/export/starexec/sandbox2/solver/bin/starexec_run_tct_rc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: U11(mark(X1),X2) -> mark(U11(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U12(ok(X)) -> ok(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U22(ok(X)) -> ok(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U32(ok(X)) -> ok(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U43(ok(X)) -> ok(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U53(ok(X)) -> ok(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U63(ok(X)) -> ok(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U81(mark(X)) -> mark(U81(X)) U81(ok(X)) -> ok(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(X)) -> U12(active(X)) active(U12(tt())) -> mark(tt()) active(U21(X1,X2)) -> U21(active(X1),X2) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(X)) -> U22(active(X)) active(U22(tt())) -> mark(tt()) active(U31(X1,X2)) -> U31(active(X1),X2) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(X)) -> U32(active(X)) active(U32(tt())) -> mark(tt()) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(X1,X2)) -> U42(active(X1),X2) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(X)) -> U43(active(X)) active(U43(tt())) -> mark(tt()) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(X1,X2)) -> U52(active(X1),X2) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(X)) -> U53(active(X)) active(U53(tt())) -> mark(tt()) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(X1,X2)) -> U62(active(X1),X2) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(X)) -> U63(active(X)) active(U63(tt())) -> mark(tt()) active(U71(X1,X2)) -> U71(active(X1),X2) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(X)) -> U81(active(X)) active(U81(tt())) -> mark(nil()) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(X1,X2)) -> and(active(X1),X2) active(and(tt(),X)) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(X)) -> length(active(X)) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) ,L)) active(length(nil())) -> mark(0()) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(X1,active(X2)) active(take(X1,X2)) -> take(active(X1),X2) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)) ,and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))) ,IL ,M ,N)) active(zeros()) -> mark(cons(0(),zeros())) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) isNatList(ok(X)) -> ok(isNatList(X)) length(mark(X)) -> mark(length(X)) length(ok(X)) -> ok(length(X)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(U12(X)) -> U12(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U81(X)) -> U81(proper(X)) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(zeros()) -> ok(zeros()) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) take(X1,mark(X2)) -> mark(take(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {U11/2,U12/1,U21/2,U22/1,U31/2,U32/1,U41/3,U42/2,U43/1,U51/3,U52/2,U53/1,U61/3,U62/2,U63/1,U71/2,U81/1,U91/4 ,active/1,and/2,cons/2,isNat/1,isNatIList/1,isNatIListKind/1,isNatKind/1,isNatList/1,length/1,proper/1,s/1 ,take/2,top/1} / {0/0,mark/1,nil/0,ok/1,tt/0,zeros/0} - Obligation: runtime complexity wrt. defined symbols {U11,U12,U21,U22,U31,U32,U41,U42,U43,U51,U52,U53,U61,U62,U63,U71 ,U81,U91,active,and,cons,isNat,isNatIList,isNatIListKind,isNatKind,isNatList,length,proper,s,take ,top} and constructors {0,mark,nil,ok,tt,zeros} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: U11(mark(X1),X2) -> mark(U11(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U12(ok(X)) -> ok(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U22(ok(X)) -> ok(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U32(ok(X)) -> ok(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U43(ok(X)) -> ok(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U53(ok(X)) -> ok(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U63(ok(X)) -> ok(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U81(mark(X)) -> mark(U81(X)) U81(ok(X)) -> ok(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(X)) -> U12(active(X)) active(U12(tt())) -> mark(tt()) active(U21(X1,X2)) -> U21(active(X1),X2) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(X)) -> U22(active(X)) active(U22(tt())) -> mark(tt()) active(U31(X1,X2)) -> U31(active(X1),X2) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(X)) -> U32(active(X)) active(U32(tt())) -> mark(tt()) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(X1,X2)) -> U42(active(X1),X2) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(X)) -> U43(active(X)) active(U43(tt())) -> mark(tt()) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(X1,X2)) -> U52(active(X1),X2) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(X)) -> U53(active(X)) active(U53(tt())) -> mark(tt()) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(X1,X2)) -> U62(active(X1),X2) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(X)) -> U63(active(X)) active(U63(tt())) -> mark(tt()) active(U71(X1,X2)) -> U71(active(X1),X2) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(X)) -> U81(active(X)) active(U81(tt())) -> mark(nil()) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(X1,X2)) -> and(active(X1),X2) active(and(tt(),X)) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(X)) -> length(active(X)) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) ,L)) active(length(nil())) -> mark(0()) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(X1,active(X2)) active(take(X1,X2)) -> take(active(X1),X2) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)) ,and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))) ,IL ,M ,N)) active(zeros()) -> mark(cons(0(),zeros())) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) isNatList(ok(X)) -> ok(isNatList(X)) length(mark(X)) -> mark(length(X)) length(ok(X)) -> ok(length(X)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(U12(X)) -> U12(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U81(X)) -> U81(proper(X)) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(zeros()) -> ok(zeros()) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) take(X1,mark(X2)) -> mark(take(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {U11/2,U12/1,U21/2,U22/1,U31/2,U32/1,U41/3,U42/2,U43/1,U51/3,U52/2,U53/1,U61/3,U62/2,U63/1,U71/2,U81/1,U91/4 ,active/1,and/2,cons/2,isNat/1,isNatIList/1,isNatIListKind/1,isNatKind/1,isNatList/1,length/1,proper/1,s/1 ,take/2,top/1} / {0/0,mark/1,nil/0,ok/1,tt/0,zeros/0} - Obligation: runtime complexity wrt. defined symbols {U11,U12,U21,U22,U31,U32,U41,U42,U43,U51,U52,U53,U61,U62,U63,U71 ,U81,U91,active,and,cons,isNat,isNatIList,isNatIListKind,isNatKind,isNatList,length,proper,s,take ,top} and constructors {0,mark,nil,ok,tt,zeros} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 3: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: U11(mark(X1),X2) -> mark(U11(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U12(ok(X)) -> ok(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U22(ok(X)) -> ok(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U32(ok(X)) -> ok(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U43(ok(X)) -> ok(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U53(ok(X)) -> ok(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U63(ok(X)) -> ok(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U81(mark(X)) -> mark(U81(X)) U81(ok(X)) -> ok(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(X)) -> U12(active(X)) active(U12(tt())) -> mark(tt()) active(U21(X1,X2)) -> U21(active(X1),X2) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(X)) -> U22(active(X)) active(U22(tt())) -> mark(tt()) active(U31(X1,X2)) -> U31(active(X1),X2) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(X)) -> U32(active(X)) active(U32(tt())) -> mark(tt()) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(X1,X2)) -> U42(active(X1),X2) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(X)) -> U43(active(X)) active(U43(tt())) -> mark(tt()) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(X1,X2)) -> U52(active(X1),X2) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(X)) -> U53(active(X)) active(U53(tt())) -> mark(tt()) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(X1,X2)) -> U62(active(X1),X2) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(X)) -> U63(active(X)) active(U63(tt())) -> mark(tt()) active(U71(X1,X2)) -> U71(active(X1),X2) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(X)) -> U81(active(X)) active(U81(tt())) -> mark(nil()) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(X1,X2)) -> and(active(X1),X2) active(and(tt(),X)) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(X)) -> length(active(X)) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) ,L)) active(length(nil())) -> mark(0()) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(X1,active(X2)) active(take(X1,X2)) -> take(active(X1),X2) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)) ,and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))) ,IL ,M ,N)) active(zeros()) -> mark(cons(0(),zeros())) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) isNatList(ok(X)) -> ok(isNatList(X)) length(mark(X)) -> mark(length(X)) length(ok(X)) -> ok(length(X)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(U12(X)) -> U12(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(U81(X)) -> U81(proper(X)) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(zeros()) -> ok(zeros()) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) take(X1,mark(X2)) -> mark(take(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {U11/2,U12/1,U21/2,U22/1,U31/2,U32/1,U41/3,U42/2,U43/1,U51/3,U52/2,U53/1,U61/3,U62/2,U63/1,U71/2,U81/1,U91/4 ,active/1,and/2,cons/2,isNat/1,isNatIList/1,isNatIListKind/1,isNatKind/1,isNatList/1,length/1,proper/1,s/1 ,take/2,top/1} / {0/0,mark/1,nil/0,ok/1,tt/0,zeros/0} - Obligation: runtime complexity wrt. defined symbols {U11,U12,U21,U22,U31,U32,U41,U42,U43,U51,U52,U53,U61,U62,U63,U71 ,U81,U91,active,and,cons,isNat,isNatIList,isNatIListKind,isNatKind,isNatList,length,proper,s,take ,top} and constructors {0,mark,nil,ok,tt,zeros} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: U11(x,y){x -> mark(x)} = U11(mark(x),y) ->^+ mark(U11(x,y)) = C[U11(x,y) = U11(x,y){}] WORST_CASE(Omega(n^1),?)