YES (VAR V2 L N V1 V X X1 X2 X3) (RULES a__zeros -> cons(0,zeros) a__U11(tt) -> tt a__U21(tt) -> tt a__U31(tt) -> tt a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U42(tt) -> tt a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U52(tt) -> tt a__U61(tt,L,N) -> a__U62(a__isNat(N),L) a__U62(tt,L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil) -> 0 a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros) -> a__zeros mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) ) Proving termination of rewriting for LengthOfFiniteLists_nokinds_noand_GM: -> Dependency pairs: nF_a__U41(tt,V2) -> nF_a__U42(a__isNatIList(V2)) nF_a__U41(tt,V2) -> nF_a__isNatIList(V2) nF_a__U51(tt,V2) -> nF_a__U52(a__isNatList(V2)) nF_a__U51(tt,V2) -> nF_a__isNatList(V2) nF_a__U61(tt,L,N) -> nF_a__U62(a__isNat(N),L) nF_a__U61(tt,L,N) -> nF_a__isNat(N) nF_a__U62(tt,L) -> nF_a__length(mark(L)) nF_a__U62(tt,L) -> nF_mark(L) nF_a__isNat(length(V1)) -> nF_a__U11(a__isNatList(V1)) nF_a__isNat(length(V1)) -> nF_a__isNatList(V1) nF_a__isNat(s(V1)) -> nF_a__U21(a__isNat(V1)) nF_a__isNat(s(V1)) -> nF_a__isNat(V1) nF_a__isNatIList(V) -> nF_a__U31(a__isNatList(V)) nF_a__isNatIList(V) -> nF_a__isNatList(V) nF_a__isNatIList(cons(V1,V2)) -> nF_a__U41(a__isNat(V1),V2) nF_a__isNatIList(cons(V1,V2)) -> nF_a__isNat(V1) nF_a__isNatList(cons(V1,V2)) -> nF_a__U51(a__isNat(V1),V2) nF_a__isNatList(cons(V1,V2)) -> nF_a__isNat(V1) nF_a__length(cons(N,L)) -> nF_a__U61(a__isNatList(L),L,N) nF_a__length(cons(N,L)) -> nF_a__isNatList(L) nF_mark(zeros) -> nF_a__zeros nF_mark(U11(X)) -> nF_a__U11(mark(X)) nF_mark(U11(X)) -> nF_mark(X) nF_mark(U21(X)) -> nF_a__U21(mark(X)) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U31(X)) -> nF_a__U31(mark(X)) nF_mark(U31(X)) -> nF_mark(X) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U42(X)) -> nF_a__U42(mark(X)) nF_mark(U42(X)) -> nF_mark(X) nF_mark(isNatIList(X)) -> nF_a__isNatIList(X) nF_mark(U51(X1,X2)) -> nF_a__U51(mark(X1),X2) nF_mark(U51(X1,X2)) -> nF_mark(X1) nF_mark(U52(X)) -> nF_a__U52(mark(X)) nF_mark(U52(X)) -> nF_mark(X) nF_mark(isNatList(X)) -> nF_a__isNatList(X) nF_mark(U61(X1,X2,X3)) -> nF_a__U61(mark(X1),X2,X3) nF_mark(U61(X1,X2,X3)) -> nF_mark(X1) nF_mark(U62(X1,X2)) -> nF_a__U62(mark(X1),X2) nF_mark(U62(X1,X2)) -> nF_mark(X1) nF_mark(isNat(X)) -> nF_a__isNat(X) nF_mark(length(X)) -> nF_a__length(mark(X)) nF_mark(length(X)) -> nF_mark(X) nF_mark(cons(X1,X2)) -> nF_mark(X1) nF_mark(s(X)) -> nF_mark(X) -> Proof of termination for LengthOfFiniteLists_nokinds_noand_GM_1_1: -> -> Dependency pairs in cycle: nF_a__U61(tt,L,N) -> nF_a__U62(a__isNat(N),L) nF_mark(U61(X1,X2,X3)) -> nF_a__U61(mark(X1),X2,X3) nF_mark(s(X)) -> nF_mark(X) nF_mark(cons(X1,X2)) -> nF_mark(X1) nF_mark(length(X)) -> nF_mark(X) nF_mark(U62(X1,X2)) -> nF_mark(X1) nF_mark(U61(X1,X2,X3)) -> nF_mark(X1) nF_mark(U52(X)) -> nF_mark(X) nF_mark(U51(X1,X2)) -> nF_mark(X1) nF_mark(U42(X)) -> nF_mark(X) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U31(X)) -> nF_mark(X) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U11(X)) -> nF_mark(X) nF_a__U62(tt,L) -> nF_mark(L) nF_mark(U62(X1,X2)) -> nF_a__U62(mark(X1),X2) nF_a__length(cons(N,L)) -> nF_a__U61(a__isNatList(L),L,N) nF_mark(length(X)) -> nF_a__length(mark(X)) nF_a__U62(tt,L) -> nF_a__length(mark(L)) UsableRules: a__zeros -> cons(0,zeros) a__U11(tt) -> tt a__U21(tt) -> tt a__U31(tt) -> tt a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U42(tt) -> tt a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U52(tt) -> tt a__U61(tt,L,N) -> a__U62(a__isNat(N),L) a__U62(tt,L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil) -> 0 a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros) -> a__zeros mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Polynomial Interpretation: [a__zeros] = 1 [cons](X1,X2) = X1 + X2 [0] = 0 [zeros] = 1 [a__U11](X) = X.X + X [tt] = 0 [a__U21](X) = X.X + X [a__U31](X) = X.X + X [a__U41](X1,X2) = X1 [a__U42](X) = X.X + X [a__isNatIList](X) = 0 [a__U51](X1,X2) = X1 [a__U52](X) = X.X + X [a__isNatList](X) = 0 [a__U61](X1,X2,X3) = X1 + X2 + 1 [a__U62](X1,X2) = X1 + X2 + 1 [a__isNat](X) = 0 [s](X) = X [a__length](X) = X + 1 [mark](X) = X [length](X) = X + 1 [nil] = 1 [U11](X) = X.X + X [U21](X) = X.X + X [U31](X) = X.X + X [U41](X1,X2) = X1 [U42](X) = X.X + X [isNatIList](X) = 0 [U51](X1,X2) = X1 [U52](X) = X.X + X [isNatList](X) = 0 [U61](X1,X2,X3) = X1 + X2 + 1 [U62](X1,X2) = X1 + X2 + 1 [isNat](X) = 0 [nF_a__U61](X1,X2,X3) = X2 [nF_mark](X) = X [nF_a__U62](X1,X2) = X2 [nF_a__length](X) = X TIME: 0.219703 -> -> Dependency pairs in cycle: nF_a__U61(tt,L,N) -> nF_a__U62(a__isNat(N),L) nF_a__length(cons(N,L)) -> nF_a__U61(a__isNatList(L),L,N) nF_a__U62(tt,L) -> nF_a__length(mark(L)) nF_mark(U62(X1,X2)) -> nF_a__U62(mark(X1),X2) nF_a__U62(tt,L) -> nF_mark(L) nF_mark(U11(X)) -> nF_mark(X) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U31(X)) -> nF_mark(X) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U42(X)) -> nF_mark(X) nF_mark(U51(X1,X2)) -> nF_mark(X1) nF_mark(U52(X)) -> nF_mark(X) nF_mark(U61(X1,X2,X3)) -> nF_mark(X1) nF_mark(U62(X1,X2)) -> nF_mark(X1) nF_mark(length(X)) -> nF_mark(X) nF_mark(cons(X1,X2)) -> nF_mark(X1) nF_mark(s(X)) -> nF_mark(X) nF_mark(U61(X1,X2,X3)) -> nF_a__U61(mark(X1),X2,X3) UsableRules: a__zeros -> cons(0,zeros) a__U11(tt) -> tt a__U21(tt) -> tt a__U31(tt) -> tt a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U42(tt) -> tt a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U52(tt) -> tt a__U61(tt,L,N) -> a__U62(a__isNat(N),L) a__U62(tt,L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil) -> 0 a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros) -> a__zeros mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Polynomial Interpretation: [a__zeros] = 0 [cons](X1,X2) = X1 + X2 [0] = 0 [zeros] = 0 [a__U11](X) = X.X + X [tt] = 0 [a__U21](X) = X.X + X [a__U31](X) = X [a__U41](X1,X2) = X1 [a__U42](X) = X.X + X [a__isNatIList](X) = 0 [a__U51](X1,X2) = X1 [a__U52](X) = X.X + X [a__isNatList](X) = 0 [a__U61](X1,X2,X3) = X1 + X2 + 1 [a__U62](X1,X2) = X1 + X2 + 1 [a__isNat](X) = 0 [s](X) = X [a__length](X) = X + 1 [mark](X) = X [length](X) = X + 1 [nil] = 0 [U11](X) = X.X + X [U21](X) = X.X + X [U31](X) = X [U41](X1,X2) = X1 [U42](X) = X.X + X [isNatIList](X) = 0 [U51](X1,X2) = X1 [U52](X) = X.X + X [isNatList](X) = 0 [U61](X1,X2,X3) = X1 + X2 + 1 [U62](X1,X2) = X1 + X2 + 1 [isNat](X) = 0 [nF_a__U61](X1,X2,X3) = X2 [nF_a__length](X) = X [nF_a__U62](X1,X2) = X2 [nF_mark](X) = X TIME: 0.229493 -> -> Dependency pairs in cycle: nF_a__U61(tt,L,N) -> nF_a__U62(a__isNat(N),L) nF_a__length(cons(N,L)) -> nF_a__U61(a__isNatList(L),L,N) nF_a__U62(tt,L) -> nF_a__length(mark(L)) nF_mark(U62(X1,X2)) -> nF_a__U62(mark(X1),X2) nF_mark(s(X)) -> nF_mark(X) nF_mark(cons(X1,X2)) -> nF_mark(X1) nF_mark(length(X)) -> nF_mark(X) nF_mark(U62(X1,X2)) -> nF_mark(X1) nF_mark(U61(X1,X2,X3)) -> nF_mark(X1) nF_mark(U52(X)) -> nF_mark(X) nF_mark(U51(X1,X2)) -> nF_mark(X1) nF_mark(U42(X)) -> nF_mark(X) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U31(X)) -> nF_mark(X) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U11(X)) -> nF_mark(X) nF_a__U62(tt,L) -> nF_mark(L) UsableRules: a__zeros -> cons(0,zeros) a__U11(tt) -> tt a__U21(tt) -> tt a__U31(tt) -> tt a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U42(tt) -> tt a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U52(tt) -> tt a__U61(tt,L,N) -> a__U62(a__isNat(N),L) a__U62(tt,L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil) -> 0 a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros) -> a__zeros mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Polynomial Interpretation: [a__zeros] = 0 [cons](X1,X2) = X1.X2 + X1 + X2 [0] = 0 [zeros] = 0 [a__U11](X) = X [tt] = 1 [a__U21](X) = X [a__U31](X) = X [a__U41](X1,X2) = X1 [a__U42](X) = X [a__isNatIList](X) = 1 [a__U51](X1,X2) = X1 [a__U52](X) = X [a__isNatList](X) = 1 [a__U61](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 [a__U62](X1,X2) = X1 + X2 [a__isNat](X) = 1 [s](X) = X [a__length](X) = X + 1 [mark](X) = X [length](X) = X + 1 [nil] = 0 [U11](X) = X [U21](X) = X [U31](X) = X [U41](X1,X2) = X1 [U42](X) = X [isNatIList](X) = 1 [U51](X1,X2) = X1 [U52](X) = X [isNatList](X) = 1 [U61](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 [U62](X1,X2) = X1 + X2 [isNat](X) = 1 [nF_a__U61](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + 1 [nF_a__length](X) = X + 1 [nF_a__U62](X1,X2) = X1 + X2 [nF_mark](X) = X TIME: 0.234398 -> -> Dependency pairs in cycle: nF_mark(s(X)) -> nF_mark(X) nF_mark(U11(X)) -> nF_mark(X) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U31(X)) -> nF_mark(X) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U42(X)) -> nF_mark(X) nF_mark(U51(X1,X2)) -> nF_mark(X1) nF_mark(U52(X)) -> nF_mark(X) nF_mark(U61(X1,X2,X3)) -> nF_mark(X1) nF_mark(U62(X1,X2)) -> nF_mark(X1) nF_mark(length(X)) -> nF_mark(X) nF_mark(cons(X1,X2)) -> nF_mark(X1) Termination proved: Cycles verify subterm criterion. -> -> Dependency pairs in cycle: nF_a__U61(tt,L,N) -> nF_a__U62(a__isNat(N),L) nF_a__length(cons(N,L)) -> nF_a__U61(a__isNatList(L),L,N) nF_a__U62(tt,L) -> nF_a__length(mark(L)) UsableRules: a__zeros -> cons(0,zeros) a__U11(tt) -> tt a__U21(tt) -> tt a__U31(tt) -> tt a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U42(tt) -> tt a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U52(tt) -> tt a__U61(tt,L,N) -> a__U62(a__isNat(N),L) a__U62(tt,L) -> s(a__length(mark(L))) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil) -> 0 a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros) -> a__zeros mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Polynomial Interpretation: [a__zeros] = 0 [cons](X1,X2) = X1.X2 + X2 [0] = 1 [zeros] = 0 [a__U11](X) = 1 [tt] = 1 [a__U21](X) = X [a__U31](X) = 1 [a__U41](X1,X2) = 1 [a__U42](X) = X.X [a__isNatIList](X) = 1 [a__U51](X1,X2) = X1.X2 [a__U52](X) = X [a__isNatList](X) = X [a__U61](X1,X2,X3) = 1 [a__U62](X1,X2) = 1 [a__isNat](X) = X [s](X) = X [a__length](X) = 1 [mark](X) = X [length](X) = 1 [nil] = 1 [U11](X) = 1 [U21](X) = X [U31](X) = 1 [U41](X1,X2) = 1 [U42](X) = X.X [isNatIList](X) = 1 [U51](X1,X2) = X1.X2 [U52](X) = X [isNatList](X) = X [U61](X1,X2,X3) = 1 [U62](X1,X2) = 1 [isNat](X) = X [nF_a__U61](X1,X2,X3) = X2.X3 + X1 [nF_a__length](X) = X [nF_a__U62](X1,X2) = X1.X2 + 1 TIME: 0.208233 -> Proof of termination for LengthOfFiniteLists_nokinds_noand_GM_1_2: -> -> Dependency pairs in cycle: nF_a__U41(tt,V2) -> nF_a__isNatIList(V2) nF_a__isNatIList(cons(V1,V2)) -> nF_a__U41(a__isNat(V1),V2) Termination proved: Cycles verify subterm criterion. -> Proof of termination for LengthOfFiniteLists_nokinds_noand_GM_1_3: -> -> Dependency pairs in cycle: nF_a__isNat(s(V1)) -> nF_a__isNat(V1) nF_a__isNatList(cons(V1,V2)) -> nF_a__isNat(V1) nF_a__isNat(length(V1)) -> nF_a__isNatList(V1) nF_a__U51(tt,V2) -> nF_a__isNatList(V2) nF_a__isNatList(cons(V1,V2)) -> nF_a__U51(a__isNat(V1),V2) Termination proved: Cycles verify subterm criterion. SETTINGS: Base ordering: Polynomial ordering Proof mode: SCCs in DG + base ordering Upper bound for coeffs: 1 Rationals below 1 for all non-replacing args: No Polynomial interpretation: Simple mixed Coeffs in polynomials: No rationals Delta: automatic Termination was proved succesfully.