1.34/1.55	YES
1.34/1.55	
1.34/1.55	Problem 1: 
1.34/1.55	
1.34/1.55	(VAR v_NonEmpty:S M:S N:S V1:S V2:S X:S X1:S X2:S X3:S)
1.34/1.55	(RULES
1.34/1.55	a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	a__and(tt,X:S) -> mark(X:S)
1.34/1.55	a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	a__isNat(0) -> tt
1.34/1.55	a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	a__isNat(X:S) -> isNat(X:S)
1.34/1.55	a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	mark(0) -> 0
1.34/1.55	mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	mark(tt) -> tt
1.34/1.55	) 
1.34/1.55	(STRATEGY INNERMOST)
1.34/1.55	
1.34/1.55	Problem 1: 
1.34/1.55	
1.34/1.55	Dependency Pairs Processor:
1.34/1.55	-> Pairs:
1.34/1.55	 A__U11(tt,N:S) -> MARK(N:S)
1.34/1.55	 A__U21(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S))
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(M:S)
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(N:S)
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__U11(a__isNat(N:S),N:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	
1.34/1.55	Problem 1: 
1.34/1.55	
1.34/1.55	SCC Processor:
1.34/1.55	-> Pairs:
1.34/1.55	 A__U11(tt,N:S) -> MARK(N:S)
1.34/1.55	 A__U21(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S))
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(M:S)
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(N:S)
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__U11(a__isNat(N:S),N:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	->Strongly Connected Components:
1.34/1.55	->->Cycle:
1.34/1.55	->->-> Pairs:
1.34/1.55	 A__U11(tt,N:S) -> MARK(N:S)
1.34/1.55	 A__U21(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S))
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(M:S)
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(N:S)
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__U11(a__isNat(N:S),N:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	->->-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	
1.34/1.55	Problem 1: 
1.34/1.55	
1.34/1.55	Reduction Pairs Processor:
1.34/1.55	-> Pairs:
1.34/1.55	 A__U11(tt,N:S) -> MARK(N:S)
1.34/1.55	 A__U21(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S))
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(M:S)
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(N:S)
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__U11(a__isNat(N:S),N:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	-> Usable rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	->Interpretation type:
1.34/1.55	 Linear
1.34/1.55	->Coefficients:
1.34/1.55	 Natural Numbers
1.34/1.55	->Dimension:
1.34/1.55	 1
1.34/1.55	->Bound:
1.34/1.55	 2
1.34/1.55	->Interpretation:
1.34/1.55	 
1.34/1.55	[a__U11](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.55	[a__U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
1.34/1.55	[a__and](X1,X2) = 2.X1 + 2.X2
1.34/1.55	[a__isNat](X) = 0
1.34/1.55	[a__plus](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.55	[mark](X) = X
1.34/1.55	[0] = 0
1.34/1.55	[U11](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.55	[U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
1.34/1.55	[and](X1,X2) = 2.X1 + 2.X2
1.34/1.55	[fSNonEmpty] = 0
1.34/1.55	[isNat](X) = 0
1.34/1.55	[plus](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.55	[s](X) = X
1.34/1.55	[tt] = 0
1.34/1.55	[A__U11](X1,X2) = 2.X2 + 1
1.34/1.55	[A__U21](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
1.34/1.55	[A__AND](X1,X2) = 2.X1 + 2.X2
1.34/1.55	[A__ISNAT](X) = 0
1.34/1.55	[A__PLUS](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.55	[MARK](X) = 2.X
1.34/1.55	
1.34/1.55	Problem 1: 
1.34/1.55	
1.34/1.55	SCC Processor:
1.34/1.55	-> Pairs:
1.34/1.55	 A__U21(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S))
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(M:S)
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(N:S)
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__U11(a__isNat(N:S),N:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	->Strongly Connected Components:
1.34/1.55	->->Cycle:
1.34/1.55	->->-> Pairs:
1.34/1.55	 A__U21(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S))
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(M:S)
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(N:S)
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	->->-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	
1.34/1.55	Problem 1: 
1.34/1.55	
1.34/1.55	Reduction Pairs Processor:
1.34/1.55	-> Pairs:
1.34/1.55	 A__U21(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S))
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(M:S)
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(N:S)
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	-> Usable rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	->Interpretation type:
1.34/1.55	 Linear
1.34/1.55	->Coefficients:
1.34/1.55	 Natural Numbers
1.34/1.55	->Dimension:
1.34/1.55	 1
1.34/1.55	->Bound:
1.34/1.55	 2
1.34/1.55	->Interpretation:
1.34/1.55	 
1.34/1.55	[a__U11](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.55	[a__U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
1.34/1.55	[a__and](X1,X2) = 2.X1 + 2.X2
1.34/1.55	[a__isNat](X) = 0
1.34/1.55	[a__plus](X1,X2) = 2.X1 + 2.X2 + 1
1.34/1.55	[mark](X) = X
1.34/1.55	[0] = 1
1.34/1.55	[U11](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.55	[U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
1.34/1.55	[and](X1,X2) = 2.X1 + 2.X2
1.34/1.55	[fSNonEmpty] = 0
1.34/1.55	[isNat](X) = 0
1.34/1.55	[plus](X1,X2) = 2.X1 + 2.X2 + 1
1.34/1.55	[s](X) = X + 1
1.34/1.55	[tt] = 0
1.34/1.55	[A__U11](X1,X2) = 0
1.34/1.55	[A__U21](X1,X2,X3) = 2.X2 + 2.X3 + 2
1.34/1.55	[A__AND](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.55	[A__ISNAT](X) = 2
1.34/1.55	[A__PLUS](X1,X2) = 2.X1 + 2.X2 + 1
1.34/1.55	[MARK](X) = 2.X + 2
1.34/1.55	
1.34/1.55	Problem 1: 
1.34/1.55	
1.34/1.55	SCC Processor:
1.34/1.55	-> Pairs:
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(M:S)
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(N:S)
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	->Strongly Connected Components:
1.34/1.55	->->Cycle:
1.34/1.55	->->-> Pairs:
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(M:S)
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(N:S)
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	->->-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	
1.34/1.55	Problem 1: 
1.34/1.55	
1.34/1.55	Reduction Pairs Processor:
1.34/1.55	-> Pairs:
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(M:S)
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(N:S)
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	-> Usable rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	->Interpretation type:
1.34/1.55	 Linear
1.34/1.55	->Coefficients:
1.34/1.55	 Natural Numbers
1.34/1.55	->Dimension:
1.34/1.55	 1
1.34/1.55	->Bound:
1.34/1.55	 2
1.34/1.55	->Interpretation:
1.34/1.55	 
1.34/1.55	[a__U11](X1,X2) = 2.X1 + X2 + 1
1.34/1.55	[a__U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1
1.34/1.55	[a__and](X1,X2) = 2.X1 + 2.X2
1.34/1.55	[a__isNat](X) = 0
1.34/1.55	[a__plus](X1,X2) = 2.X1 + 2.X2
1.34/1.55	[mark](X) = X
1.34/1.55	[0] = 2
1.34/1.55	[U11](X1,X2) = 2.X1 + X2 + 1
1.34/1.55	[U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1
1.34/1.55	[and](X1,X2) = 2.X1 + 2.X2
1.34/1.55	[fSNonEmpty] = 0
1.34/1.55	[isNat](X) = 0
1.34/1.55	[plus](X1,X2) = 2.X1 + 2.X2
1.34/1.55	[s](X) = X + 1
1.34/1.55	[tt] = 0
1.34/1.55	[A__U11](X1,X2) = 0
1.34/1.55	[A__U21](X1,X2,X3) = 2.X2 + 2.X3 + 2
1.34/1.55	[A__AND](X1,X2) = 2.X1 + 2.X2 + 1
1.34/1.55	[A__ISNAT](X) = 1
1.34/1.55	[A__PLUS](X1,X2) = 2.X1 + 2.X2 + 1
1.34/1.55	[MARK](X) = 2.X + 1
1.34/1.55	
1.34/1.55	Problem 1: 
1.34/1.55	
1.34/1.55	SCC Processor:
1.34/1.55	-> Pairs:
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(N:S)
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	->Strongly Connected Components:
1.34/1.55	->->Cycle:
1.34/1.55	->->-> Pairs:
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(N:S)
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	->->-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	
1.34/1.55	Problem 1: 
1.34/1.55	
1.34/1.55	Reduction Pairs Processor:
1.34/1.55	-> Pairs:
1.34/1.55	 A__U21(tt,M:S,N:S) -> MARK(N:S)
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	-> Usable rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	->Interpretation type:
1.34/1.55	 Linear
1.34/1.55	->Coefficients:
1.34/1.55	 Natural Numbers
1.34/1.55	->Dimension:
1.34/1.55	 1
1.34/1.55	->Bound:
1.34/1.55	 2
1.34/1.55	->Interpretation:
1.34/1.55	 
1.34/1.55	[a__U11](X1,X2) = 2.X1 + 2.X2 + 1
1.34/1.55	[a__U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
1.34/1.55	[a__and](X1,X2) = 2.X1 + 2.X2
1.34/1.55	[a__isNat](X) = 0
1.34/1.55	[a__plus](X1,X2) = 2.X1 + 2.X2
1.34/1.55	[mark](X) = X
1.34/1.55	[0] = 1
1.34/1.55	[U11](X1,X2) = 2.X1 + 2.X2 + 1
1.34/1.55	[U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
1.34/1.55	[and](X1,X2) = 2.X1 + 2.X2
1.34/1.55	[fSNonEmpty] = 0
1.34/1.55	[isNat](X) = 0
1.34/1.55	[plus](X1,X2) = 2.X1 + 2.X2
1.34/1.55	[s](X) = X + 2
1.34/1.55	[tt] = 0
1.34/1.55	[A__U11](X1,X2) = 0
1.34/1.55	[A__U21](X1,X2,X3) = 2.X3 + 2
1.34/1.55	[A__AND](X1,X2) = 2.X2 + 1
1.34/1.55	[A__ISNAT](X) = 1
1.34/1.55	[A__PLUS](X1,X2) = 2.X1 + 2.X2
1.34/1.55	[MARK](X) = 2.X + 1
1.34/1.55	
1.34/1.55	Problem 1: 
1.34/1.55	
1.34/1.55	SCC Processor:
1.34/1.55	-> Pairs:
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	->Strongly Connected Components:
1.34/1.55	->->Cycle:
1.34/1.55	->->-> Pairs:
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	->->-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	
1.34/1.55	Problem 1: 
1.34/1.55	
1.34/1.55	Reduction Pairs Processor:
1.34/1.55	-> Pairs:
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,0) -> A__ISNAT(N:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	-> Usable rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	->Interpretation type:
1.34/1.55	 Linear
1.34/1.55	->Coefficients:
1.34/1.55	 Natural Numbers
1.34/1.55	->Dimension:
1.34/1.55	 1
1.34/1.55	->Bound:
1.34/1.55	 2
1.34/1.55	->Interpretation:
1.34/1.55	 
1.34/1.55	[a__U11](X1,X2) = X1 + 2.X2 + 1
1.34/1.55	[a__U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
1.34/1.55	[a__and](X1,X2) = 2.X1 + X2
1.34/1.55	[a__isNat](X) = 0
1.34/1.55	[a__plus](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.55	[mark](X) = X
1.34/1.55	[0] = 2
1.34/1.55	[U11](X1,X2) = X1 + 2.X2 + 1
1.34/1.55	[U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
1.34/1.55	[and](X1,X2) = 2.X1 + X2
1.34/1.55	[fSNonEmpty] = 0
1.34/1.55	[isNat](X) = 0
1.34/1.55	[plus](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.55	[s](X) = X
1.34/1.55	[tt] = 0
1.34/1.55	[A__U11](X1,X2) = 0
1.34/1.55	[A__U21](X1,X2,X3) = 0
1.34/1.55	[A__AND](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.55	[A__ISNAT](X) = 2
1.34/1.55	[A__PLUS](X1,X2) = 2.X1 + X2 + 2
1.34/1.55	[MARK](X) = 2.X + 2
1.34/1.55	
1.34/1.55	Problem 1: 
1.34/1.55	
1.34/1.55	SCC Processor:
1.34/1.55	-> Pairs:
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.55	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.55	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.55	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.55	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.55	-> Rules:
1.34/1.55	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.55	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.55	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.55	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.55	 a__and(tt,X:S) -> mark(X:S)
1.34/1.55	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.55	 a__isNat(0) -> tt
1.34/1.55	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.55	 a__isNat(X:S) -> isNat(X:S)
1.34/1.55	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.55	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.55	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.55	 mark(0) -> 0
1.34/1.55	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.55	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.55	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.55	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.55	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.55	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.55	 mark(tt) -> tt
1.34/1.55	->Strongly Connected Components:
1.34/1.55	->->Cycle:
1.34/1.55	->->-> Pairs:
1.34/1.55	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.55	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.55	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.56	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.56	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	->->-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	
1.34/1.56	Problem 1: 
1.34/1.56	
1.34/1.56	Reduction Pairs Processor:
1.34/1.56	-> Pairs:
1.34/1.56	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__PLUS(N:S,s(M:S)) -> A__AND(a__isNat(M:S),isNat(N:S))
1.34/1.56	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.56	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	-> Usable rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	->Interpretation type:
1.34/1.56	 Linear
1.34/1.56	->Coefficients:
1.34/1.56	 Natural Numbers
1.34/1.56	->Dimension:
1.34/1.56	 1
1.34/1.56	->Bound:
1.34/1.56	 2
1.34/1.56	->Interpretation:
1.34/1.56	 
1.34/1.56	[a__U11](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.56	[a__U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
1.34/1.56	[a__and](X1,X2) = X1 + 2.X2
1.34/1.56	[a__isNat](X) = 0
1.34/1.56	[a__plus](X1,X2) = 2.X1 + 2.X2 + 1
1.34/1.56	[mark](X) = X
1.34/1.56	[0] = 1
1.34/1.56	[U11](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.56	[U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
1.34/1.56	[and](X1,X2) = X1 + 2.X2
1.34/1.56	[fSNonEmpty] = 0
1.34/1.56	[isNat](X) = 0
1.34/1.56	[plus](X1,X2) = 2.X1 + 2.X2 + 1
1.34/1.56	[s](X) = X + 1
1.34/1.56	[tt] = 0
1.34/1.56	[A__U11](X1,X2) = 0
1.34/1.56	[A__U21](X1,X2,X3) = 0
1.34/1.56	[A__AND](X1,X2) = X1 + 2.X2 + 2
1.34/1.56	[A__ISNAT](X) = 2
1.34/1.56	[A__PLUS](X1,X2) = X2 + 2
1.34/1.56	[MARK](X) = 2.X + 2
1.34/1.56	
1.34/1.56	Problem 1: 
1.34/1.56	
1.34/1.56	SCC Processor:
1.34/1.56	-> Pairs:
1.34/1.56	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.56	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	->Strongly Connected Components:
1.34/1.56	->->Cycle:
1.34/1.56	->->-> Pairs:
1.34/1.56	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.56	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	->->-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	
1.34/1.56	Problem 1: 
1.34/1.56	
1.34/1.56	Reduction Pairs Processor:
1.34/1.56	-> Pairs:
1.34/1.56	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S)
1.34/1.56	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	-> Usable rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	->Interpretation type:
1.34/1.56	 Linear
1.34/1.56	->Coefficients:
1.34/1.56	 Natural Numbers
1.34/1.56	->Dimension:
1.34/1.56	 1
1.34/1.56	->Bound:
1.34/1.56	 2
1.34/1.56	->Interpretation:
1.34/1.56	 
1.34/1.56	[a__U11](X1,X2) = X1 + 2.X2 + 1
1.34/1.56	[a__U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
1.34/1.56	[a__and](X1,X2) = X1 + X2
1.34/1.56	[a__isNat](X) = 0
1.34/1.56	[a__plus](X1,X2) = 2.X1 + 2.X2 + 1
1.34/1.56	[mark](X) = X
1.34/1.56	[0] = 2
1.34/1.56	[U11](X1,X2) = X1 + 2.X2 + 1
1.34/1.56	[U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
1.34/1.56	[and](X1,X2) = X1 + X2
1.34/1.56	[fSNonEmpty] = 0
1.34/1.56	[isNat](X) = 0
1.34/1.56	[plus](X1,X2) = 2.X1 + 2.X2 + 1
1.34/1.56	[s](X) = X + 1
1.34/1.56	[tt] = 0
1.34/1.56	[A__U11](X1,X2) = 0
1.34/1.56	[A__U21](X1,X2,X3) = 0
1.34/1.56	[A__AND](X1,X2) = 2.X2
1.34/1.56	[A__ISNAT](X) = 0
1.34/1.56	[A__PLUS](X1,X2) = 2.X2 + 2
1.34/1.56	[MARK](X) = 2.X
1.34/1.56	
1.34/1.56	Problem 1: 
1.34/1.56	
1.34/1.56	SCC Processor:
1.34/1.56	-> Pairs:
1.34/1.56	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.56	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S))
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	->Strongly Connected Components:
1.34/1.56	->->Cycle:
1.34/1.56	->->-> Pairs:
1.34/1.56	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.56	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	->->-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	
1.34/1.56	Problem 1: 
1.34/1.56	
1.34/1.56	Reduction Pairs Processor:
1.34/1.56	-> Pairs:
1.34/1.56	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.56	 MARK(U11(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	-> Usable rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	->Interpretation type:
1.34/1.56	 Linear
1.34/1.56	->Coefficients:
1.34/1.56	 Natural Numbers
1.34/1.56	->Dimension:
1.34/1.56	 1
1.34/1.56	->Bound:
1.34/1.56	 2
1.34/1.56	->Interpretation:
1.34/1.56	 
1.34/1.56	[a__U11](X1,X2) = 2.X1 + X2 + 2
1.34/1.56	[a__U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
1.34/1.56	[a__and](X1,X2) = X1 + 2.X2
1.34/1.56	[a__isNat](X) = 0
1.34/1.56	[a__plus](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.56	[mark](X) = X
1.34/1.56	[0] = 2
1.34/1.56	[U11](X1,X2) = 2.X1 + X2 + 2
1.34/1.56	[U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
1.34/1.56	[and](X1,X2) = X1 + 2.X2
1.34/1.56	[fSNonEmpty] = 0
1.34/1.56	[isNat](X) = 0
1.34/1.56	[plus](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.56	[s](X) = X
1.34/1.56	[tt] = 0
1.34/1.56	[A__U11](X1,X2) = 0
1.34/1.56	[A__U21](X1,X2,X3) = 0
1.34/1.56	[A__AND](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.56	[A__ISNAT](X) = 2
1.34/1.56	[A__PLUS](X1,X2) = 0
1.34/1.56	[MARK](X) = 2.X + 2
1.34/1.56	
1.34/1.56	Problem 1: 
1.34/1.56	
1.34/1.56	SCC Processor:
1.34/1.56	-> Pairs:
1.34/1.56	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.56	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	->Strongly Connected Components:
1.34/1.56	->->Cycle:
1.34/1.56	->->-> Pairs:
1.34/1.56	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.56	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	->->-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	
1.34/1.56	Problem 1: 
1.34/1.56	
1.34/1.56	Reduction Pairs Processor:
1.34/1.56	-> Pairs:
1.34/1.56	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.56	 MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	-> Usable rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	->Interpretation type:
1.34/1.56	 Linear
1.34/1.56	->Coefficients:
1.34/1.56	 Natural Numbers
1.34/1.56	->Dimension:
1.34/1.56	 1
1.34/1.56	->Bound:
1.34/1.56	 2
1.34/1.56	->Interpretation:
1.34/1.56	 
1.34/1.56	[a__U11](X1,X2) = 2.X1 + X2
1.34/1.56	[a__U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
1.34/1.56	[a__and](X1,X2) = 2.X1 + 2.X2
1.34/1.56	[a__isNat](X) = 0
1.34/1.56	[a__plus](X1,X2) = 2.X1 + 2.X2
1.34/1.56	[mark](X) = X
1.34/1.56	[0] = 2
1.34/1.56	[U11](X1,X2) = 2.X1 + X2
1.34/1.56	[U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
1.34/1.56	[and](X1,X2) = 2.X1 + 2.X2
1.34/1.56	[fSNonEmpty] = 0
1.34/1.56	[isNat](X) = 0
1.34/1.56	[plus](X1,X2) = 2.X1 + 2.X2
1.34/1.56	[s](X) = X + 2
1.34/1.56	[tt] = 0
1.34/1.56	[A__U11](X1,X2) = 0
1.34/1.56	[A__U21](X1,X2,X3) = 0
1.34/1.56	[A__AND](X1,X2) = 2.X2
1.34/1.56	[A__ISNAT](X) = 0
1.34/1.56	[A__PLUS](X1,X2) = 0
1.34/1.56	[MARK](X) = 2.X
1.34/1.56	
1.34/1.56	Problem 1: 
1.34/1.56	
1.34/1.56	SCC Processor:
1.34/1.56	-> Pairs:
1.34/1.56	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	->Strongly Connected Components:
1.34/1.56	->->Cycle:
1.34/1.56	->->-> Pairs:
1.34/1.56	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	->->-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	
1.34/1.56	Problem 1: 
1.34/1.56	
1.34/1.56	Reduction Pairs Processor:
1.34/1.56	-> Pairs:
1.34/1.56	 A__AND(tt,X:S) -> MARK(X:S)
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	-> Usable rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	->Interpretation type:
1.34/1.56	 Linear
1.34/1.56	->Coefficients:
1.34/1.56	 Natural Numbers
1.34/1.56	->Dimension:
1.34/1.56	 1
1.34/1.56	->Bound:
1.34/1.56	 2
1.34/1.56	->Interpretation:
1.34/1.56	 
1.34/1.56	[a__U11](X1,X2) = X2 + 2
1.34/1.56	[a__U21](X1,X2,X3) = 2.X2 + 2.X3 + 2
1.34/1.56	[a__and](X1,X2) = 2.X1 + 2.X2 + 1
1.34/1.56	[a__isNat](X) = 2.X + 1
1.34/1.56	[a__plus](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.56	[mark](X) = X
1.34/1.56	[0] = 2
1.34/1.56	[U11](X1,X2) = X2 + 2
1.34/1.56	[U21](X1,X2,X3) = 2.X2 + 2.X3 + 2
1.34/1.56	[and](X1,X2) = 2.X1 + 2.X2 + 1
1.34/1.56	[fSNonEmpty] = 0
1.34/1.56	[isNat](X) = 2.X + 1
1.34/1.56	[plus](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.56	[s](X) = X
1.34/1.56	[tt] = 2
1.34/1.56	[A__U11](X1,X2) = 0
1.34/1.56	[A__U21](X1,X2,X3) = 0
1.34/1.56	[A__AND](X1,X2) = 2.X1 + 2.X2 + 2
1.34/1.56	[A__ISNAT](X) = 2.X + 2
1.34/1.56	[A__PLUS](X1,X2) = 0
1.34/1.56	[MARK](X) = 2.X + 2
1.34/1.56	
1.34/1.56	Problem 1: 
1.34/1.56	
1.34/1.56	SCC Processor:
1.34/1.56	-> Pairs:
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S)
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(isNat(X:S)) -> A__ISNAT(X:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	->Strongly Connected Components:
1.34/1.56	->->Cycle:
1.34/1.56	->->-> Pairs:
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.56	->->-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	->->Cycle:
1.34/1.56	->->-> Pairs:
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	->->-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	
1.34/1.56	
1.34/1.56	The problem is decomposed in 2 subproblems.
1.34/1.56	
1.34/1.56	Problem 1.1: 
1.34/1.56	
1.34/1.56	Subterm Processor:
1.34/1.56	-> Pairs:
1.34/1.56	 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S)
1.34/1.56	 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S)
1.34/1.56	-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	->Projection:
1.34/1.56	 pi(A__ISNAT) = 1
1.34/1.56	
1.34/1.56	Problem 1.1: 
1.34/1.56	
1.34/1.56	SCC Processor:
1.34/1.56	-> Pairs:
1.34/1.56	 Empty
1.34/1.56	-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	->Strongly Connected Components:
1.34/1.56	 There is no strongly connected component
1.34/1.56	
1.34/1.56	The problem is finite.
1.34/1.56	
1.34/1.56	Problem 1.2: 
1.34/1.56	
1.34/1.56	Subterm Processor:
1.34/1.56	-> Pairs:
1.34/1.56	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
1.34/1.56	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
1.34/1.56	 MARK(s(X:S)) -> MARK(X:S)
1.34/1.56	-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	->Projection:
1.34/1.56	 pi(MARK) = 1
1.34/1.56	
1.34/1.56	Problem 1.2: 
1.34/1.56	
1.34/1.56	SCC Processor:
1.34/1.56	-> Pairs:
1.34/1.56	 Empty
1.34/1.56	-> Rules:
1.34/1.56	 a__U11(tt,N:S) -> mark(N:S)
1.34/1.56	 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S)
1.34/1.56	 a__U21(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S)))
1.34/1.56	 a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S)
1.34/1.56	 a__and(tt,X:S) -> mark(X:S)
1.34/1.56	 a__and(X1:S,X2:S) -> and(X1:S,X2:S)
1.34/1.56	 a__isNat(0) -> tt
1.34/1.56	 a__isNat(plus(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNat(V2:S))
1.34/1.56	 a__isNat(s(V1:S)) -> a__isNat(V1:S)
1.34/1.56	 a__isNat(X:S) -> isNat(X:S)
1.34/1.56	 a__plus(N:S,0) -> a__U11(a__isNat(N:S),N:S)
1.34/1.56	 a__plus(N:S,s(M:S)) -> a__U21(a__and(a__isNat(M:S),isNat(N:S)),M:S,N:S)
1.34/1.56	 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S)
1.34/1.56	 mark(0) -> 0
1.34/1.56	 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S)
1.34/1.56	 mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S)
1.34/1.56	 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S)
1.34/1.56	 mark(isNat(X:S)) -> a__isNat(X:S)
1.34/1.56	 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S))
1.34/1.56	 mark(s(X:S)) -> s(mark(X:S))
1.34/1.56	 mark(tt) -> tt
1.34/1.56	->Strongly Connected Components:
1.34/1.56	 There is no strongly connected component
1.34/1.56	
1.34/1.56	The problem is finite.
1.34/1.56	EOF