
107.67/111.38	[A__U52](X1,X2,X3) = X2 + X3 + 2
107.67/111.38	[A__U61](X) = 0
107.67/111.38	[A__U71](X1,X2,X3) = 2.X1.X2.X3 + X1.X3 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2
107.67/111.38	[A__U72](X1,X2,X3) = X1.X2.X3 + 2.X2.X3 + 2.X2 + 2.X3 + 2
107.67/111.38	[A__ISNAT](X) = 0
107.67/111.38	[A__PLUS](X1,X2) = X1 + X2 + 2
107.67/111.38	[A__X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1
107.67/111.38	[MARK](X) = X + 1
107.67/111.38	
107.67/111.38	Problem 1.2: 
107.67/111.38	
107.67/111.38	SCC Processor:
107.67/111.38	-> Pairs:
107.67/111.38	 A__U51(tt,M:S,N:S) -> A__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 A__U52(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S))
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__U71(tt,M:S,N:S) -> A__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__PLUS(N:S,0) -> A__U41(a__isNat(N:S),N:S)
107.67/111.38	 A__PLUS(N:S,s(M:S)) -> A__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U41(X1:S,X2:S)) -> A__U41(mark(X1:S),X2:S)
107.67/111.38	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> A__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> A__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	-> Rules:
107.67/111.38	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.38	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.38	 a__U12(tt) -> tt
107.67/111.38	 a__U12(X:S) -> U12(X:S)
107.67/111.38	 a__U21(tt) -> tt
107.67/111.38	 a__U21(X:S) -> U21(X:S)
107.67/111.38	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.38	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.38	 a__U32(tt) -> tt
107.67/111.38	 a__U32(X:S) -> U32(X:S)
107.67/111.38	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.38	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.38	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.38	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.38	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.38	 a__U61(tt) -> 0
107.67/111.38	 a__U61(X:S) -> U61(X:S)
107.67/111.38	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.38	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.38	 a__isNat(0) -> tt
107.67/111.38	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.38	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(X:S) -> isNat(X:S)
107.67/111.38	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.38	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.38	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.38	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.38	 mark(0) -> 0
107.67/111.38	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.38	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.38	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.38	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.38	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.38	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.38	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.38	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.38	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.38	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.38	 mark(tt) -> tt
107.67/111.38	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.38	->Strongly Connected Components:
107.67/111.38	->->Cycle:
107.67/111.38	->->-> Pairs:
107.67/111.38	 A__U51(tt,M:S,N:S) -> A__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 A__U52(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S))
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__U71(tt,M:S,N:S) -> A__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__PLUS(N:S,s(M:S)) -> A__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> A__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> A__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	->->-> Rules:
107.67/111.38	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.38	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.38	 a__U12(tt) -> tt
107.67/111.38	 a__U12(X:S) -> U12(X:S)
107.67/111.38	 a__U21(tt) -> tt
107.67/111.38	 a__U21(X:S) -> U21(X:S)
107.67/111.38	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.38	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.38	 a__U32(tt) -> tt
107.67/111.38	 a__U32(X:S) -> U32(X:S)
107.67/111.38	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.38	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.38	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.38	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.38	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.38	 a__U61(tt) -> 0
107.67/111.38	 a__U61(X:S) -> U61(X:S)
107.67/111.38	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.38	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.38	 a__isNat(0) -> tt
107.67/111.38	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.38	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(X:S) -> isNat(X:S)
107.67/111.38	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.38	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.38	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.38	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.38	 mark(0) -> 0
107.67/111.38	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.38	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.38	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.38	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.38	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.38	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.38	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.38	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.38	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.38	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.38	 mark(tt) -> tt
107.67/111.38	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.38	
107.67/111.38	Problem 1.2: 
107.67/111.38	
107.67/111.38	Reduction Pairs Processor:
107.67/111.38	-> Pairs:
107.67/111.38	 A__U51(tt,M:S,N:S) -> A__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 A__U52(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S))
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__U71(tt,M:S,N:S) -> A__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__PLUS(N:S,s(M:S)) -> A__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> A__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> A__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	-> Rules:
107.67/111.38	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.38	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.38	 a__U12(tt) -> tt
107.67/111.38	 a__U12(X:S) -> U12(X:S)
107.67/111.38	 a__U21(tt) -> tt
107.67/111.38	 a__U21(X:S) -> U21(X:S)
107.67/111.38	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.38	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.38	 a__U32(tt) -> tt
107.67/111.38	 a__U32(X:S) -> U32(X:S)
107.67/111.38	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.38	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.38	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.38	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.38	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.38	 a__U61(tt) -> 0
107.67/111.38	 a__U61(X:S) -> U61(X:S)
107.67/111.38	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.38	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.38	 a__isNat(0) -> tt
107.67/111.38	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.38	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(X:S) -> isNat(X:S)
107.67/111.38	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.38	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.38	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.38	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.38	 mark(0) -> 0
107.67/111.38	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.38	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.38	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.38	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.38	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.38	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.38	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.38	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.38	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.38	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.38	 mark(tt) -> tt
107.67/111.38	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.38	-> Usable rules:
107.67/111.38	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.38	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.38	 a__U12(tt) -> tt
107.67/111.38	 a__U12(X:S) -> U12(X:S)
107.67/111.38	 a__U21(tt) -> tt
107.67/111.38	 a__U21(X:S) -> U21(X:S)
107.67/111.38	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.38	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.38	 a__U32(tt) -> tt
107.67/111.38	 a__U32(X:S) -> U32(X:S)
107.67/111.38	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.38	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.38	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.38	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.38	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.38	 a__U61(tt) -> 0
107.67/111.38	 a__U61(X:S) -> U61(X:S)
107.67/111.38	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.38	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.38	 a__isNat(0) -> tt
107.67/111.38	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.38	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(X:S) -> isNat(X:S)
107.67/111.38	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.38	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.38	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.38	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.38	 mark(0) -> 0
107.67/111.38	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.38	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.38	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.38	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.38	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.38	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.38	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.38	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.38	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.38	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.38	 mark(tt) -> tt
107.67/111.38	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.38	->Interpretation type:
107.67/111.38	 Simple mixed
107.67/111.38	->Coefficients:
107.67/111.38	 Natural Numbers
107.67/111.38	->Dimension:
107.67/111.38	 1
107.67/111.38	->Bound:
107.67/111.38	 2
107.67/111.38	->Interpretation:
107.67/111.38	 
107.67/111.38	[a__U11](X1,X2) = X1
107.67/111.38	[a__U12](X) = X
107.67/111.38	[a__U21](X) = X
107.67/111.38	[a__U31](X1,X2) = X1
107.67/111.38	[a__U32](X) = X
107.67/111.38	[a__U41](X1,X2) = 2.X1 + X2 + 2
107.67/111.38	[a__U51](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2
107.67/111.38	[a__U52](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2
107.67/111.38	[a__U61](X) = X + 2
107.67/111.38	[a__U71](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + 2.X3 + 2
107.67/111.38	[a__U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + 2.X3 + 2
107.67/111.38	[a__isNat](X) = 2
107.67/111.38	[a__plus](X1,X2) = X1 + 2.X2 + 2
107.67/111.38	[a__x](X1,X2) = X1.X2 + 2.X1 + X2 + 2
107.67/111.38	[mark](X) = X
107.67/111.38	[0] = 2
107.67/111.38	[U11](X1,X2) = X1
107.67/111.38	[U12](X) = X
107.67/111.38	[U21](X) = X
107.67/111.38	[U31](X1,X2) = X1
107.67/111.38	[U32](X) = X
107.67/111.38	[U41](X1,X2) = 2.X1 + X2 + 2
107.67/111.38	[U51](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2
107.67/111.38	[U52](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2
107.67/111.38	[U61](X) = X + 2
107.67/111.38	[U71](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + 2.X3 + 2
107.67/111.38	[U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + 2.X3 + 2
107.67/111.38	[fSNonEmpty] = 0
107.67/111.38	[isNat](X) = 2
107.67/111.38	[plus](X1,X2) = X1 + 2.X2 + 2
107.67/111.38	[s](X) = X + 2
107.67/111.38	[tt] = 2
107.67/111.38	[x](X1,X2) = X1.X2 + 2.X1 + X2 + 2
107.67/111.38	[A__U11](X1,X2) = 0
107.67/111.38	[A__U12](X) = 0
107.67/111.38	[A__U21](X) = 0
107.67/111.38	[A__U31](X1,X2) = 0
107.67/111.38	[A__U32](X) = 0
107.67/111.38	[A__U41](X1,X2) = 0
107.67/111.38	[A__U51](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
107.67/111.38	[A__U52](X1,X2,X3) = X1.X2 + X1 + 2.X3 + 2
107.67/111.38	[A__U61](X) = 0
107.67/111.38	[A__U71](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
107.67/111.38	[A__U72](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
107.67/111.38	[A__ISNAT](X) = 0
107.67/111.38	[A__PLUS](X1,X2) = 2.X1 + 2.X2 + 2
107.67/111.38	[A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2
107.67/111.38	[MARK](X) = 2.X + 2
107.67/111.38	
107.67/111.38	Problem 1.2: 
107.67/111.38	
107.67/111.38	SCC Processor:
107.67/111.38	-> Pairs:
107.67/111.38	 A__U52(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S))
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__U71(tt,M:S,N:S) -> A__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__PLUS(N:S,s(M:S)) -> A__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> A__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> A__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	-> Rules:
107.67/111.38	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.38	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.38	 a__U12(tt) -> tt
107.67/111.38	 a__U12(X:S) -> U12(X:S)
107.67/111.38	 a__U21(tt) -> tt
107.67/111.38	 a__U21(X:S) -> U21(X:S)
107.67/111.38	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.38	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.38	 a__U32(tt) -> tt
107.67/111.38	 a__U32(X:S) -> U32(X:S)
107.67/111.38	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.38	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.38	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.38	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.38	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.38	 a__U61(tt) -> 0
107.67/111.38	 a__U61(X:S) -> U61(X:S)
107.67/111.38	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.38	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.38	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.38	 a__isNat(0) -> tt
107.67/111.38	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.38	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.38	 a__isNat(X:S) -> isNat(X:S)
107.67/111.38	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.38	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.38	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.38	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.38	 mark(0) -> 0
107.67/111.38	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.38	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.38	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.38	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.38	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.38	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.38	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.38	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.38	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.38	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.38	 mark(tt) -> tt
107.67/111.38	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.38	->Strongly Connected Components:
107.67/111.38	->->Cycle:
107.67/111.38	->->-> Pairs:
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U52(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__U71(tt,M:S,N:S) -> A__U72(a__isNat(N:S),M:S,N:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.38	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.38	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.38	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> A__U52(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.38	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.38	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.38	->->-> Rules:
107.67/111.38	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.38	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.38	 a__U12(tt) -> tt
107.67/111.38	 a__U12(X:S) -> U12(X:S)
107.67/111.38	 a__U21(tt) -> tt
107.67/111.38	 a__U21(X:S) -> U21(X:S)
107.67/111.38	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.38	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.38	 a__U32(tt) -> tt
107.67/111.38	 a__U32(X:S) -> U32(X:S)
107.67/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.39	 a__U61(tt) -> 0
107.67/111.39	 a__U61(X:S) -> U61(X:S)
107.67/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.39	 a__isNat(0) -> tt
107.67/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(X:S) -> isNat(X:S)
107.67/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.39	 mark(0) -> 0
107.67/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.39	 mark(tt) -> tt
107.67/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.39	
107.67/111.39	Problem 1.2: 
107.67/111.39	
107.67/111.39	Reduction Pairs Processor:
107.67/111.39	-> Pairs:
107.67/111.39	 A__U52(tt,M:S,N:S) -> MARK(M:S)
107.67/111.39	 A__U52(tt,M:S,N:S) -> MARK(N:S)
107.67/111.39	 A__U71(tt,M:S,N:S) -> A__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.39	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> A__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	-> Rules:
107.67/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.39	 a__U12(tt) -> tt
107.67/111.39	 a__U12(X:S) -> U12(X:S)
107.67/111.39	 a__U21(tt) -> tt
107.67/111.39	 a__U21(X:S) -> U21(X:S)
107.67/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.39	 a__U32(tt) -> tt
107.67/111.39	 a__U32(X:S) -> U32(X:S)
107.67/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.39	 a__U61(tt) -> 0
107.67/111.39	 a__U61(X:S) -> U61(X:S)
107.67/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.39	 a__isNat(0) -> tt
107.67/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(X:S) -> isNat(X:S)
107.67/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.39	 mark(0) -> 0
107.67/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.39	 mark(tt) -> tt
107.67/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.39	-> Usable rules:
107.67/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.39	 a__U12(tt) -> tt
107.67/111.39	 a__U12(X:S) -> U12(X:S)
107.67/111.39	 a__U21(tt) -> tt
107.67/111.39	 a__U21(X:S) -> U21(X:S)
107.67/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.39	 a__U32(tt) -> tt
107.67/111.39	 a__U32(X:S) -> U32(X:S)
107.67/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.39	 a__U61(tt) -> 0
107.67/111.39	 a__U61(X:S) -> U61(X:S)
107.67/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.39	 a__isNat(0) -> tt
107.67/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(X:S) -> isNat(X:S)
107.67/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.39	 mark(0) -> 0
107.67/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.39	 mark(tt) -> tt
107.67/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.39	->Interpretation type:
107.67/111.39	 Simple mixed
107.67/111.39	->Coefficients:
107.67/111.39	 Natural Numbers
107.67/111.39	->Dimension:
107.67/111.39	 1
107.67/111.39	->Bound:
107.67/111.39	 2
107.67/111.39	->Interpretation:
107.67/111.39	 
107.67/111.39	[a__U11](X1,X2) = X1
107.67/111.39	[a__U12](X) = X
107.67/111.39	[a__U21](X) = X
107.67/111.39	[a__U31](X1,X2) = X1
107.67/111.39	[a__U32](X) = X
107.67/111.39	[a__U41](X1,X2) = 2.X1 + X2 + 1
107.67/111.39	[a__U51](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2
107.67/111.39	[a__U52](X1,X2,X3) = X1.X2 + X1 + X3 + 2
107.67/111.39	[a__U61](X) = X + 2
107.67/111.39	[a__U71](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
107.67/111.39	[a__U72](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
107.67/111.39	[a__isNat](X) = 2
107.67/111.39	[a__plus](X1,X2) = X1 + 2.X2 + 2
107.67/111.39	[a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2
107.67/111.39	[mark](X) = X
107.67/111.39	[0] = 2
107.67/111.39	[U11](X1,X2) = X1
107.67/111.39	[U12](X) = X
107.67/111.39	[U21](X) = X
107.67/111.39	[U31](X1,X2) = X1
107.67/111.39	[U32](X) = X
107.67/111.39	[U41](X1,X2) = 2.X1 + X2 + 1
107.67/111.39	[U51](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2
107.67/111.39	[U52](X1,X2,X3) = X1.X2 + X1 + X3 + 2
107.67/111.39	[U61](X) = X + 2
107.67/111.39	[U71](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
107.67/111.39	[U72](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
107.67/111.39	[fSNonEmpty] = 0
107.67/111.39	[isNat](X) = 2
107.67/111.39	[plus](X1,X2) = X1 + 2.X2 + 2
107.67/111.39	[s](X) = X + 2
107.67/111.39	[tt] = 2
107.67/111.39	[x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2
107.67/111.39	[A__U11](X1,X2) = 0
107.67/111.39	[A__U12](X) = 0
107.67/111.39	[A__U21](X) = 0
107.67/111.39	[A__U31](X1,X2) = 0
107.67/111.39	[A__U32](X) = 0
107.67/111.39	[A__U41](X1,X2) = 0
107.67/111.39	[A__U51](X1,X2,X3) = 0
107.67/111.39	[A__U52](X1,X2,X3) = 2.X1.X2 + 2.X3 + 2
107.67/111.39	[A__U61](X) = 0
107.67/111.39	[A__U71](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
107.67/111.39	[A__U72](X1,X2,X3) = 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2
107.67/111.39	[A__ISNAT](X) = 0
107.67/111.39	[A__PLUS](X1,X2) = 0
107.67/111.39	[A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2
107.67/111.39	[MARK](X) = 2.X + 1
107.67/111.39	
107.67/111.39	Problem 1.2: 
107.67/111.39	
107.67/111.39	SCC Processor:
107.67/111.39	-> Pairs:
107.67/111.39	 A__U52(tt,M:S,N:S) -> MARK(N:S)
107.67/111.39	 A__U71(tt,M:S,N:S) -> A__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.39	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> A__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	-> Rules:
107.67/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.39	 a__U12(tt) -> tt
107.67/111.39	 a__U12(X:S) -> U12(X:S)
107.67/111.39	 a__U21(tt) -> tt
107.67/111.39	 a__U21(X:S) -> U21(X:S)
107.67/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.39	 a__U32(tt) -> tt
107.67/111.39	 a__U32(X:S) -> U32(X:S)
107.67/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.39	 a__U61(tt) -> 0
107.67/111.39	 a__U61(X:S) -> U61(X:S)
107.67/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.39	 a__isNat(0) -> tt
107.67/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(X:S) -> isNat(X:S)
107.67/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.39	 mark(0) -> 0
107.67/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.39	 mark(tt) -> tt
107.67/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.39	->Strongly Connected Components:
107.67/111.39	->->Cycle:
107.67/111.39	->->-> Pairs:
107.67/111.39	 A__U52(tt,M:S,N:S) -> MARK(N:S)
107.67/111.39	 A__U71(tt,M:S,N:S) -> A__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.39	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> A__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	->->-> Rules:
107.67/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.39	 a__U12(tt) -> tt
107.67/111.39	 a__U12(X:S) -> U12(X:S)
107.67/111.39	 a__U21(tt) -> tt
107.67/111.39	 a__U21(X:S) -> U21(X:S)
107.67/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.39	 a__U32(tt) -> tt
107.67/111.39	 a__U32(X:S) -> U32(X:S)
107.67/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.39	 a__U61(tt) -> 0
107.67/111.39	 a__U61(X:S) -> U61(X:S)
107.67/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.39	 a__isNat(0) -> tt
107.67/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(X:S) -> isNat(X:S)
107.67/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.39	 mark(0) -> 0
107.67/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.39	 mark(tt) -> tt
107.67/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.39	
107.67/111.39	Problem 1.2: 
107.67/111.39	
107.67/111.39	Reduction Pairs Processor:
107.67/111.39	-> Pairs:
107.67/111.39	 A__U52(tt,M:S,N:S) -> MARK(N:S)
107.67/111.39	 A__U71(tt,M:S,N:S) -> A__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.39	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> A__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	-> Rules:
107.67/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.39	 a__U12(tt) -> tt
107.67/111.39	 a__U12(X:S) -> U12(X:S)
107.67/111.39	 a__U21(tt) -> tt
107.67/111.39	 a__U21(X:S) -> U21(X:S)
107.67/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.39	 a__U32(tt) -> tt
107.67/111.39	 a__U32(X:S) -> U32(X:S)
107.67/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.39	 a__U61(tt) -> 0
107.67/111.39	 a__U61(X:S) -> U61(X:S)
107.67/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.39	 a__isNat(0) -> tt
107.67/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(X:S) -> isNat(X:S)
107.67/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.39	 mark(0) -> 0
107.67/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.39	 mark(tt) -> tt
107.67/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.39	-> Usable rules:
107.67/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.39	 a__U12(tt) -> tt
107.67/111.39	 a__U12(X:S) -> U12(X:S)
107.67/111.39	 a__U21(tt) -> tt
107.67/111.39	 a__U21(X:S) -> U21(X:S)
107.67/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.39	 a__U32(tt) -> tt
107.67/111.39	 a__U32(X:S) -> U32(X:S)
107.67/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.39	 a__U61(tt) -> 0
107.67/111.39	 a__U61(X:S) -> U61(X:S)
107.67/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.39	 a__isNat(0) -> tt
107.67/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(X:S) -> isNat(X:S)
107.67/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.39	 mark(0) -> 0
107.67/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.39	 mark(tt) -> tt
107.67/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.39	->Interpretation type:
107.67/111.39	 Simple mixed
107.67/111.39	->Coefficients:
107.67/111.39	 Natural Numbers
107.67/111.39	->Dimension:
107.67/111.39	 1
107.67/111.39	->Bound:
107.67/111.39	 2
107.67/111.39	->Interpretation:
107.67/111.39	 
107.67/111.39	[a__U11](X1,X2) = 2.X1.X2 + 2.X1
107.67/111.39	[a__U12](X) = X
107.67/111.39	[a__U21](X) = 2.X.X + 2.X
107.67/111.39	[a__U31](X1,X2) = 2.X1.X2 + 2.X1
107.67/111.39	[a__U32](X) = 2.X.X + X
107.67/111.39	[a__U41](X1,X2) = X1.X2 + 2.X1 + X2
107.67/111.39	[a__U51](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2
107.67/111.39	[a__U52](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2
107.67/111.39	[a__U61](X) = X + 2
107.67/111.39	[a__U71](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
107.67/111.39	[a__U72](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
107.67/111.39	[a__isNat](X) = 0
107.67/111.39	[a__plus](X1,X2) = X1 + X2
107.67/111.39	[a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2
107.67/111.39	[mark](X) = X
107.67/111.39	[0] = 2
107.67/111.39	[U11](X1,X2) = 2.X1.X2 + 2.X1
107.67/111.39	[U12](X) = X
107.67/111.39	[U21](X) = 2.X.X + 2.X
107.67/111.39	[U31](X1,X2) = 2.X1.X2 + 2.X1
107.67/111.39	[U32](X) = 2.X.X + X
107.67/111.39	[U41](X1,X2) = X1.X2 + 2.X1 + X2
107.67/111.39	[U51](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2
107.67/111.39	[U52](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2
107.67/111.39	[U61](X) = X + 2
107.67/111.39	[U71](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
107.67/111.39	[U72](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
107.67/111.39	[fSNonEmpty] = 0
107.67/111.39	[isNat](X) = 0
107.67/111.39	[plus](X1,X2) = X1 + X2
107.67/111.39	[s](X) = X + 2
107.67/111.39	[tt] = 0
107.67/111.39	[x](X1,X2) = 2.X1.X2 + X1 + 2.X2
107.67/111.39	[A__U11](X1,X2) = 0
107.67/111.39	[A__U12](X) = 0
107.67/111.39	[A__U21](X) = 0
107.67/111.39	[A__U31](X1,X2) = 0
107.67/111.39	[A__U32](X) = 0
107.67/111.39	[A__U41](X1,X2) = 0
107.67/111.39	[A__U51](X1,X2,X3) = 0
107.67/111.39	[A__U52](X1,X2,X3) = 2.X1.X2.X3 + 2.X3 + 2
107.67/111.39	[A__U61](X) = 0
107.67/111.39	[A__U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2
107.67/111.39	[A__U72](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X2 + 2.X3 + 2
107.67/111.39	[A__ISNAT](X) = 0
107.67/111.39	[A__PLUS](X1,X2) = 0
107.67/111.39	[A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2
107.67/111.39	[MARK](X) = 2.X
107.67/111.39	
107.67/111.39	Problem 1.2: 
107.67/111.39	
107.67/111.39	SCC Processor:
107.67/111.39	-> Pairs:
107.67/111.39	 A__U71(tt,M:S,N:S) -> A__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.39	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> A__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	-> Rules:
107.67/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.39	 a__U12(tt) -> tt
107.67/111.39	 a__U12(X:S) -> U12(X:S)
107.67/111.39	 a__U21(tt) -> tt
107.67/111.39	 a__U21(X:S) -> U21(X:S)
107.67/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.39	 a__U32(tt) -> tt
107.67/111.39	 a__U32(X:S) -> U32(X:S)
107.67/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.39	 a__U61(tt) -> 0
107.67/111.39	 a__U61(X:S) -> U61(X:S)
107.67/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.39	 a__isNat(0) -> tt
107.67/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(X:S) -> isNat(X:S)
107.67/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.39	 mark(0) -> 0
107.67/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.39	 mark(tt) -> tt
107.67/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.39	->Strongly Connected Components:
107.67/111.39	->->Cycle:
107.67/111.39	->->-> Pairs:
107.67/111.39	 A__U71(tt,M:S,N:S) -> A__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.39	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	->->-> Rules:
107.67/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.39	 a__U12(tt) -> tt
107.67/111.39	 a__U12(X:S) -> U12(X:S)
107.67/111.39	 a__U21(tt) -> tt
107.67/111.39	 a__U21(X:S) -> U21(X:S)
107.67/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.39	 a__U32(tt) -> tt
107.67/111.39	 a__U32(X:S) -> U32(X:S)
107.67/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.39	 a__U61(tt) -> 0
107.67/111.39	 a__U61(X:S) -> U61(X:S)
107.67/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.39	 a__isNat(0) -> tt
107.67/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(X:S) -> isNat(X:S)
107.67/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.39	 mark(0) -> 0
107.67/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.39	 mark(tt) -> tt
107.67/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.39	
107.67/111.39	Problem 1.2: 
107.67/111.39	
107.67/111.39	Reduction Pairs Processor:
107.67/111.39	-> Pairs:
107.67/111.39	 A__U71(tt,M:S,N:S) -> A__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.39	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	-> Rules:
107.67/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.39	 a__U12(tt) -> tt
107.67/111.39	 a__U12(X:S) -> U12(X:S)
107.67/111.39	 a__U21(tt) -> tt
107.67/111.39	 a__U21(X:S) -> U21(X:S)
107.67/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.39	 a__U32(tt) -> tt
107.67/111.39	 a__U32(X:S) -> U32(X:S)
107.67/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.39	 a__U61(tt) -> 0
107.67/111.39	 a__U61(X:S) -> U61(X:S)
107.67/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.39	 a__isNat(0) -> tt
107.67/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(X:S) -> isNat(X:S)
107.67/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.39	 mark(0) -> 0
107.67/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.39	 mark(tt) -> tt
107.67/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.39	-> Usable rules:
107.67/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.39	 a__U12(tt) -> tt
107.67/111.39	 a__U12(X:S) -> U12(X:S)
107.67/111.39	 a__U21(tt) -> tt
107.67/111.39	 a__U21(X:S) -> U21(X:S)
107.67/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.39	 a__U32(tt) -> tt
107.67/111.39	 a__U32(X:S) -> U32(X:S)
107.67/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.39	 a__U61(tt) -> 0
107.67/111.39	 a__U61(X:S) -> U61(X:S)
107.67/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.39	 a__isNat(0) -> tt
107.67/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(X:S) -> isNat(X:S)
107.67/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.39	 mark(0) -> 0
107.67/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.39	 mark(tt) -> tt
107.67/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.39	->Interpretation type:
107.67/111.39	 Simple mixed
107.67/111.39	->Coefficients:
107.67/111.39	 Natural Numbers
107.67/111.39	->Dimension:
107.67/111.39	 1
107.67/111.39	->Bound:
107.67/111.39	 2
107.67/111.39	->Interpretation:
107.67/111.39	 
107.67/111.39	[a__U11](X1,X2) = X1
107.67/111.39	[a__U12](X) = X
107.67/111.39	[a__U21](X) = X
107.67/111.39	[a__U31](X1,X2) = X1
107.67/111.39	[a__U32](X) = X
107.67/111.39	[a__U41](X1,X2) = 2.X1 + X2 + 2
107.67/111.39	[a__U51](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2
107.67/111.39	[a__U52](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2
107.67/111.39	[a__U61](X) = X.X + X
107.67/111.39	[a__U71](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2
107.67/111.39	[a__U72](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
107.67/111.39	[a__isNat](X) = 2
107.67/111.39	[a__plus](X1,X2) = X1 + 2.X2 + 2
107.67/111.39	[a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2
107.67/111.39	[mark](X) = X
107.67/111.39	[0] = 2
107.67/111.39	[U11](X1,X2) = X1
107.67/111.39	[U12](X) = X
107.67/111.39	[U21](X) = X
107.67/111.39	[U31](X1,X2) = X1
107.67/111.39	[U32](X) = X
107.67/111.39	[U41](X1,X2) = 2.X1 + X2 + 2
107.67/111.39	[U51](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2
107.67/111.39	[U52](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2
107.67/111.39	[U61](X) = X.X + X
107.67/111.39	[U71](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2
107.67/111.39	[U72](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
107.67/111.39	[fSNonEmpty] = 0
107.67/111.39	[isNat](X) = 2
107.67/111.39	[plus](X1,X2) = X1 + 2.X2 + 2
107.67/111.39	[s](X) = X + 2
107.67/111.39	[tt] = 2
107.67/111.39	[x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2
107.67/111.39	[A__U11](X1,X2) = 0
107.67/111.39	[A__U12](X) = 0
107.67/111.39	[A__U21](X) = 0
107.67/111.39	[A__U31](X1,X2) = 0
107.67/111.39	[A__U32](X) = 0
107.67/111.39	[A__U41](X1,X2) = 0
107.67/111.39	[A__U51](X1,X2,X3) = 0
107.67/111.39	[A__U52](X1,X2,X3) = 0
107.67/111.39	[A__U61](X) = 0
107.67/111.39	[A__U71](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3
107.67/111.39	[A__U72](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X2 + X3 + 2
107.67/111.39	[A__ISNAT](X) = 0
107.67/111.39	[A__PLUS](X1,X2) = 0
107.67/111.39	[A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2
107.67/111.39	[MARK](X) = 2.X + 2
107.67/111.39	
107.67/111.39	Problem 1.2: 
107.67/111.39	
107.67/111.39	SCC Processor:
107.67/111.39	-> Pairs:
107.67/111.39	 A__U72(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S))
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.39	 A__X(N:S,s(M:S)) -> A__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S))
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	-> Rules:
107.67/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.39	 a__U12(tt) -> tt
107.67/111.39	 a__U12(X:S) -> U12(X:S)
107.67/111.39	 a__U21(tt) -> tt
107.67/111.39	 a__U21(X:S) -> U21(X:S)
107.67/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.39	 a__U32(tt) -> tt
107.67/111.39	 a__U32(X:S) -> U32(X:S)
107.67/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.39	 a__U61(tt) -> 0
107.67/111.39	 a__U61(X:S) -> U61(X:S)
107.67/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.39	 a__isNat(0) -> tt
107.67/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(X:S) -> isNat(X:S)
107.67/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.39	 mark(0) -> 0
107.67/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.39	 mark(tt) -> tt
107.67/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.39	->Strongly Connected Components:
107.67/111.39	->->Cycle:
107.67/111.39	->->-> Pairs:
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.39	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	->->-> Rules:
107.67/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.39	 a__U12(tt) -> tt
107.67/111.39	 a__U12(X:S) -> U12(X:S)
107.67/111.39	 a__U21(tt) -> tt
107.67/111.39	 a__U21(X:S) -> U21(X:S)
107.67/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.39	 a__U32(tt) -> tt
107.67/111.39	 a__U32(X:S) -> U32(X:S)
107.67/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.39	 a__U61(tt) -> 0
107.67/111.39	 a__U61(X:S) -> U61(X:S)
107.67/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.39	 a__isNat(0) -> tt
107.67/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(X:S) -> isNat(X:S)
107.67/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.39	 mark(0) -> 0
107.67/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.39	 mark(tt) -> tt
107.67/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.39	
107.67/111.39	Problem 1.2: 
107.67/111.39	
107.67/111.39	Reduction Pairs Processor:
107.67/111.39	-> Pairs:
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(M:S)
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.39	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U61(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	 MARK(s(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.67/111.39	-> Rules:
107.67/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.39	 a__U12(tt) -> tt
107.67/111.39	 a__U12(X:S) -> U12(X:S)
107.67/111.39	 a__U21(tt) -> tt
107.67/111.39	 a__U21(X:S) -> U21(X:S)
107.67/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.39	 a__U32(tt) -> tt
107.67/111.39	 a__U32(X:S) -> U32(X:S)
107.67/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.39	 a__U61(tt) -> 0
107.67/111.39	 a__U61(X:S) -> U61(X:S)
107.67/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.39	 a__isNat(0) -> tt
107.67/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(X:S) -> isNat(X:S)
107.67/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.39	 mark(0) -> 0
107.67/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.39	 mark(tt) -> tt
107.67/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.39	-> Usable rules:
107.67/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.67/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.67/111.39	 a__U12(tt) -> tt
107.67/111.39	 a__U12(X:S) -> U12(X:S)
107.67/111.39	 a__U21(tt) -> tt
107.67/111.39	 a__U21(X:S) -> U21(X:S)
107.67/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.67/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.67/111.39	 a__U32(tt) -> tt
107.67/111.39	 a__U32(X:S) -> U32(X:S)
107.67/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.67/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.67/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.67/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.67/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.67/111.39	 a__U61(tt) -> 0
107.67/111.39	 a__U61(X:S) -> U61(X:S)
107.67/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.67/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.67/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.67/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.67/111.39	 a__isNat(0) -> tt
107.67/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.67/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.67/111.39	 a__isNat(X:S) -> isNat(X:S)
107.67/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.67/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.67/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.67/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.67/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.67/111.39	 mark(0) -> 0
107.67/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.67/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.67/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.67/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.67/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.67/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.67/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.67/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.67/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.67/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.67/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.67/111.39	 mark(tt) -> tt
107.67/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.67/111.39	->Interpretation type:
107.67/111.39	 Simple mixed
107.67/111.39	->Coefficients:
107.67/111.39	 Natural Numbers
107.67/111.39	->Dimension:
107.67/111.39	 1
107.67/111.39	->Bound:
107.67/111.39	 2
107.67/111.39	->Interpretation:
107.67/111.39	 
107.67/111.39	[a__U11](X1,X2) = X1
107.67/111.39	[a__U12](X) = X
107.67/111.39	[a__U21](X) = X
107.67/111.39	[a__U31](X1,X2) = X1
107.67/111.39	[a__U32](X) = X
107.67/111.39	[a__U41](X1,X2) = X1 + X2
107.67/111.39	[a__U51](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2
107.67/111.39	[a__U52](X1,X2,X3) = X1.X2 + X1 + X3 + 2
107.67/111.39	[a__U61](X) = X
107.67/111.39	[a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1.X3 + 2.X1 + 2.X3 + 2
107.67/111.39	[a__U72](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2
107.67/111.39	[a__isNat](X) = 2
107.67/111.39	[a__plus](X1,X2) = X1 + 2.X2 + 2
107.67/111.39	[a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2
107.67/111.39	[mark](X) = X
107.67/111.39	[0] = 0
107.67/111.39	[U11](X1,X2) = X1
107.67/111.39	[U12](X) = X
107.67/111.39	[U21](X) = X
107.67/111.39	[U31](X1,X2) = X1
107.67/111.39	[U32](X) = X
107.67/111.39	[U41](X1,X2) = X1 + X2
107.67/111.39	[U51](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2
107.67/111.39	[U52](X1,X2,X3) = X1.X2 + X1 + X3 + 2
107.67/111.39	[U61](X) = X
107.67/111.39	[U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1.X3 + 2.X1 + 2.X3 + 2
107.67/111.39	[U72](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2
107.67/111.39	[fSNonEmpty] = 0
107.67/111.39	[isNat](X) = 2
107.67/111.39	[plus](X1,X2) = X1 + 2.X2 + 2
107.67/111.39	[s](X) = X + 2
107.67/111.39	[tt] = 2
107.67/111.39	[x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2
107.67/111.39	[A__U11](X1,X2) = 0
107.67/111.39	[A__U12](X) = 0
107.67/111.39	[A__U21](X) = 0
107.67/111.39	[A__U31](X1,X2) = 0
107.67/111.39	[A__U32](X) = 0
107.67/111.39	[A__U41](X1,X2) = 0
107.67/111.39	[A__U51](X1,X2,X3) = 0
107.67/111.39	[A__U52](X1,X2,X3) = 0
107.67/111.39	[A__U61](X) = 0
107.67/111.39	[A__U71](X1,X2,X3) = 0
107.67/111.39	[A__U72](X1,X2,X3) = X1.X2 + X2.X3 + X1 + 2.X3 + 2
107.67/111.39	[A__ISNAT](X) = 0
107.67/111.39	[A__PLUS](X1,X2) = 0
107.67/111.39	[A__X](X1,X2) = 0
107.67/111.39	[MARK](X) = 2.X + 2
107.67/111.39	
107.67/111.39	Problem 1.2: 
107.67/111.39	
107.67/111.39	SCC Processor:
107.67/111.39	-> Pairs:
107.67/111.39	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.67/111.39	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U12(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U21(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U32(X:S)) -> MARK(X:S)
107.67/111.39	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.67/111.39	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.71/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.71/111.39	 MARK(U61(X:S)) -> MARK(X:S)
107.71/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.71/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.71/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.71/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.71/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.71/111.39	 MARK(s(X:S)) -> MARK(X:S)
107.71/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.71/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.71/111.39	-> Rules:
107.71/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.71/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.71/111.39	 a__U12(tt) -> tt
107.71/111.39	 a__U12(X:S) -> U12(X:S)
107.71/111.39	 a__U21(tt) -> tt
107.71/111.39	 a__U21(X:S) -> U21(X:S)
107.71/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.71/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.71/111.39	 a__U32(tt) -> tt
107.71/111.39	 a__U32(X:S) -> U32(X:S)
107.71/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.71/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.71/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.71/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.71/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.71/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.71/111.39	 a__U61(tt) -> 0
107.71/111.39	 a__U61(X:S) -> U61(X:S)
107.71/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.71/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.71/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.71/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.71/111.39	 a__isNat(0) -> tt
107.71/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.71/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.71/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.71/111.39	 a__isNat(X:S) -> isNat(X:S)
107.71/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.71/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.71/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.71/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.71/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.71/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.71/111.39	 mark(0) -> 0
107.71/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.71/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.71/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.71/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.71/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.71/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.71/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.71/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.71/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.71/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.71/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.71/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.71/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.71/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.71/111.39	 mark(tt) -> tt
107.71/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.71/111.39	->Strongly Connected Components:
107.71/111.39	->->Cycle:
107.71/111.39	->->-> Pairs:
107.71/111.39	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.71/111.39	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.71/111.39	 MARK(U12(X:S)) -> MARK(X:S)
107.71/111.39	 MARK(U21(X:S)) -> MARK(X:S)
107.71/111.39	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.71/111.39	 MARK(U32(X:S)) -> MARK(X:S)
107.71/111.39	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.71/111.39	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.71/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.71/111.39	 MARK(U61(X:S)) -> MARK(X:S)
107.71/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.71/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.71/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.71/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.71/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.71/111.39	 MARK(s(X:S)) -> MARK(X:S)
107.71/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.71/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.71/111.39	->->-> Rules:
107.71/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.71/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.71/111.39	 a__U12(tt) -> tt
107.71/111.39	 a__U12(X:S) -> U12(X:S)
107.71/111.39	 a__U21(tt) -> tt
107.71/111.39	 a__U21(X:S) -> U21(X:S)
107.71/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.71/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.71/111.39	 a__U32(tt) -> tt
107.71/111.39	 a__U32(X:S) -> U32(X:S)
107.71/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.71/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.71/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.71/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.71/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.71/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.71/111.39	 a__U61(tt) -> 0
107.71/111.39	 a__U61(X:S) -> U61(X:S)
107.71/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.71/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.71/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.71/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.71/111.39	 a__isNat(0) -> tt
107.71/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.71/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.71/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.71/111.39	 a__isNat(X:S) -> isNat(X:S)
107.71/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.71/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.71/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.71/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.71/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.71/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.71/111.39	 mark(0) -> 0
107.71/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.71/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.71/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.71/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.71/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.71/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.71/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.71/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.71/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.71/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.71/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.71/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.71/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.71/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.71/111.39	 mark(tt) -> tt
107.71/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.71/111.39	
107.71/111.39	Problem 1.2: 
107.71/111.39	
107.71/111.39	Subterm Processor:
107.71/111.39	-> Pairs:
107.71/111.39	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.71/111.39	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
107.71/111.39	 MARK(U12(X:S)) -> MARK(X:S)
107.71/111.39	 MARK(U21(X:S)) -> MARK(X:S)
107.71/111.39	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
107.71/111.39	 MARK(U32(X:S)) -> MARK(X:S)
107.71/111.39	 MARK(U41(X1:S,X2:S)) -> MARK(X1:S)
107.71/111.39	 MARK(U51(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.71/111.39	 MARK(U52(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.71/111.39	 MARK(U61(X:S)) -> MARK(X:S)
107.71/111.39	 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.71/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> A__U72(mark(X1:S),X2:S,X3:S)
107.71/111.39	 MARK(U72(X1:S,X2:S,X3:S)) -> MARK(X1:S)
107.71/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
107.71/111.39	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
107.71/111.39	 MARK(s(X:S)) -> MARK(X:S)
107.71/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
107.71/111.39	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
107.71/111.39	-> Rules:
107.71/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.71/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.71/111.39	 a__U12(tt) -> tt
107.71/111.39	 a__U12(X:S) -> U12(X:S)
107.71/111.39	 a__U21(tt) -> tt
107.71/111.39	 a__U21(X:S) -> U21(X:S)
107.71/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.71/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.71/111.39	 a__U32(tt) -> tt
107.71/111.39	 a__U32(X:S) -> U32(X:S)
107.71/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.71/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.71/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.71/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.71/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.71/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.71/111.39	 a__U61(tt) -> 0
107.71/111.39	 a__U61(X:S) -> U61(X:S)
107.71/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.71/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.71/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.71/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.71/111.39	 a__isNat(0) -> tt
107.71/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.71/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.71/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.71/111.39	 a__isNat(X:S) -> isNat(X:S)
107.71/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.71/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.71/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.71/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.71/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.71/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.71/111.39	 mark(0) -> 0
107.71/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.71/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.71/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.71/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.71/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.71/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.71/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.71/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.71/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.71/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.71/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.71/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.71/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.71/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.71/111.39	 mark(tt) -> tt
107.71/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.71/111.39	->Projection:
107.71/111.39	 pi(A__U72) = 3
107.71/111.39	 pi(MARK) = 1
107.71/111.39	
107.71/111.39	Problem 1.2: 
107.71/111.39	
107.71/111.39	SCC Processor:
107.71/111.39	-> Pairs:
107.71/111.39	 A__U72(tt,M:S,N:S) -> MARK(N:S)
107.71/111.39	-> Rules:
107.71/111.39	 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S))
107.71/111.39	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
107.71/111.39	 a__U12(tt) -> tt
107.71/111.39	 a__U12(X:S) -> U12(X:S)
107.71/111.39	 a__U21(tt) -> tt
107.71/111.39	 a__U21(X:S) -> U21(X:S)
107.71/111.39	 a__U31(tt,V2:S) -> a__U32(a__isNat(V2:S))
107.71/111.39	 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S)
107.71/111.39	 a__U32(tt) -> tt
107.71/111.39	 a__U32(X:S) -> U32(X:S)
107.71/111.39	 a__U41(tt,N:S) -> mark(N:S)
107.71/111.39	 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S)
107.71/111.39	 a__U51(tt,M:S,N:S) -> a__U52(a__isNat(N:S),M:S,N:S)
107.71/111.39	 a__U51(X1:S,X2:S,X3:S) -> U51(X1:S,X2:S,X3:S)
107.71/111.39	 a__U52(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
107.71/111.39	 a__U52(X1:S,X2:S,X3:S) -> U52(X1:S,X2:S,X3:S)
107.71/111.39	 a__U61(tt) -> 0
107.71/111.39	 a__U61(X:S) -> U61(X:S)
107.71/111.39	 a__U71(tt,M:S,N:S) -> a__U72(a__isNat(N:S),M:S,N:S)
107.71/111.39	 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S)
107.71/111.39	 a__U72(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S))
107.71/111.39	 a__U72(X1:S,X2:S,X3:S) -> U72(X1:S,X2:S,X3:S)
107.71/111.39	 a__isNat(0) -> tt
107.71/111.39	 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S)
107.71/111.39	 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S))
107.71/111.39	 a__isNat(x(V1:S,V2:S)) -> a__U31(a__isNat(V1:S),V2:S)
107.71/111.39	 a__isNat(X:S) -> isNat(X:S)
107.71/111.39	 a__plus(N:S,0) -> a__U41(a__isNat(N:S),N:S)
107.71/111.39	 a__plus(N:S,s(M:S)) -> a__U51(a__isNat(M:S),M:S,N:S)
107.71/111.39	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
107.71/111.39	 a__x(N:S,0) -> a__U61(a__isNat(N:S))
107.71/111.39	 a__x(N:S,s(M:S)) -> a__U71(a__isNat(M:S),M:S,N:S)
107.71/111.39	 a__x(X1:S,X2:S) -> x(X1:S,X2:S)
107.71/111.39	 mark(0) -> 0
107.71/111.39	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
107.71/111.39	 mark(U12(X:S)) -> a__U12(mark(X:S))
107.71/111.39	 mark(U21(X:S)) -> a__U21(mark(X:S))
107.71/111.39	 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S)
107.71/111.39	 mark(U32(X:S)) -> a__U32(mark(X:S))
107.71/111.39	 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S)
107.71/111.39	 mark(U51(X1:S,X2:S,X3:S)) -> a__U51(mark(X1:S),X2:S,X3:S)
107.71/111.39	 mark(U52(X1:S,X2:S,X3:S)) -> a__U52(mark(X1:S),X2:S,X3:S)
107.71/111.39	 mark(U61(X:S)) -> a__U61(mark(X:S))
107.71/111.39	 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S)
107.71/111.39	 mark(U72(X1:S,X2:S,X3:S)) -> a__U72(mark(X1:S),X2:S,X3:S)
107.71/111.39	 mark(isNat(X:S)) -> a__isNat(X:S)
107.71/111.39	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
107.71/111.39	 mark(s(X:S)) -> s(mark(X:S))
107.71/111.39	 mark(tt) -> tt
107.71/111.39	 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S))
107.71/111.39	->Strongly Connected Components:
107.71/111.39	 There is no strongly connected component
107.71/111.39	
107.71/111.39	The problem is finite.
107.71/111.39	EOF