0.18/0.60 YES 0.18/0.60 0.18/0.60 Problem 1: 0.18/0.60 0.18/0.60 (VAR v_NonEmpty:S L:S N:S V:S V1:S V2:S X:S X1:S X2:S) 0.18/0.60 (RULES 0.18/0.60 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.60 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.60 a__and(tt,X:S) -> mark(X:S) 0.18/0.60 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.60 a__isNat(0) -> tt 0.18/0.60 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.60 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.60 a__isNat(X:S) -> isNat(X:S) 0.18/0.60 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.60 a__isNatIList(zeros) -> tt 0.18/0.60 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.60 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.60 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.60 a__isNatList(nil) -> tt 0.18/0.60 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.60 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.60 a__length(nil) -> 0 0.18/0.60 a__length(X:S) -> length(X:S) 0.18/0.60 a__zeros -> cons(0,zeros) 0.18/0.60 a__zeros -> zeros 0.18/0.60 mark(0) -> 0 0.18/0.60 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.60 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.60 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.60 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.60 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.60 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.60 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.60 mark(nil) -> nil 0.18/0.60 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.60 mark(tt) -> tt 0.18/0.60 mark(zeros) -> a__zeros 0.18/0.60 ) 0.18/0.60 (STRATEGY INNERMOST) 0.18/0.60 0.18/0.60 Problem 1: 0.18/0.60 0.18/0.60 Dependency Pairs Processor: 0.18/0.60 -> Pairs: 0.18/0.60 A__U11(tt,L:S) -> A__LENGTH(mark(L:S)) 0.18/0.60 A__U11(tt,L:S) -> MARK(L:S) 0.18/0.60 A__AND(tt,X:S) -> MARK(X:S) 0.18/0.60 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 0.18/0.60 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.18/0.60 A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.60 A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.60 A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) 0.18/0.60 A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.60 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.60 A__LENGTH(cons(N:S,L:S)) -> A__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.60 A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) 0.18/0.60 A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) 0.18/0.60 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) 0.18/0.60 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.60 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) 0.18/0.60 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.60 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.60 MARK(isNat(X:S)) -> A__ISNAT(X:S) 0.18/0.60 MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) 0.18/0.60 MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) 0.18/0.60 MARK(length(X:S)) -> A__LENGTH(mark(X:S)) 0.18/0.60 MARK(length(X:S)) -> MARK(X:S) 0.18/0.60 MARK(s(X:S)) -> MARK(X:S) 0.18/0.60 MARK(zeros) -> A__ZEROS 0.18/0.60 -> Rules: 0.18/0.60 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.60 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.60 a__and(tt,X:S) -> mark(X:S) 0.18/0.60 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.60 a__isNat(0) -> tt 0.18/0.60 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.60 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.60 a__isNat(X:S) -> isNat(X:S) 0.18/0.60 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.60 a__isNatIList(zeros) -> tt 0.18/0.60 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.60 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.60 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.60 a__isNatList(nil) -> tt 0.18/0.60 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.60 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.60 a__length(nil) -> 0 0.18/0.60 a__length(X:S) -> length(X:S) 0.18/0.60 a__zeros -> cons(0,zeros) 0.18/0.60 a__zeros -> zeros 0.18/0.60 mark(0) -> 0 0.18/0.60 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.60 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.60 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.60 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.60 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.60 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.60 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.60 mark(nil) -> nil 0.18/0.60 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.60 mark(tt) -> tt 0.18/0.60 mark(zeros) -> a__zeros 0.18/0.60 0.18/0.60 Problem 1: 0.18/0.60 0.18/0.60 SCC Processor: 0.18/0.60 -> Pairs: 0.18/0.60 A__U11(tt,L:S) -> A__LENGTH(mark(L:S)) 0.18/0.60 A__U11(tt,L:S) -> MARK(L:S) 0.18/0.60 A__AND(tt,X:S) -> MARK(X:S) 0.18/0.60 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 0.18/0.60 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.18/0.60 A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.60 A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.60 A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) 0.18/0.60 A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.60 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.60 A__LENGTH(cons(N:S,L:S)) -> A__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.60 A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) 0.18/0.60 A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) 0.18/0.60 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) 0.18/0.60 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.60 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) 0.18/0.60 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.60 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.60 MARK(isNat(X:S)) -> A__ISNAT(X:S) 0.18/0.60 MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) 0.18/0.60 MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) 0.18/0.60 MARK(length(X:S)) -> A__LENGTH(mark(X:S)) 0.18/0.60 MARK(length(X:S)) -> MARK(X:S) 0.18/0.60 MARK(s(X:S)) -> MARK(X:S) 0.18/0.60 MARK(zeros) -> A__ZEROS 0.18/0.60 -> Rules: 0.18/0.60 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.60 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.60 a__and(tt,X:S) -> mark(X:S) 0.18/0.60 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.60 a__isNat(0) -> tt 0.18/0.60 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.60 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.60 a__isNat(X:S) -> isNat(X:S) 0.18/0.60 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.60 a__isNatIList(zeros) -> tt 0.18/0.60 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.60 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.60 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.60 a__isNatList(nil) -> tt 0.18/0.60 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.60 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.60 a__length(nil) -> 0 0.18/0.60 a__length(X:S) -> length(X:S) 0.18/0.60 a__zeros -> cons(0,zeros) 0.18/0.60 a__zeros -> zeros 0.18/0.60 mark(0) -> 0 0.18/0.60 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.60 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.60 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.60 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.60 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.60 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.60 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.60 mark(nil) -> nil 0.18/0.60 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.60 mark(tt) -> tt 0.18/0.60 mark(zeros) -> a__zeros 0.18/0.60 ->Strongly Connected Components: 0.18/0.60 ->->Cycle: 0.18/0.60 ->->-> Pairs: 0.18/0.60 A__U11(tt,L:S) -> A__LENGTH(mark(L:S)) 0.18/0.60 A__U11(tt,L:S) -> MARK(L:S) 0.18/0.60 A__AND(tt,X:S) -> MARK(X:S) 0.18/0.60 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 0.18/0.60 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.18/0.60 A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.60 A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.60 A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) 0.18/0.60 A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.60 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.60 A__LENGTH(cons(N:S,L:S)) -> A__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.60 A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) 0.18/0.60 A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) 0.18/0.60 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) 0.18/0.60 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.60 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) 0.18/0.60 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.60 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.60 MARK(isNat(X:S)) -> A__ISNAT(X:S) 0.18/0.60 MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) 0.18/0.60 MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) 0.18/0.60 MARK(length(X:S)) -> A__LENGTH(mark(X:S)) 0.18/0.60 MARK(length(X:S)) -> MARK(X:S) 0.18/0.60 MARK(s(X:S)) -> MARK(X:S) 0.18/0.60 ->->-> Rules: 0.18/0.60 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.60 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.60 a__and(tt,X:S) -> mark(X:S) 0.18/0.60 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.60 a__isNat(0) -> tt 0.18/0.60 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.60 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.60 a__isNat(X:S) -> isNat(X:S) 0.18/0.60 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.60 a__isNatIList(zeros) -> tt 0.18/0.60 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.60 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.60 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.60 a__isNatList(nil) -> tt 0.18/0.60 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.60 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.60 a__length(nil) -> 0 0.18/0.60 a__length(X:S) -> length(X:S) 0.18/0.60 a__zeros -> cons(0,zeros) 0.18/0.60 a__zeros -> zeros 0.18/0.60 mark(0) -> 0 0.18/0.60 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.60 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.60 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.60 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.60 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.60 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.60 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.60 mark(nil) -> nil 0.18/0.60 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.60 mark(tt) -> tt 0.18/0.60 mark(zeros) -> a__zeros 0.18/0.60 0.18/0.60 Problem 1: 0.18/0.60 0.18/0.60 Reduction Pairs Processor: 0.18/0.60 -> Pairs: 0.18/0.60 A__U11(tt,L:S) -> A__LENGTH(mark(L:S)) 0.18/0.60 A__U11(tt,L:S) -> MARK(L:S) 0.18/0.60 A__AND(tt,X:S) -> MARK(X:S) 0.18/0.60 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 0.18/0.60 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.18/0.60 A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.60 A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.60 A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) 0.18/0.60 A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.60 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.60 A__LENGTH(cons(N:S,L:S)) -> A__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.60 A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) 0.18/0.60 A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) 0.18/0.60 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) 0.18/0.60 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.60 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) 0.18/0.60 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.60 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.60 MARK(isNat(X:S)) -> A__ISNAT(X:S) 0.18/0.60 MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) 0.18/0.60 MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) 0.18/0.60 MARK(length(X:S)) -> A__LENGTH(mark(X:S)) 0.18/0.60 MARK(length(X:S)) -> MARK(X:S) 0.18/0.60 MARK(s(X:S)) -> MARK(X:S) 0.18/0.60 -> Rules: 0.18/0.60 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.60 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.60 a__and(tt,X:S) -> mark(X:S) 0.18/0.60 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.60 a__isNat(0) -> tt 0.18/0.60 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.60 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.60 a__isNat(X:S) -> isNat(X:S) 0.18/0.60 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.60 a__isNatIList(zeros) -> tt 0.18/0.60 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.60 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.60 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.60 a__isNatList(nil) -> tt 0.18/0.60 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.60 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.60 a__length(nil) -> 0 0.18/0.60 a__length(X:S) -> length(X:S) 0.18/0.60 a__zeros -> cons(0,zeros) 0.18/0.60 a__zeros -> zeros 0.18/0.60 mark(0) -> 0 0.18/0.60 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.60 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.60 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.60 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.60 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.60 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.60 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.60 mark(nil) -> nil 0.18/0.60 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.60 mark(tt) -> tt 0.18/0.60 mark(zeros) -> a__zeros 0.18/0.60 -> Usable rules: 0.18/0.60 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.60 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.60 a__and(tt,X:S) -> mark(X:S) 0.18/0.60 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.60 a__isNat(0) -> tt 0.18/0.60 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.60 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.60 a__isNat(X:S) -> isNat(X:S) 0.18/0.60 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.60 a__isNatIList(zeros) -> tt 0.18/0.60 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.60 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.60 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.60 a__isNatList(nil) -> tt 0.18/0.60 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.60 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.60 a__length(nil) -> 0 0.18/0.60 a__length(X:S) -> length(X:S) 0.18/0.60 a__zeros -> cons(0,zeros) 0.18/0.60 a__zeros -> zeros 0.18/0.60 mark(0) -> 0 0.18/0.60 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.60 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.60 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.60 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.60 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.60 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.60 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.60 mark(nil) -> nil 0.18/0.60 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.60 mark(tt) -> tt 0.18/0.60 mark(zeros) -> a__zeros 0.18/0.60 ->Interpretation type: 0.18/0.60 Linear 0.18/0.60 ->Coefficients: 0.18/0.60 Natural Numbers 0.18/0.60 ->Dimension: 0.18/0.60 1 0.18/0.60 ->Bound: 0.18/0.60 2 0.18/0.60 ->Interpretation: 0.18/0.60 0.18/0.60 [a__U11](X1,X2) = X1 + X2 + 2 0.18/0.60 [a__and](X1,X2) = X1 + X2 0.18/0.60 [a__isNat](X) = X + 2 0.18/0.60 [a__isNatIList](X) = 2.X + 2 0.18/0.60 [a__isNatList](X) = X + 1 0.18/0.60 [a__length](X) = X + 2 0.18/0.60 [a__zeros] = 2 0.18/0.60 [mark](X) = X + 2 0.18/0.60 [0] = 0 0.18/0.60 [U11](X1,X2) = X1 + X2 + 2 0.18/0.60 [and](X1,X2) = X1 + X2 0.18/0.60 [cons](X1,X2) = X1 + 2.X2 + 2 0.18/0.60 [fSNonEmpty] = 0 0.18/0.60 [isNat](X) = X 0.18/0.61 [isNatIList](X) = 2.X + 2 0.18/0.61 [isNatList](X) = X + 1 0.18/0.61 [length](X) = X + 2 0.18/0.61 [nil] = 1 0.18/0.61 [s](X) = X 0.18/0.61 [tt] = 2 0.18/0.61 [zeros] = 0 0.18/0.61 [A__U11](X1,X2) = 2.X1 + 2.X2 + 1 0.18/0.61 [A__AND](X1,X2) = 2.X2 + 1 0.18/0.61 [A__ISNAT](X) = X + 1 0.18/0.61 [A__ISNATILIST](X) = 2.X + 2 0.18/0.61 [A__ISNATLIST](X) = X + 2 0.18/0.61 [A__LENGTH](X) = 2.X 0.18/0.61 [A__ZEROS] = 0 0.18/0.61 [MARK](X) = 2.X + 1 0.18/0.61 0.18/0.61 Problem 1: 0.18/0.61 0.18/0.61 SCC Processor: 0.18/0.61 -> Pairs: 0.18/0.61 A__U11(tt,L:S) -> MARK(L:S) 0.18/0.61 A__AND(tt,X:S) -> MARK(X:S) 0.18/0.61 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 0.18/0.61 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) 0.18/0.61 A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) 0.18/0.61 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) 0.18/0.61 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) 0.18/0.61 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(isNat(X:S)) -> A__ISNAT(X:S) 0.18/0.61 MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) 0.18/0.61 MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) 0.18/0.61 MARK(length(X:S)) -> A__LENGTH(mark(X:S)) 0.18/0.61 MARK(length(X:S)) -> MARK(X:S) 0.18/0.61 MARK(s(X:S)) -> MARK(X:S) 0.18/0.61 -> Rules: 0.18/0.61 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.61 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.61 a__and(tt,X:S) -> mark(X:S) 0.18/0.61 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.61 a__isNat(0) -> tt 0.18/0.61 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.61 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.61 a__isNat(X:S) -> isNat(X:S) 0.18/0.61 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 a__isNatIList(zeros) -> tt 0.18/0.61 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.61 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.61 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 a__isNatList(nil) -> tt 0.18/0.61 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.61 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 a__length(nil) -> 0 0.18/0.61 a__length(X:S) -> length(X:S) 0.18/0.61 a__zeros -> cons(0,zeros) 0.18/0.61 a__zeros -> zeros 0.18/0.61 mark(0) -> 0 0.18/0.61 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.61 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.61 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.61 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.61 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.61 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.61 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.61 mark(nil) -> nil 0.18/0.61 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.61 mark(tt) -> tt 0.18/0.61 mark(zeros) -> a__zeros 0.18/0.61 ->Strongly Connected Components: 0.18/0.61 ->->Cycle: 0.18/0.61 ->->-> Pairs: 0.18/0.61 A__U11(tt,L:S) -> MARK(L:S) 0.18/0.61 A__AND(tt,X:S) -> MARK(X:S) 0.18/0.61 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 0.18/0.61 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) 0.18/0.61 A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) 0.18/0.61 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) 0.18/0.61 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) 0.18/0.61 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(isNat(X:S)) -> A__ISNAT(X:S) 0.18/0.61 MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) 0.18/0.61 MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) 0.18/0.61 MARK(length(X:S)) -> A__LENGTH(mark(X:S)) 0.18/0.61 MARK(length(X:S)) -> MARK(X:S) 0.18/0.61 MARK(s(X:S)) -> MARK(X:S) 0.18/0.61 ->->-> Rules: 0.18/0.61 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.61 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.61 a__and(tt,X:S) -> mark(X:S) 0.18/0.61 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.61 a__isNat(0) -> tt 0.18/0.61 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.61 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.61 a__isNat(X:S) -> isNat(X:S) 0.18/0.61 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 a__isNatIList(zeros) -> tt 0.18/0.61 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.61 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.61 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 a__isNatList(nil) -> tt 0.18/0.61 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.61 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 a__length(nil) -> 0 0.18/0.61 a__length(X:S) -> length(X:S) 0.18/0.61 a__zeros -> cons(0,zeros) 0.18/0.61 a__zeros -> zeros 0.18/0.61 mark(0) -> 0 0.18/0.61 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.61 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.61 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.61 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.61 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.61 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.61 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.61 mark(nil) -> nil 0.18/0.61 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.61 mark(tt) -> tt 0.18/0.61 mark(zeros) -> a__zeros 0.18/0.61 0.18/0.61 Problem 1: 0.18/0.61 0.18/0.61 Reduction Pairs Processor: 0.18/0.61 -> Pairs: 0.18/0.61 A__U11(tt,L:S) -> MARK(L:S) 0.18/0.61 A__AND(tt,X:S) -> MARK(X:S) 0.18/0.61 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 0.18/0.61 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) 0.18/0.61 A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) 0.18/0.61 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) 0.18/0.61 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) 0.18/0.61 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(isNat(X:S)) -> A__ISNAT(X:S) 0.18/0.61 MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) 0.18/0.61 MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) 0.18/0.61 MARK(length(X:S)) -> A__LENGTH(mark(X:S)) 0.18/0.61 MARK(length(X:S)) -> MARK(X:S) 0.18/0.61 MARK(s(X:S)) -> MARK(X:S) 0.18/0.61 -> Rules: 0.18/0.61 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.61 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.61 a__and(tt,X:S) -> mark(X:S) 0.18/0.61 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.61 a__isNat(0) -> tt 0.18/0.61 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.61 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.61 a__isNat(X:S) -> isNat(X:S) 0.18/0.61 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 a__isNatIList(zeros) -> tt 0.18/0.61 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.61 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.61 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 a__isNatList(nil) -> tt 0.18/0.61 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.61 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 a__length(nil) -> 0 0.18/0.61 a__length(X:S) -> length(X:S) 0.18/0.61 a__zeros -> cons(0,zeros) 0.18/0.61 a__zeros -> zeros 0.18/0.61 mark(0) -> 0 0.18/0.61 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.61 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.61 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.61 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.61 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.61 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.61 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.61 mark(nil) -> nil 0.18/0.61 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.61 mark(tt) -> tt 0.18/0.61 mark(zeros) -> a__zeros 0.18/0.61 -> Usable rules: 0.18/0.61 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.61 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.61 a__and(tt,X:S) -> mark(X:S) 0.18/0.61 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.61 a__isNat(0) -> tt 0.18/0.61 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.61 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.61 a__isNat(X:S) -> isNat(X:S) 0.18/0.61 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 a__isNatIList(zeros) -> tt 0.18/0.61 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.61 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.61 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 a__isNatList(nil) -> tt 0.18/0.61 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.61 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 a__length(nil) -> 0 0.18/0.61 a__length(X:S) -> length(X:S) 0.18/0.61 a__zeros -> cons(0,zeros) 0.18/0.61 a__zeros -> zeros 0.18/0.61 mark(0) -> 0 0.18/0.61 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.61 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.61 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.61 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.61 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.61 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.61 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.61 mark(nil) -> nil 0.18/0.61 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.61 mark(tt) -> tt 0.18/0.61 mark(zeros) -> a__zeros 0.18/0.61 ->Interpretation type: 0.18/0.61 Linear 0.18/0.61 ->Coefficients: 0.18/0.61 Natural Numbers 0.18/0.61 ->Dimension: 0.18/0.61 1 0.18/0.61 ->Bound: 0.18/0.61 2 0.18/0.61 ->Interpretation: 0.18/0.61 0.18/0.61 [a__U11](X1,X2) = X1 + 2.X2 + 2 0.18/0.61 [a__and](X1,X2) = X1 + 2.X2 0.18/0.61 [a__isNat](X) = 2 0.18/0.61 [a__isNatIList](X) = 2.X + 2 0.18/0.61 [a__isNatList](X) = 2 0.18/0.61 [a__length](X) = X + 2 0.18/0.61 [a__zeros] = 2 0.18/0.61 [mark](X) = 2.X + 2 0.18/0.61 [0] = 0 0.18/0.61 [U11](X1,X2) = X1 + X2 + 1 0.18/0.61 [and](X1,X2) = X1 + X2 0.18/0.61 [cons](X1,X2) = 2.X1 + 2.X2 + 2 0.18/0.61 [fSNonEmpty] = 0 0.18/0.61 [isNat](X) = 0 0.18/0.61 [isNatIList](X) = 2.X + 2 0.18/0.61 [isNatList](X) = 0 0.18/0.61 [length](X) = X + 2 0.18/0.61 [nil] = 0 0.18/0.61 [s](X) = X 0.18/0.61 [tt] = 2 0.18/0.61 [zeros] = 0 0.18/0.61 [A__U11](X1,X2) = X1 + 2.X2 0.18/0.61 [A__AND](X1,X2) = 2.X2 + 1 0.18/0.61 [A__ISNAT](X) = 1 0.18/0.61 [A__ISNATILIST](X) = 2.X + 2 0.18/0.61 [A__ISNATLIST](X) = 1 0.18/0.61 [A__LENGTH](X) = X 0.18/0.61 [A__ZEROS] = 0 0.18/0.61 [MARK](X) = 2.X + 1 0.18/0.61 0.18/0.61 Problem 1: 0.18/0.61 0.18/0.61 SCC Processor: 0.18/0.61 -> Pairs: 0.18/0.61 A__AND(tt,X:S) -> MARK(X:S) 0.18/0.61 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 0.18/0.61 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) 0.18/0.61 A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) 0.18/0.61 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) 0.18/0.61 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) 0.18/0.61 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(isNat(X:S)) -> A__ISNAT(X:S) 0.18/0.61 MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) 0.18/0.61 MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) 0.18/0.61 MARK(length(X:S)) -> A__LENGTH(mark(X:S)) 0.18/0.61 MARK(length(X:S)) -> MARK(X:S) 0.18/0.61 MARK(s(X:S)) -> MARK(X:S) 0.18/0.61 -> Rules: 0.18/0.61 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.61 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.61 a__and(tt,X:S) -> mark(X:S) 0.18/0.61 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.61 a__isNat(0) -> tt 0.18/0.61 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.61 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.61 a__isNat(X:S) -> isNat(X:S) 0.18/0.61 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 a__isNatIList(zeros) -> tt 0.18/0.61 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.61 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.61 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 a__isNatList(nil) -> tt 0.18/0.61 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.61 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 a__length(nil) -> 0 0.18/0.61 a__length(X:S) -> length(X:S) 0.18/0.61 a__zeros -> cons(0,zeros) 0.18/0.61 a__zeros -> zeros 0.18/0.61 mark(0) -> 0 0.18/0.61 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.61 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.61 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.61 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.61 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.61 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.61 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.61 mark(nil) -> nil 0.18/0.61 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.61 mark(tt) -> tt 0.18/0.61 mark(zeros) -> a__zeros 0.18/0.61 ->Strongly Connected Components: 0.18/0.61 ->->Cycle: 0.18/0.61 ->->-> Pairs: 0.18/0.61 A__AND(tt,X:S) -> MARK(X:S) 0.18/0.61 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 0.18/0.61 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) 0.18/0.61 A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) 0.18/0.61 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) 0.18/0.61 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(isNat(X:S)) -> A__ISNAT(X:S) 0.18/0.61 MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) 0.18/0.61 MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) 0.18/0.61 MARK(length(X:S)) -> A__LENGTH(mark(X:S)) 0.18/0.61 MARK(length(X:S)) -> MARK(X:S) 0.18/0.61 MARK(s(X:S)) -> MARK(X:S) 0.18/0.61 ->->-> Rules: 0.18/0.61 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.61 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.61 a__and(tt,X:S) -> mark(X:S) 0.18/0.61 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.61 a__isNat(0) -> tt 0.18/0.61 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.61 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.61 a__isNat(X:S) -> isNat(X:S) 0.18/0.61 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 a__isNatIList(zeros) -> tt 0.18/0.61 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.61 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.61 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 a__isNatList(nil) -> tt 0.18/0.61 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.61 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 a__length(nil) -> 0 0.18/0.61 a__length(X:S) -> length(X:S) 0.18/0.61 a__zeros -> cons(0,zeros) 0.18/0.61 a__zeros -> zeros 0.18/0.61 mark(0) -> 0 0.18/0.61 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.61 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.61 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.61 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.61 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.61 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.61 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.61 mark(nil) -> nil 0.18/0.61 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.61 mark(tt) -> tt 0.18/0.61 mark(zeros) -> a__zeros 0.18/0.61 0.18/0.61 Problem 1: 0.18/0.61 0.18/0.61 Reduction Pairs Processor: 0.18/0.61 -> Pairs: 0.18/0.61 A__AND(tt,X:S) -> MARK(X:S) 0.18/0.61 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 0.18/0.61 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) 0.18/0.61 A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) 0.18/0.61 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) 0.18/0.61 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(isNat(X:S)) -> A__ISNAT(X:S) 0.18/0.61 MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) 0.18/0.61 MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) 0.18/0.61 MARK(length(X:S)) -> A__LENGTH(mark(X:S)) 0.18/0.61 MARK(length(X:S)) -> MARK(X:S) 0.18/0.61 MARK(s(X:S)) -> MARK(X:S) 0.18/0.61 -> Rules: 0.18/0.61 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.61 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.61 a__and(tt,X:S) -> mark(X:S) 0.18/0.61 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.61 a__isNat(0) -> tt 0.18/0.61 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.61 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.61 a__isNat(X:S) -> isNat(X:S) 0.18/0.61 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 a__isNatIList(zeros) -> tt 0.18/0.61 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.61 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.61 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 a__isNatList(nil) -> tt 0.18/0.61 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.61 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 a__length(nil) -> 0 0.18/0.61 a__length(X:S) -> length(X:S) 0.18/0.61 a__zeros -> cons(0,zeros) 0.18/0.61 a__zeros -> zeros 0.18/0.61 mark(0) -> 0 0.18/0.61 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.61 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.61 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.61 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.61 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.61 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.61 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.61 mark(nil) -> nil 0.18/0.61 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.61 mark(tt) -> tt 0.18/0.61 mark(zeros) -> a__zeros 0.18/0.61 -> Usable rules: 0.18/0.61 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.61 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.61 a__and(tt,X:S) -> mark(X:S) 0.18/0.61 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.61 a__isNat(0) -> tt 0.18/0.61 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.61 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.61 a__isNat(X:S) -> isNat(X:S) 0.18/0.61 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 a__isNatIList(zeros) -> tt 0.18/0.61 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.61 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.61 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 a__isNatList(nil) -> tt 0.18/0.61 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.61 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 a__length(nil) -> 0 0.18/0.61 a__length(X:S) -> length(X:S) 0.18/0.61 a__zeros -> cons(0,zeros) 0.18/0.61 a__zeros -> zeros 0.18/0.61 mark(0) -> 0 0.18/0.61 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.61 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.61 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.61 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.61 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.61 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.61 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.61 mark(nil) -> nil 0.18/0.61 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.61 mark(tt) -> tt 0.18/0.61 mark(zeros) -> a__zeros 0.18/0.61 ->Interpretation type: 0.18/0.61 Linear 0.18/0.61 ->Coefficients: 0.18/0.61 Natural Numbers 0.18/0.61 ->Dimension: 0.18/0.61 1 0.18/0.61 ->Bound: 0.18/0.61 2 0.18/0.61 ->Interpretation: 0.18/0.61 0.18/0.61 [a__U11](X1,X2) = X1 + X2 + 2 0.18/0.61 [a__and](X1,X2) = X1 + X2 + 1 0.18/0.61 [a__isNat](X) = X + 2 0.18/0.61 [a__isNatIList](X) = 2.X + 2 0.18/0.61 [a__isNatList](X) = X + 1 0.18/0.61 [a__length](X) = X + 2 0.18/0.61 [a__zeros] = 2 0.18/0.61 [mark](X) = X + 2 0.18/0.61 [0] = 0 0.18/0.61 [U11](X1,X2) = X1 + X2 + 2 0.18/0.61 [and](X1,X2) = X1 + X2 + 1 0.18/0.61 [cons](X1,X2) = X1 + 2.X2 + 2 0.18/0.61 [fSNonEmpty] = 0 0.18/0.61 [isNat](X) = X 0.18/0.61 [isNatIList](X) = 2.X + 1 0.18/0.61 [isNatList](X) = X 0.18/0.61 [length](X) = X + 2 0.18/0.61 [nil] = 1 0.18/0.61 [s](X) = X 0.18/0.61 [tt] = 2 0.18/0.61 [zeros] = 0 0.18/0.61 [A__U11](X1,X2) = 0 0.18/0.61 [A__AND](X1,X2) = X1 + 2.X2 + 2 0.18/0.61 [A__ISNAT](X) = 2.X 0.18/0.61 [A__ISNATILIST](X) = 2.X + 2 0.18/0.61 [A__ISNATLIST](X) = 2.X + 1 0.18/0.61 [A__LENGTH](X) = 2.X + 1 0.18/0.61 [A__ZEROS] = 0 0.18/0.61 [MARK](X) = 2.X + 2 0.18/0.61 0.18/0.61 Problem 1: 0.18/0.61 0.18/0.61 SCC Processor: 0.18/0.61 -> Pairs: 0.18/0.61 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 0.18/0.61 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) 0.18/0.61 A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) 0.18/0.61 A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) 0.18/0.61 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) 0.18/0.61 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(isNat(X:S)) -> A__ISNAT(X:S) 0.18/0.61 MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) 0.18/0.61 MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) 0.18/0.61 MARK(length(X:S)) -> A__LENGTH(mark(X:S)) 0.18/0.61 MARK(length(X:S)) -> MARK(X:S) 0.18/0.61 MARK(s(X:S)) -> MARK(X:S) 0.18/0.61 -> Rules: 0.18/0.61 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.61 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.61 a__and(tt,X:S) -> mark(X:S) 0.18/0.61 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.61 a__isNat(0) -> tt 0.18/0.61 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.61 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.61 a__isNat(X:S) -> isNat(X:S) 0.18/0.61 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 a__isNatIList(zeros) -> tt 0.18/0.61 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.61 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.61 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 a__isNatList(nil) -> tt 0.18/0.61 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.61 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 a__length(nil) -> 0 0.18/0.61 a__length(X:S) -> length(X:S) 0.18/0.61 a__zeros -> cons(0,zeros) 0.18/0.61 a__zeros -> zeros 0.18/0.61 mark(0) -> 0 0.18/0.61 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.61 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.61 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.61 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.61 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.61 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.61 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.61 mark(nil) -> nil 0.18/0.61 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.61 mark(tt) -> tt 0.18/0.61 mark(zeros) -> a__zeros 0.18/0.61 ->Strongly Connected Components: 0.18/0.61 ->->Cycle: 0.18/0.61 ->->-> Pairs: 0.18/0.61 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 0.18/0.61 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.61 ->->-> Rules: 0.18/0.61 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.61 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.61 a__and(tt,X:S) -> mark(X:S) 0.18/0.61 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.61 a__isNat(0) -> tt 0.18/0.61 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.61 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.61 a__isNat(X:S) -> isNat(X:S) 0.18/0.61 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 a__isNatIList(zeros) -> tt 0.18/0.61 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.61 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.61 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 a__isNatList(nil) -> tt 0.18/0.61 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.61 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 a__length(nil) -> 0 0.18/0.61 a__length(X:S) -> length(X:S) 0.18/0.61 a__zeros -> cons(0,zeros) 0.18/0.61 a__zeros -> zeros 0.18/0.61 mark(0) -> 0 0.18/0.61 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.61 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.61 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.61 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.61 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.61 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.61 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.61 mark(nil) -> nil 0.18/0.61 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.61 mark(tt) -> tt 0.18/0.61 mark(zeros) -> a__zeros 0.18/0.61 ->->Cycle: 0.18/0.61 ->->-> Pairs: 0.18/0.61 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(length(X:S)) -> MARK(X:S) 0.18/0.61 MARK(s(X:S)) -> MARK(X:S) 0.18/0.61 ->->-> Rules: 0.18/0.61 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.61 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.61 a__and(tt,X:S) -> mark(X:S) 0.18/0.61 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.61 a__isNat(0) -> tt 0.18/0.61 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.61 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.61 a__isNat(X:S) -> isNat(X:S) 0.18/0.61 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 a__isNatIList(zeros) -> tt 0.18/0.61 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.61 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.61 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 a__isNatList(nil) -> tt 0.18/0.61 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.61 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 a__length(nil) -> 0 0.18/0.61 a__length(X:S) -> length(X:S) 0.18/0.61 a__zeros -> cons(0,zeros) 0.18/0.61 a__zeros -> zeros 0.18/0.61 mark(0) -> 0 0.18/0.61 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.61 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.61 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.61 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.61 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.61 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.61 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.61 mark(nil) -> nil 0.18/0.61 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.61 mark(tt) -> tt 0.18/0.61 mark(zeros) -> a__zeros 0.18/0.61 0.18/0.61 0.18/0.61 The problem is decomposed in 2 subproblems. 0.18/0.61 0.18/0.61 Problem 1.1: 0.18/0.61 0.18/0.61 Subterm Processor: 0.18/0.61 -> Pairs: 0.18/0.61 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 0.18/0.61 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.18/0.61 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.18/0.61 -> Rules: 0.18/0.61 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.61 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.61 a__and(tt,X:S) -> mark(X:S) 0.18/0.61 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.61 a__isNat(0) -> tt 0.18/0.61 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.61 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.61 a__isNat(X:S) -> isNat(X:S) 0.18/0.61 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 a__isNatIList(zeros) -> tt 0.18/0.61 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.61 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.61 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 a__isNatList(nil) -> tt 0.18/0.61 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.61 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 a__length(nil) -> 0 0.18/0.61 a__length(X:S) -> length(X:S) 0.18/0.61 a__zeros -> cons(0,zeros) 0.18/0.61 a__zeros -> zeros 0.18/0.61 mark(0) -> 0 0.18/0.61 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.61 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.61 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.61 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.61 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.61 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.61 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.61 mark(nil) -> nil 0.18/0.61 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.61 mark(tt) -> tt 0.18/0.61 mark(zeros) -> a__zeros 0.18/0.61 ->Projection: 0.18/0.61 pi(A__ISNAT) = 1 0.18/0.61 pi(A__ISNATLIST) = 1 0.18/0.61 0.18/0.61 Problem 1.1: 0.18/0.61 0.18/0.61 SCC Processor: 0.18/0.61 -> Pairs: 0.18/0.61 Empty 0.18/0.61 -> Rules: 0.18/0.61 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.61 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.61 a__and(tt,X:S) -> mark(X:S) 0.18/0.61 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.61 a__isNat(0) -> tt 0.18/0.61 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.61 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.61 a__isNat(X:S) -> isNat(X:S) 0.18/0.61 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 a__isNatIList(zeros) -> tt 0.18/0.61 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.61 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.61 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 a__isNatList(nil) -> tt 0.18/0.61 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.61 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 a__length(nil) -> 0 0.18/0.61 a__length(X:S) -> length(X:S) 0.18/0.61 a__zeros -> cons(0,zeros) 0.18/0.61 a__zeros -> zeros 0.18/0.61 mark(0) -> 0 0.18/0.61 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.61 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.61 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.61 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.61 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.61 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.61 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.61 mark(nil) -> nil 0.18/0.61 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.61 mark(tt) -> tt 0.18/0.61 mark(zeros) -> a__zeros 0.18/0.61 ->Strongly Connected Components: 0.18/0.61 There is no strongly connected component 0.18/0.61 0.18/0.61 The problem is finite. 0.18/0.61 0.18/0.61 Problem 1.2: 0.18/0.61 0.18/0.61 Subterm Processor: 0.18/0.61 -> Pairs: 0.18/0.61 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.18/0.61 MARK(length(X:S)) -> MARK(X:S) 0.18/0.61 MARK(s(X:S)) -> MARK(X:S) 0.18/0.61 -> Rules: 0.18/0.61 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.61 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.61 a__and(tt,X:S) -> mark(X:S) 0.18/0.61 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.61 a__isNat(0) -> tt 0.18/0.61 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.61 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.61 a__isNat(X:S) -> isNat(X:S) 0.18/0.61 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 a__isNatIList(zeros) -> tt 0.18/0.61 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.61 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.61 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 a__isNatList(nil) -> tt 0.18/0.61 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.61 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 a__length(nil) -> 0 0.18/0.61 a__length(X:S) -> length(X:S) 0.18/0.61 a__zeros -> cons(0,zeros) 0.18/0.61 a__zeros -> zeros 0.18/0.61 mark(0) -> 0 0.18/0.61 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.61 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.61 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.61 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.61 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.61 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.61 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.61 mark(nil) -> nil 0.18/0.61 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.61 mark(tt) -> tt 0.18/0.61 mark(zeros) -> a__zeros 0.18/0.61 ->Projection: 0.18/0.61 pi(MARK) = 1 0.18/0.61 0.18/0.61 Problem 1.2: 0.18/0.61 0.18/0.61 SCC Processor: 0.18/0.61 -> Pairs: 0.18/0.61 Empty 0.18/0.61 -> Rules: 0.18/0.61 a__U11(tt,L:S) -> s(a__length(mark(L:S))) 0.18/0.61 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.18/0.61 a__and(tt,X:S) -> mark(X:S) 0.18/0.61 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 0.18/0.61 a__isNat(0) -> tt 0.18/0.61 a__isNat(length(V1:S)) -> a__isNatList(V1:S) 0.18/0.61 a__isNat(s(V1:S)) -> a__isNat(V1:S) 0.18/0.61 a__isNat(X:S) -> isNat(X:S) 0.18/0.61 a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) 0.18/0.61 a__isNatIList(zeros) -> tt 0.18/0.61 a__isNatIList(V:S) -> a__isNatList(V:S) 0.18/0.61 a__isNatIList(X:S) -> isNatIList(X:S) 0.18/0.61 a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) 0.18/0.61 a__isNatList(nil) -> tt 0.18/0.61 a__isNatList(X:S) -> isNatList(X:S) 0.18/0.61 a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) 0.18/0.61 a__length(nil) -> 0 0.18/0.61 a__length(X:S) -> length(X:S) 0.18/0.61 a__zeros -> cons(0,zeros) 0.18/0.61 a__zeros -> zeros 0.18/0.61 mark(0) -> 0 0.18/0.61 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.18/0.61 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 0.18/0.61 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.18/0.61 mark(isNat(X:S)) -> a__isNat(X:S) 0.18/0.61 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 0.18/0.61 mark(isNatList(X:S)) -> a__isNatList(X:S) 0.18/0.61 mark(length(X:S)) -> a__length(mark(X:S)) 0.18/0.61 mark(nil) -> nil 0.18/0.61 mark(s(X:S)) -> s(mark(X:S)) 0.18/0.61 mark(tt) -> tt 0.18/0.61 mark(zeros) -> a__zeros 0.18/0.61 ->Strongly Connected Components: 0.18/0.61 There is no strongly connected component 0.18/0.61 0.18/0.61 The problem is finite. 0.18/0.61 EOF