1.07/1.13 YES 1.07/1.13 1.07/1.13 Problem 1: 1.07/1.13 1.07/1.13 (VAR v_NonEmpty:S IL:S L:S M:S N:S V:S V1:S V2:S X:S X1:S X2:S X3:S X4:S) 1.07/1.13 (RULES 1.07/1.13 a__U11(tt) -> tt 1.07/1.13 a__U11(X:S) -> U11(X:S) 1.07/1.13 a__U21(tt) -> tt 1.07/1.13 a__U21(X:S) -> U21(X:S) 1.07/1.13 a__U31(tt) -> tt 1.07/1.13 a__U31(X:S) -> U31(X:S) 1.07/1.13 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.13 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.13 a__U42(tt) -> tt 1.07/1.13 a__U42(X:S) -> U42(X:S) 1.07/1.13 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.13 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.13 a__U52(tt) -> tt 1.07/1.13 a__U52(X:S) -> U52(X:S) 1.07/1.13 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.13 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.13 a__U62(tt) -> tt 1.07/1.13 a__U62(X:S) -> U62(X:S) 1.07/1.13 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.13 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.13 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.13 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.13 a__U81(tt) -> nil 1.07/1.13 a__U81(X:S) -> U81(X:S) 1.07/1.13 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.13 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.13 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.13 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.13 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.13 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.13 a__isNat(0) -> tt 1.07/1.13 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.13 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.13 a__isNat(X:S) -> isNat(X:S) 1.07/1.13 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.13 a__isNatIList(zeros) -> tt 1.07/1.13 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.13 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.13 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.13 a__isNatList(nil) -> tt 1.07/1.13 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.13 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.13 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.13 a__length(nil) -> 0 1.07/1.13 a__length(X:S) -> length(X:S) 1.07/1.13 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.13 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.13 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.13 a__zeros -> cons(0,zeros) 1.07/1.13 a__zeros -> zeros 1.07/1.13 mark(0) -> 0 1.07/1.13 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.13 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.13 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.13 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.13 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.13 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.13 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.13 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.13 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.13 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.13 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.13 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.13 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.13 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.13 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.13 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.13 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.13 mark(nil) -> nil 1.07/1.13 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.13 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.13 mark(tt) -> tt 1.07/1.13 mark(zeros) -> a__zeros 1.07/1.13 ) 1.07/1.13 (STRATEGY INNERMOST) 1.07/1.13 1.07/1.13 Problem 1: 1.07/1.13 1.07/1.13 Dependency Pairs Processor: 1.07/1.13 -> Pairs: 1.07/1.13 A__U41(tt,V2:S) -> A__U42(a__isNatIList(V2:S)) 1.07/1.13 A__U41(tt,V2:S) -> A__ISNATILIST(V2:S) 1.07/1.13 A__U51(tt,V2:S) -> A__U52(a__isNatList(V2:S)) 1.07/1.13 A__U51(tt,V2:S) -> A__ISNATLIST(V2:S) 1.07/1.13 A__U61(tt,V2:S) -> A__U62(a__isNatIList(V2:S)) 1.07/1.13 A__U61(tt,V2:S) -> A__ISNATILIST(V2:S) 1.07/1.13 A__U71(tt,L:S,N:S) -> A__U72(a__isNat(N:S),L:S) 1.07/1.13 A__U71(tt,L:S,N:S) -> A__ISNAT(N:S) 1.07/1.13 A__U72(tt,L:S) -> A__LENGTH(mark(L:S)) 1.07/1.13 A__U72(tt,L:S) -> MARK(L:S) 1.07/1.13 A__U91(tt,IL:S,M:S,N:S) -> A__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.13 A__U91(tt,IL:S,M:S,N:S) -> A__ISNAT(M:S) 1.07/1.13 A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.13 A__U92(tt,IL:S,M:S,N:S) -> A__ISNAT(N:S) 1.07/1.13 A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) 1.07/1.13 A__ISNAT(length(V1:S)) -> A__U11(a__isNatList(V1:S)) 1.07/1.13 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 1.07/1.13 A__ISNAT(s(V1:S)) -> A__U21(a__isNat(V1:S)) 1.07/1.13 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 1.07/1.13 A__ISNATILIST(cons(V1:S,V2:S)) -> A__U41(a__isNat(V1:S),V2:S) 1.07/1.13 A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 1.07/1.13 A__ISNATILIST(V:S) -> A__U31(a__isNatList(V:S)) 1.07/1.13 A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) 1.07/1.13 A__ISNATLIST(cons(V1:S,V2:S)) -> A__U51(a__isNat(V1:S),V2:S) 1.07/1.13 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 1.07/1.13 A__ISNATLIST(take(V1:S,V2:S)) -> A__U61(a__isNat(V1:S),V2:S) 1.07/1.13 A__ISNATLIST(take(V1:S,V2:S)) -> A__ISNAT(V1:S) 1.07/1.13 A__LENGTH(cons(N:S,L:S)) -> A__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.13 A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) 1.07/1.13 A__TAKE(0,IL:S) -> A__U81(a__isNatIList(IL:S)) 1.07/1.13 A__TAKE(0,IL:S) -> A__ISNATILIST(IL:S) 1.07/1.13 A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.13 A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__ISNATILIST(IL:S) 1.07/1.13 MARK(U11(X:S)) -> A__U11(mark(X:S)) 1.07/1.13 MARK(U11(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U21(X:S)) -> A__U21(mark(X:S)) 1.07/1.13 MARK(U21(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U31(X:S)) -> A__U31(mark(X:S)) 1.07/1.13 MARK(U31(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U41(X1:S,X2:S)) -> A__U41(mark(X1:S),X2:S) 1.07/1.13 MARK(U41(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(U42(X:S)) -> A__U42(mark(X:S)) 1.07/1.13 MARK(U42(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U51(X1:S,X2:S)) -> A__U51(mark(X1:S),X2:S) 1.07/1.13 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(U52(X:S)) -> A__U52(mark(X:S)) 1.07/1.13 MARK(U52(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U61(X1:S,X2:S)) -> A__U61(mark(X1:S),X2:S) 1.07/1.13 MARK(U61(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(U62(X:S)) -> A__U62(mark(X:S)) 1.07/1.13 MARK(U62(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S) 1.07/1.13 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.07/1.13 MARK(U72(X1:S,X2:S)) -> A__U72(mark(X1:S),X2:S) 1.07/1.13 MARK(U72(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(U81(X:S)) -> A__U81(mark(X:S)) 1.07/1.13 MARK(U81(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> A__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.13 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.13 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.13 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(isNat(X:S)) -> A__ISNAT(X:S) 1.07/1.13 MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) 1.07/1.13 MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) 1.07/1.13 MARK(length(X:S)) -> A__LENGTH(mark(X:S)) 1.07/1.13 MARK(length(X:S)) -> MARK(X:S) 1.07/1.13 MARK(s(X:S)) -> MARK(X:S) 1.07/1.13 MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S)) 1.07/1.13 MARK(take(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(take(X1:S,X2:S)) -> MARK(X2:S) 1.07/1.13 MARK(zeros) -> A__ZEROS 1.07/1.13 -> Rules: 1.07/1.13 a__U11(tt) -> tt 1.07/1.13 a__U11(X:S) -> U11(X:S) 1.07/1.13 a__U21(tt) -> tt 1.07/1.13 a__U21(X:S) -> U21(X:S) 1.07/1.13 a__U31(tt) -> tt 1.07/1.13 a__U31(X:S) -> U31(X:S) 1.07/1.13 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.13 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.13 a__U42(tt) -> tt 1.07/1.13 a__U42(X:S) -> U42(X:S) 1.07/1.13 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.13 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.13 a__U52(tt) -> tt 1.07/1.13 a__U52(X:S) -> U52(X:S) 1.07/1.13 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.13 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.13 a__U62(tt) -> tt 1.07/1.13 a__U62(X:S) -> U62(X:S) 1.07/1.13 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.13 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.13 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.13 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.13 a__U81(tt) -> nil 1.07/1.13 a__U81(X:S) -> U81(X:S) 1.07/1.13 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.13 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.13 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.13 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.13 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.13 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.13 a__isNat(0) -> tt 1.07/1.13 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.13 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.13 a__isNat(X:S) -> isNat(X:S) 1.07/1.13 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.13 a__isNatIList(zeros) -> tt 1.07/1.13 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.13 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.13 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.13 a__isNatList(nil) -> tt 1.07/1.13 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.13 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.13 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.13 a__length(nil) -> 0 1.07/1.13 a__length(X:S) -> length(X:S) 1.07/1.13 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.13 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.13 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.13 a__zeros -> cons(0,zeros) 1.07/1.13 a__zeros -> zeros 1.07/1.13 mark(0) -> 0 1.07/1.13 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.13 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.13 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.13 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.13 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.13 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.13 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.13 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.13 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.13 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.13 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.13 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.13 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.13 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.13 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.13 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.13 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.13 mark(nil) -> nil 1.07/1.13 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.13 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.13 mark(tt) -> tt 1.07/1.13 mark(zeros) -> a__zeros 1.07/1.13 1.07/1.13 Problem 1: 1.07/1.13 1.07/1.13 SCC Processor: 1.07/1.13 -> Pairs: 1.07/1.13 A__U41(tt,V2:S) -> A__U42(a__isNatIList(V2:S)) 1.07/1.13 A__U41(tt,V2:S) -> A__ISNATILIST(V2:S) 1.07/1.13 A__U51(tt,V2:S) -> A__U52(a__isNatList(V2:S)) 1.07/1.13 A__U51(tt,V2:S) -> A__ISNATLIST(V2:S) 1.07/1.13 A__U61(tt,V2:S) -> A__U62(a__isNatIList(V2:S)) 1.07/1.13 A__U61(tt,V2:S) -> A__ISNATILIST(V2:S) 1.07/1.13 A__U71(tt,L:S,N:S) -> A__U72(a__isNat(N:S),L:S) 1.07/1.13 A__U71(tt,L:S,N:S) -> A__ISNAT(N:S) 1.07/1.13 A__U72(tt,L:S) -> A__LENGTH(mark(L:S)) 1.07/1.13 A__U72(tt,L:S) -> MARK(L:S) 1.07/1.13 A__U91(tt,IL:S,M:S,N:S) -> A__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.13 A__U91(tt,IL:S,M:S,N:S) -> A__ISNAT(M:S) 1.07/1.13 A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.13 A__U92(tt,IL:S,M:S,N:S) -> A__ISNAT(N:S) 1.07/1.13 A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) 1.07/1.13 A__ISNAT(length(V1:S)) -> A__U11(a__isNatList(V1:S)) 1.07/1.13 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 1.07/1.13 A__ISNAT(s(V1:S)) -> A__U21(a__isNat(V1:S)) 1.07/1.13 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 1.07/1.13 A__ISNATILIST(cons(V1:S,V2:S)) -> A__U41(a__isNat(V1:S),V2:S) 1.07/1.13 A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 1.07/1.13 A__ISNATILIST(V:S) -> A__U31(a__isNatList(V:S)) 1.07/1.13 A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) 1.07/1.13 A__ISNATLIST(cons(V1:S,V2:S)) -> A__U51(a__isNat(V1:S),V2:S) 1.07/1.13 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 1.07/1.13 A__ISNATLIST(take(V1:S,V2:S)) -> A__U61(a__isNat(V1:S),V2:S) 1.07/1.13 A__ISNATLIST(take(V1:S,V2:S)) -> A__ISNAT(V1:S) 1.07/1.13 A__LENGTH(cons(N:S,L:S)) -> A__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.13 A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) 1.07/1.13 A__TAKE(0,IL:S) -> A__U81(a__isNatIList(IL:S)) 1.07/1.13 A__TAKE(0,IL:S) -> A__ISNATILIST(IL:S) 1.07/1.13 A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.13 A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__ISNATILIST(IL:S) 1.07/1.13 MARK(U11(X:S)) -> A__U11(mark(X:S)) 1.07/1.13 MARK(U11(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U21(X:S)) -> A__U21(mark(X:S)) 1.07/1.13 MARK(U21(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U31(X:S)) -> A__U31(mark(X:S)) 1.07/1.13 MARK(U31(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U41(X1:S,X2:S)) -> A__U41(mark(X1:S),X2:S) 1.07/1.13 MARK(U41(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(U42(X:S)) -> A__U42(mark(X:S)) 1.07/1.13 MARK(U42(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U51(X1:S,X2:S)) -> A__U51(mark(X1:S),X2:S) 1.07/1.13 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(U52(X:S)) -> A__U52(mark(X:S)) 1.07/1.13 MARK(U52(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U61(X1:S,X2:S)) -> A__U61(mark(X1:S),X2:S) 1.07/1.13 MARK(U61(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(U62(X:S)) -> A__U62(mark(X:S)) 1.07/1.13 MARK(U62(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S) 1.07/1.13 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.07/1.13 MARK(U72(X1:S,X2:S)) -> A__U72(mark(X1:S),X2:S) 1.07/1.13 MARK(U72(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(U81(X:S)) -> A__U81(mark(X:S)) 1.07/1.13 MARK(U81(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> A__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.13 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.13 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.13 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(isNat(X:S)) -> A__ISNAT(X:S) 1.07/1.13 MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) 1.07/1.13 MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) 1.07/1.13 MARK(length(X:S)) -> A__LENGTH(mark(X:S)) 1.07/1.13 MARK(length(X:S)) -> MARK(X:S) 1.07/1.13 MARK(s(X:S)) -> MARK(X:S) 1.07/1.13 MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S)) 1.07/1.13 MARK(take(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(take(X1:S,X2:S)) -> MARK(X2:S) 1.07/1.13 MARK(zeros) -> A__ZEROS 1.07/1.13 -> Rules: 1.07/1.13 a__U11(tt) -> tt 1.07/1.13 a__U11(X:S) -> U11(X:S) 1.07/1.13 a__U21(tt) -> tt 1.07/1.13 a__U21(X:S) -> U21(X:S) 1.07/1.13 a__U31(tt) -> tt 1.07/1.13 a__U31(X:S) -> U31(X:S) 1.07/1.13 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.13 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.13 a__U42(tt) -> tt 1.07/1.13 a__U42(X:S) -> U42(X:S) 1.07/1.13 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.13 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.13 a__U52(tt) -> tt 1.07/1.13 a__U52(X:S) -> U52(X:S) 1.07/1.13 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.13 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.13 a__U62(tt) -> tt 1.07/1.13 a__U62(X:S) -> U62(X:S) 1.07/1.13 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.13 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.13 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.13 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.13 a__U81(tt) -> nil 1.07/1.13 a__U81(X:S) -> U81(X:S) 1.07/1.13 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.13 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.13 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.13 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.13 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.13 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.13 a__isNat(0) -> tt 1.07/1.13 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.13 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.13 a__isNat(X:S) -> isNat(X:S) 1.07/1.13 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.13 a__isNatIList(zeros) -> tt 1.07/1.13 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.13 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.13 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.13 a__isNatList(nil) -> tt 1.07/1.13 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.13 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.13 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.13 a__length(nil) -> 0 1.07/1.13 a__length(X:S) -> length(X:S) 1.07/1.13 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.13 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.13 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.13 a__zeros -> cons(0,zeros) 1.07/1.13 a__zeros -> zeros 1.07/1.13 mark(0) -> 0 1.07/1.13 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.13 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.13 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.13 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.13 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.13 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.13 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.13 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.13 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.13 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.13 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.13 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.13 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.13 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.13 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.13 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.13 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.13 mark(nil) -> nil 1.07/1.13 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.13 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.13 mark(tt) -> tt 1.07/1.13 mark(zeros) -> a__zeros 1.07/1.13 ->Strongly Connected Components: 1.07/1.13 ->->Cycle: 1.07/1.13 ->->-> Pairs: 1.07/1.13 A__U41(tt,V2:S) -> A__ISNATILIST(V2:S) 1.07/1.13 A__U51(tt,V2:S) -> A__ISNATLIST(V2:S) 1.07/1.13 A__U61(tt,V2:S) -> A__ISNATILIST(V2:S) 1.07/1.13 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 1.07/1.13 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 1.07/1.13 A__ISNATILIST(cons(V1:S,V2:S)) -> A__U41(a__isNat(V1:S),V2:S) 1.07/1.13 A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 1.07/1.13 A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) 1.07/1.13 A__ISNATLIST(cons(V1:S,V2:S)) -> A__U51(a__isNat(V1:S),V2:S) 1.07/1.13 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 1.07/1.13 A__ISNATLIST(take(V1:S,V2:S)) -> A__U61(a__isNat(V1:S),V2:S) 1.07/1.13 A__ISNATLIST(take(V1:S,V2:S)) -> A__ISNAT(V1:S) 1.07/1.13 ->->-> Rules: 1.07/1.13 a__U11(tt) -> tt 1.07/1.13 a__U11(X:S) -> U11(X:S) 1.07/1.13 a__U21(tt) -> tt 1.07/1.13 a__U21(X:S) -> U21(X:S) 1.07/1.13 a__U31(tt) -> tt 1.07/1.13 a__U31(X:S) -> U31(X:S) 1.07/1.13 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.13 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.13 a__U42(tt) -> tt 1.07/1.13 a__U42(X:S) -> U42(X:S) 1.07/1.13 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.13 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.13 a__U52(tt) -> tt 1.07/1.13 a__U52(X:S) -> U52(X:S) 1.07/1.13 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.13 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.13 a__U62(tt) -> tt 1.07/1.13 a__U62(X:S) -> U62(X:S) 1.07/1.13 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.13 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.13 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.13 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.13 a__U81(tt) -> nil 1.07/1.13 a__U81(X:S) -> U81(X:S) 1.07/1.13 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.13 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.13 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.13 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.13 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.13 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.13 a__isNat(0) -> tt 1.07/1.13 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.13 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.13 a__isNat(X:S) -> isNat(X:S) 1.07/1.13 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.13 a__isNatIList(zeros) -> tt 1.07/1.13 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.13 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.13 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.13 a__isNatList(nil) -> tt 1.07/1.13 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.13 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.13 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.13 a__length(nil) -> 0 1.07/1.13 a__length(X:S) -> length(X:S) 1.07/1.13 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.13 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.13 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.13 a__zeros -> cons(0,zeros) 1.07/1.13 a__zeros -> zeros 1.07/1.13 mark(0) -> 0 1.07/1.13 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.13 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.13 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.13 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.13 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.13 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.13 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.13 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.13 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.13 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.13 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.13 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.13 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.13 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.13 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.13 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.13 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.13 mark(nil) -> nil 1.07/1.13 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.13 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.13 mark(tt) -> tt 1.07/1.13 mark(zeros) -> a__zeros 1.07/1.13 ->->Cycle: 1.07/1.13 ->->-> Pairs: 1.07/1.13 A__U71(tt,L:S,N:S) -> A__U72(a__isNat(N:S),L:S) 1.07/1.13 A__U72(tt,L:S) -> A__LENGTH(mark(L:S)) 1.07/1.13 A__U72(tt,L:S) -> MARK(L:S) 1.07/1.13 A__U91(tt,IL:S,M:S,N:S) -> A__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.13 A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.13 A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) 1.07/1.13 A__LENGTH(cons(N:S,L:S)) -> A__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.13 A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.13 MARK(U11(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U21(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U31(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U41(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(U42(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(U52(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U61(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(U62(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S) 1.07/1.13 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.07/1.13 MARK(U72(X1:S,X2:S)) -> A__U72(mark(X1:S),X2:S) 1.07/1.13 MARK(U72(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(U81(X:S)) -> MARK(X:S) 1.07/1.13 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> A__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.13 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.13 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.13 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(length(X:S)) -> A__LENGTH(mark(X:S)) 1.07/1.13 MARK(length(X:S)) -> MARK(X:S) 1.07/1.13 MARK(s(X:S)) -> MARK(X:S) 1.07/1.13 MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S)) 1.07/1.13 MARK(take(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.13 MARK(take(X1:S,X2:S)) -> MARK(X2:S) 1.07/1.13 ->->-> Rules: 1.07/1.13 a__U11(tt) -> tt 1.07/1.13 a__U11(X:S) -> U11(X:S) 1.07/1.13 a__U21(tt) -> tt 1.07/1.13 a__U21(X:S) -> U21(X:S) 1.07/1.13 a__U31(tt) -> tt 1.07/1.13 a__U31(X:S) -> U31(X:S) 1.07/1.13 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.13 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.13 a__U42(tt) -> tt 1.07/1.13 a__U42(X:S) -> U42(X:S) 1.07/1.13 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.13 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.13 a__U52(tt) -> tt 1.07/1.13 a__U52(X:S) -> U52(X:S) 1.07/1.13 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.13 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.13 a__U62(tt) -> tt 1.07/1.13 a__U62(X:S) -> U62(X:S) 1.07/1.13 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.13 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.13 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.13 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.13 a__U81(tt) -> nil 1.07/1.13 a__U81(X:S) -> U81(X:S) 1.07/1.13 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.13 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.13 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.13 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.13 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.13 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.13 a__isNat(0) -> tt 1.07/1.13 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.13 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.13 a__isNat(X:S) -> isNat(X:S) 1.07/1.13 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.13 a__isNatIList(zeros) -> tt 1.07/1.13 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.13 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.13 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.13 a__isNatList(nil) -> tt 1.07/1.13 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.13 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.13 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.13 a__length(nil) -> 0 1.07/1.13 a__length(X:S) -> length(X:S) 1.07/1.13 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.13 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.13 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.13 a__zeros -> cons(0,zeros) 1.07/1.13 a__zeros -> zeros 1.07/1.13 mark(0) -> 0 1.07/1.13 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.13 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.13 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.13 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.13 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.13 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.13 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.13 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.13 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.13 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.13 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.13 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.13 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.13 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.13 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.13 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.13 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.13 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.13 mark(nil) -> nil 1.07/1.13 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.13 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.13 mark(tt) -> tt 1.07/1.13 mark(zeros) -> a__zeros 1.07/1.13 1.07/1.13 1.07/1.13 The problem is decomposed in 2 subproblems. 1.07/1.13 1.07/1.13 Problem 1.1: 1.07/1.13 1.07/1.13 Subterm Processor: 1.07/1.13 -> Pairs: 1.07/1.13 A__U41(tt,V2:S) -> A__ISNATILIST(V2:S) 1.07/1.13 A__U51(tt,V2:S) -> A__ISNATLIST(V2:S) 1.07/1.13 A__U61(tt,V2:S) -> A__ISNATILIST(V2:S) 1.07/1.13 A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) 1.07/1.13 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 1.07/1.13 A__ISNATILIST(cons(V1:S,V2:S)) -> A__U41(a__isNat(V1:S),V2:S) 1.07/1.13 A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 1.07/1.13 A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) 1.07/1.13 A__ISNATLIST(cons(V1:S,V2:S)) -> A__U51(a__isNat(V1:S),V2:S) 1.07/1.13 A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) 1.07/1.13 A__ISNATLIST(take(V1:S,V2:S)) -> A__U61(a__isNat(V1:S),V2:S) 1.07/1.13 A__ISNATLIST(take(V1:S,V2:S)) -> A__ISNAT(V1:S) 1.07/1.13 -> Rules: 1.07/1.13 a__U11(tt) -> tt 1.07/1.13 a__U11(X:S) -> U11(X:S) 1.07/1.13 a__U21(tt) -> tt 1.07/1.13 a__U21(X:S) -> U21(X:S) 1.07/1.13 a__U31(tt) -> tt 1.07/1.13 a__U31(X:S) -> U31(X:S) 1.07/1.13 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.13 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.13 a__U42(tt) -> tt 1.07/1.13 a__U42(X:S) -> U42(X:S) 1.07/1.13 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.13 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.13 a__U52(tt) -> tt 1.07/1.13 a__U52(X:S) -> U52(X:S) 1.07/1.13 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.13 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.13 a__U62(tt) -> tt 1.07/1.13 a__U62(X:S) -> U62(X:S) 1.07/1.13 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.13 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.13 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.13 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.13 a__U81(tt) -> nil 1.07/1.13 a__U81(X:S) -> U81(X:S) 1.07/1.13 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.13 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.13 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.14 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__isNat(0) -> tt 1.07/1.14 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.14 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.14 a__isNat(X:S) -> isNat(X:S) 1.07/1.14 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatIList(zeros) -> tt 1.07/1.14 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.14 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.14 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(nil) -> tt 1.07/1.14 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.14 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 a__length(nil) -> 0 1.07/1.14 a__length(X:S) -> length(X:S) 1.07/1.14 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.14 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.14 a__zeros -> cons(0,zeros) 1.07/1.14 a__zeros -> zeros 1.07/1.14 mark(0) -> 0 1.07/1.14 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.14 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.14 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.14 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.14 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.14 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.14 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.14 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.14 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.14 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.14 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.14 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.14 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.14 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.14 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.14 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.14 mark(nil) -> nil 1.07/1.14 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.14 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.14 mark(tt) -> tt 1.07/1.14 mark(zeros) -> a__zeros 1.07/1.14 ->Projection: 1.07/1.14 pi(A__U41) = 2 1.07/1.14 pi(A__U51) = 2 1.07/1.14 pi(A__U61) = 2 1.07/1.14 pi(A__ISNAT) = 1 1.07/1.14 pi(A__ISNATILIST) = 1 1.07/1.14 pi(A__ISNATLIST) = 1 1.07/1.14 1.07/1.14 Problem 1.1: 1.07/1.14 1.07/1.14 SCC Processor: 1.07/1.14 -> Pairs: 1.07/1.14 A__U41(tt,V2:S) -> A__ISNATILIST(V2:S) 1.07/1.14 A__U51(tt,V2:S) -> A__ISNATLIST(V2:S) 1.07/1.14 A__U61(tt,V2:S) -> A__ISNATILIST(V2:S) 1.07/1.14 A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) 1.07/1.14 -> Rules: 1.07/1.14 a__U11(tt) -> tt 1.07/1.14 a__U11(X:S) -> U11(X:S) 1.07/1.14 a__U21(tt) -> tt 1.07/1.14 a__U21(X:S) -> U21(X:S) 1.07/1.14 a__U31(tt) -> tt 1.07/1.14 a__U31(X:S) -> U31(X:S) 1.07/1.14 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.14 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.14 a__U42(tt) -> tt 1.07/1.14 a__U42(X:S) -> U42(X:S) 1.07/1.14 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.14 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.14 a__U52(tt) -> tt 1.07/1.14 a__U52(X:S) -> U52(X:S) 1.07/1.14 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.14 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.14 a__U62(tt) -> tt 1.07/1.14 a__U62(X:S) -> U62(X:S) 1.07/1.14 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.14 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.14 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.14 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.14 a__U81(tt) -> nil 1.07/1.14 a__U81(X:S) -> U81(X:S) 1.07/1.14 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.14 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__isNat(0) -> tt 1.07/1.14 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.14 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.14 a__isNat(X:S) -> isNat(X:S) 1.07/1.14 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatIList(zeros) -> tt 1.07/1.14 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.14 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.14 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(nil) -> tt 1.07/1.14 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.14 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 a__length(nil) -> 0 1.07/1.14 a__length(X:S) -> length(X:S) 1.07/1.14 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.14 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.14 a__zeros -> cons(0,zeros) 1.07/1.14 a__zeros -> zeros 1.07/1.14 mark(0) -> 0 1.07/1.14 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.14 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.14 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.14 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.14 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.14 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.14 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.14 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.14 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.14 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.14 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.14 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.14 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.14 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.14 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.14 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.14 mark(nil) -> nil 1.07/1.14 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.14 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.14 mark(tt) -> tt 1.07/1.14 mark(zeros) -> a__zeros 1.07/1.14 ->Strongly Connected Components: 1.07/1.14 There is no strongly connected component 1.07/1.14 1.07/1.14 The problem is finite. 1.07/1.14 1.07/1.14 Problem 1.2: 1.07/1.14 1.07/1.14 Reduction Pairs Processor: 1.07/1.14 -> Pairs: 1.07/1.14 A__U71(tt,L:S,N:S) -> A__U72(a__isNat(N:S),L:S) 1.07/1.14 A__U72(tt,L:S) -> A__LENGTH(mark(L:S)) 1.07/1.14 A__U72(tt,L:S) -> MARK(L:S) 1.07/1.14 A__U91(tt,IL:S,M:S,N:S) -> A__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) 1.07/1.14 A__LENGTH(cons(N:S,L:S)) -> A__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 MARK(U11(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U21(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U31(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U41(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U42(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U52(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U61(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U62(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.07/1.14 MARK(U72(X1:S,X2:S)) -> A__U72(mark(X1:S),X2:S) 1.07/1.14 MARK(U72(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U81(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> A__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(length(X:S)) -> A__LENGTH(mark(X:S)) 1.07/1.14 MARK(length(X:S)) -> MARK(X:S) 1.07/1.14 MARK(s(X:S)) -> MARK(X:S) 1.07/1.14 MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S)) 1.07/1.14 MARK(take(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(take(X1:S,X2:S)) -> MARK(X2:S) 1.07/1.14 -> Rules: 1.07/1.14 a__U11(tt) -> tt 1.07/1.14 a__U11(X:S) -> U11(X:S) 1.07/1.14 a__U21(tt) -> tt 1.07/1.14 a__U21(X:S) -> U21(X:S) 1.07/1.14 a__U31(tt) -> tt 1.07/1.14 a__U31(X:S) -> U31(X:S) 1.07/1.14 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.14 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.14 a__U42(tt) -> tt 1.07/1.14 a__U42(X:S) -> U42(X:S) 1.07/1.14 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.14 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.14 a__U52(tt) -> tt 1.07/1.14 a__U52(X:S) -> U52(X:S) 1.07/1.14 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.14 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.14 a__U62(tt) -> tt 1.07/1.14 a__U62(X:S) -> U62(X:S) 1.07/1.14 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.14 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.14 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.14 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.14 a__U81(tt) -> nil 1.07/1.14 a__U81(X:S) -> U81(X:S) 1.07/1.14 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.14 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__isNat(0) -> tt 1.07/1.14 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.14 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.14 a__isNat(X:S) -> isNat(X:S) 1.07/1.14 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatIList(zeros) -> tt 1.07/1.14 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.14 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.14 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(nil) -> tt 1.07/1.14 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.14 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 a__length(nil) -> 0 1.07/1.14 a__length(X:S) -> length(X:S) 1.07/1.14 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.14 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.14 a__zeros -> cons(0,zeros) 1.07/1.14 a__zeros -> zeros 1.07/1.14 mark(0) -> 0 1.07/1.14 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.14 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.14 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.14 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.14 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.14 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.14 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.14 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.14 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.14 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.14 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.14 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.14 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.14 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.14 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.14 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.14 mark(nil) -> nil 1.07/1.14 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.14 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.14 mark(tt) -> tt 1.07/1.14 mark(zeros) -> a__zeros 1.07/1.14 -> Usable rules: 1.07/1.14 a__U11(tt) -> tt 1.07/1.14 a__U11(X:S) -> U11(X:S) 1.07/1.14 a__U21(tt) -> tt 1.07/1.14 a__U21(X:S) -> U21(X:S) 1.07/1.14 a__U31(tt) -> tt 1.07/1.14 a__U31(X:S) -> U31(X:S) 1.07/1.14 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.14 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.14 a__U42(tt) -> tt 1.07/1.14 a__U42(X:S) -> U42(X:S) 1.07/1.14 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.14 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.14 a__U52(tt) -> tt 1.07/1.14 a__U52(X:S) -> U52(X:S) 1.07/1.14 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.14 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.14 a__U62(tt) -> tt 1.07/1.14 a__U62(X:S) -> U62(X:S) 1.07/1.14 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.14 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.14 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.14 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.14 a__U81(tt) -> nil 1.07/1.14 a__U81(X:S) -> U81(X:S) 1.07/1.14 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.14 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__isNat(0) -> tt 1.07/1.14 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.14 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.14 a__isNat(X:S) -> isNat(X:S) 1.07/1.14 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatIList(zeros) -> tt 1.07/1.14 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.14 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.14 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(nil) -> tt 1.07/1.14 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.14 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 a__length(nil) -> 0 1.07/1.14 a__length(X:S) -> length(X:S) 1.07/1.14 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.14 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.14 a__zeros -> cons(0,zeros) 1.07/1.14 a__zeros -> zeros 1.07/1.14 mark(0) -> 0 1.07/1.14 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.14 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.14 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.14 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.14 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.14 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.14 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.14 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.14 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.14 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.14 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.14 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.14 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.14 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.14 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.14 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.14 mark(nil) -> nil 1.07/1.14 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.14 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.14 mark(tt) -> tt 1.07/1.14 mark(zeros) -> a__zeros 1.07/1.14 ->Interpretation type: 1.07/1.14 Linear 1.07/1.14 ->Coefficients: 1.07/1.14 Natural Numbers 1.07/1.14 ->Dimension: 1.07/1.14 1 1.07/1.14 ->Bound: 1.07/1.14 2 1.07/1.14 ->Interpretation: 1.07/1.14 1.07/1.14 [a__U11](X) = X 1.07/1.14 [a__U21](X) = X 1.07/1.14 [a__U31](X) = X 1.07/1.14 [a__U41](X1,X2) = X1 1.07/1.14 [a__U42](X) = X 1.07/1.14 [a__U51](X1,X2) = X1 1.07/1.14 [a__U52](X) = X 1.07/1.14 [a__U61](X1,X2) = X1 1.07/1.14 [a__U62](X) = X 1.07/1.14 [a__U71](X1,X2,X3) = X1 + 2.X2 + X3 1.07/1.14 [a__U72](X1,X2) = X1 + 2.X2 1.07/1.14 [a__U81](X) = X 1.07/1.14 [a__U91](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 1.07/1.14 [a__U92](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 1.07/1.14 [a__U93](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 1.07/1.14 [a__isNat](X) = 2 1.07/1.14 [a__isNatIList](X) = 2 1.07/1.14 [a__isNatList](X) = 2 1.07/1.14 [a__length](X) = 2.X + 2 1.07/1.14 [a__take](X1,X2) = 2.X1 + 2.X2 + 2 1.07/1.14 [a__zeros] = 2 1.07/1.14 [mark](X) = X 1.07/1.14 [0] = 0 1.07/1.14 [U11](X) = X 1.07/1.14 [U21](X) = X 1.07/1.14 [U31](X) = X 1.07/1.14 [U41](X1,X2) = X1 1.07/1.14 [U42](X) = X 1.07/1.14 [U51](X1,X2) = X1 1.07/1.14 [U52](X) = X 1.07/1.14 [U61](X1,X2) = X1 1.07/1.14 [U62](X) = X 1.07/1.14 [U71](X1,X2,X3) = X1 + 2.X2 + X3 1.07/1.14 [U72](X1,X2) = X1 + 2.X2 1.07/1.14 [U81](X) = X 1.07/1.14 [U91](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 1.07/1.14 [U92](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 1.07/1.14 [U93](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 1.07/1.14 [cons](X1,X2) = 2.X1 + X2 1.07/1.14 [fSNonEmpty] = 0 1.07/1.14 [isNat](X) = 2 1.07/1.14 [isNatIList](X) = 2 1.07/1.14 [isNatList](X) = 2 1.07/1.14 [length](X) = 2.X + 2 1.07/1.14 [nil] = 2 1.07/1.14 [s](X) = X 1.07/1.14 [take](X1,X2) = 2.X1 + 2.X2 + 2 1.07/1.14 [tt] = 2 1.07/1.14 [zeros] = 2 1.07/1.14 [A__U11](X) = 0 1.07/1.14 [A__U21](X) = 0 1.07/1.14 [A__U31](X) = 0 1.07/1.14 [A__U41](X1,X2) = 0 1.07/1.14 [A__U42](X) = 0 1.07/1.14 [A__U51](X1,X2) = 0 1.07/1.14 [A__U52](X) = 0 1.07/1.14 [A__U61](X1,X2) = 0 1.07/1.14 [A__U62](X) = 0 1.07/1.14 [A__U71](X1,X2,X3) = X1 + 2.X2 + 2.X3 1.07/1.14 [A__U72](X1,X2) = X1 + 2.X2 1.07/1.14 [A__U81](X) = 0 1.07/1.14 [A__U91](X1,X2,X3,X4) = 2.X2 + 2.X4 1.07/1.14 [A__U92](X1,X2,X3,X4) = 2.X2 + 2.X4 1.07/1.14 [A__U93](X1,X2,X3,X4) = 2.X2 + 2.X4 1.07/1.14 [A__ISNAT](X) = 0 1.07/1.14 [A__ISNATILIST](X) = 0 1.07/1.14 [A__ISNATLIST](X) = 0 1.07/1.14 [A__LENGTH](X) = 2.X + 2 1.07/1.14 [A__TAKE](X1,X2) = 2.X2 1.07/1.14 [A__ZEROS] = 0 1.07/1.14 [MARK](X) = 2.X 1.07/1.14 1.07/1.14 Problem 1.2: 1.07/1.14 1.07/1.14 SCC Processor: 1.07/1.14 -> Pairs: 1.07/1.14 A__U71(tt,L:S,N:S) -> A__U72(a__isNat(N:S),L:S) 1.07/1.14 A__U72(tt,L:S) -> A__LENGTH(mark(L:S)) 1.07/1.14 A__U91(tt,IL:S,M:S,N:S) -> A__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) 1.07/1.14 A__LENGTH(cons(N:S,L:S)) -> A__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 MARK(U11(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U21(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U31(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U41(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U42(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U52(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U61(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U62(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.07/1.14 MARK(U72(X1:S,X2:S)) -> A__U72(mark(X1:S),X2:S) 1.07/1.14 MARK(U72(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U81(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> A__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(length(X:S)) -> A__LENGTH(mark(X:S)) 1.07/1.14 MARK(length(X:S)) -> MARK(X:S) 1.07/1.14 MARK(s(X:S)) -> MARK(X:S) 1.07/1.14 MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S)) 1.07/1.14 MARK(take(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(take(X1:S,X2:S)) -> MARK(X2:S) 1.07/1.14 -> Rules: 1.07/1.14 a__U11(tt) -> tt 1.07/1.14 a__U11(X:S) -> U11(X:S) 1.07/1.14 a__U21(tt) -> tt 1.07/1.14 a__U21(X:S) -> U21(X:S) 1.07/1.14 a__U31(tt) -> tt 1.07/1.14 a__U31(X:S) -> U31(X:S) 1.07/1.14 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.14 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.14 a__U42(tt) -> tt 1.07/1.14 a__U42(X:S) -> U42(X:S) 1.07/1.14 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.14 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.14 a__U52(tt) -> tt 1.07/1.14 a__U52(X:S) -> U52(X:S) 1.07/1.14 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.14 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.14 a__U62(tt) -> tt 1.07/1.14 a__U62(X:S) -> U62(X:S) 1.07/1.14 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.14 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.14 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.14 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.14 a__U81(tt) -> nil 1.07/1.14 a__U81(X:S) -> U81(X:S) 1.07/1.14 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.14 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__isNat(0) -> tt 1.07/1.14 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.14 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.14 a__isNat(X:S) -> isNat(X:S) 1.07/1.14 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatIList(zeros) -> tt 1.07/1.14 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.14 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.14 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(nil) -> tt 1.07/1.14 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.14 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 a__length(nil) -> 0 1.07/1.14 a__length(X:S) -> length(X:S) 1.07/1.14 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.14 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.14 a__zeros -> cons(0,zeros) 1.07/1.14 a__zeros -> zeros 1.07/1.14 mark(0) -> 0 1.07/1.14 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.14 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.14 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.14 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.14 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.14 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.14 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.14 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.14 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.14 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.14 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.14 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.14 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.14 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.14 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.14 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.14 mark(nil) -> nil 1.07/1.14 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.14 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.14 mark(tt) -> tt 1.07/1.14 mark(zeros) -> a__zeros 1.07/1.14 ->Strongly Connected Components: 1.07/1.14 ->->Cycle: 1.07/1.14 ->->-> Pairs: 1.07/1.14 A__U71(tt,L:S,N:S) -> A__U72(a__isNat(N:S),L:S) 1.07/1.14 A__U72(tt,L:S) -> A__LENGTH(mark(L:S)) 1.07/1.14 A__LENGTH(cons(N:S,L:S)) -> A__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 ->->-> Rules: 1.07/1.14 a__U11(tt) -> tt 1.07/1.14 a__U11(X:S) -> U11(X:S) 1.07/1.14 a__U21(tt) -> tt 1.07/1.14 a__U21(X:S) -> U21(X:S) 1.07/1.14 a__U31(tt) -> tt 1.07/1.14 a__U31(X:S) -> U31(X:S) 1.07/1.14 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.14 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.14 a__U42(tt) -> tt 1.07/1.14 a__U42(X:S) -> U42(X:S) 1.07/1.14 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.14 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.14 a__U52(tt) -> tt 1.07/1.14 a__U52(X:S) -> U52(X:S) 1.07/1.14 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.14 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.14 a__U62(tt) -> tt 1.07/1.14 a__U62(X:S) -> U62(X:S) 1.07/1.14 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.14 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.14 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.14 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.14 a__U81(tt) -> nil 1.07/1.14 a__U81(X:S) -> U81(X:S) 1.07/1.14 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.14 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__isNat(0) -> tt 1.07/1.14 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.14 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.14 a__isNat(X:S) -> isNat(X:S) 1.07/1.14 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatIList(zeros) -> tt 1.07/1.14 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.14 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.14 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(nil) -> tt 1.07/1.14 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.14 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 a__length(nil) -> 0 1.07/1.14 a__length(X:S) -> length(X:S) 1.07/1.14 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.14 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.14 a__zeros -> cons(0,zeros) 1.07/1.14 a__zeros -> zeros 1.07/1.14 mark(0) -> 0 1.07/1.14 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.14 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.14 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.14 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.14 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.14 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.14 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.14 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.14 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.14 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.14 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.14 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.14 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.14 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.14 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.14 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.14 mark(nil) -> nil 1.07/1.14 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.14 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.14 mark(tt) -> tt 1.07/1.14 mark(zeros) -> a__zeros 1.07/1.14 ->->Cycle: 1.07/1.14 ->->-> Pairs: 1.07/1.14 A__U91(tt,IL:S,M:S,N:S) -> A__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) 1.07/1.14 A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 MARK(U11(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U21(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U31(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U41(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U42(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U52(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U61(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U62(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.07/1.14 MARK(U72(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U81(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> A__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(length(X:S)) -> MARK(X:S) 1.07/1.14 MARK(s(X:S)) -> MARK(X:S) 1.07/1.14 MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S)) 1.07/1.14 MARK(take(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(take(X1:S,X2:S)) -> MARK(X2:S) 1.07/1.14 ->->-> Rules: 1.07/1.14 a__U11(tt) -> tt 1.07/1.14 a__U11(X:S) -> U11(X:S) 1.07/1.14 a__U21(tt) -> tt 1.07/1.14 a__U21(X:S) -> U21(X:S) 1.07/1.14 a__U31(tt) -> tt 1.07/1.14 a__U31(X:S) -> U31(X:S) 1.07/1.14 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.14 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.14 a__U42(tt) -> tt 1.07/1.14 a__U42(X:S) -> U42(X:S) 1.07/1.14 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.14 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.14 a__U52(tt) -> tt 1.07/1.14 a__U52(X:S) -> U52(X:S) 1.07/1.14 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.14 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.14 a__U62(tt) -> tt 1.07/1.14 a__U62(X:S) -> U62(X:S) 1.07/1.14 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.14 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.14 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.14 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.14 a__U81(tt) -> nil 1.07/1.14 a__U81(X:S) -> U81(X:S) 1.07/1.14 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.14 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__isNat(0) -> tt 1.07/1.14 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.14 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.14 a__isNat(X:S) -> isNat(X:S) 1.07/1.14 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatIList(zeros) -> tt 1.07/1.14 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.14 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.14 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(nil) -> tt 1.07/1.14 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.14 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 a__length(nil) -> 0 1.07/1.14 a__length(X:S) -> length(X:S) 1.07/1.14 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.14 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.14 a__zeros -> cons(0,zeros) 1.07/1.14 a__zeros -> zeros 1.07/1.14 mark(0) -> 0 1.07/1.14 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.14 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.14 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.14 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.14 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.14 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.14 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.14 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.14 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.14 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.14 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.14 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.14 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.14 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.14 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.14 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.14 mark(nil) -> nil 1.07/1.14 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.14 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.14 mark(tt) -> tt 1.07/1.14 mark(zeros) -> a__zeros 1.07/1.14 1.07/1.14 1.07/1.14 The problem is decomposed in 2 subproblems. 1.07/1.14 1.07/1.14 Problem 1.2.1: 1.07/1.14 1.07/1.14 Reduction Pairs Processor: 1.07/1.14 -> Pairs: 1.07/1.14 A__U71(tt,L:S,N:S) -> A__U72(a__isNat(N:S),L:S) 1.07/1.14 A__U72(tt,L:S) -> A__LENGTH(mark(L:S)) 1.07/1.14 A__LENGTH(cons(N:S,L:S)) -> A__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 -> Rules: 1.07/1.14 a__U11(tt) -> tt 1.07/1.14 a__U11(X:S) -> U11(X:S) 1.07/1.14 a__U21(tt) -> tt 1.07/1.14 a__U21(X:S) -> U21(X:S) 1.07/1.14 a__U31(tt) -> tt 1.07/1.14 a__U31(X:S) -> U31(X:S) 1.07/1.14 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.14 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.14 a__U42(tt) -> tt 1.07/1.14 a__U42(X:S) -> U42(X:S) 1.07/1.14 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.14 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.14 a__U52(tt) -> tt 1.07/1.14 a__U52(X:S) -> U52(X:S) 1.07/1.14 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.14 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.14 a__U62(tt) -> tt 1.07/1.14 a__U62(X:S) -> U62(X:S) 1.07/1.14 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.14 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.14 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.14 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.14 a__U81(tt) -> nil 1.07/1.14 a__U81(X:S) -> U81(X:S) 1.07/1.14 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.14 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__isNat(0) -> tt 1.07/1.14 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.14 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.14 a__isNat(X:S) -> isNat(X:S) 1.07/1.14 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatIList(zeros) -> tt 1.07/1.14 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.14 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.14 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(nil) -> tt 1.07/1.14 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.14 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 a__length(nil) -> 0 1.07/1.14 a__length(X:S) -> length(X:S) 1.07/1.14 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.14 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.14 a__zeros -> cons(0,zeros) 1.07/1.14 a__zeros -> zeros 1.07/1.14 mark(0) -> 0 1.07/1.14 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.14 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.14 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.14 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.14 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.14 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.14 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.14 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.14 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.14 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.14 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.14 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.14 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.14 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.14 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.14 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.14 mark(nil) -> nil 1.07/1.14 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.14 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.14 mark(tt) -> tt 1.07/1.14 mark(zeros) -> a__zeros 1.07/1.14 -> Usable rules: 1.07/1.14 a__U11(tt) -> tt 1.07/1.14 a__U11(X:S) -> U11(X:S) 1.07/1.14 a__U21(tt) -> tt 1.07/1.14 a__U21(X:S) -> U21(X:S) 1.07/1.14 a__U31(tt) -> tt 1.07/1.14 a__U31(X:S) -> U31(X:S) 1.07/1.14 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.14 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.14 a__U42(tt) -> tt 1.07/1.14 a__U42(X:S) -> U42(X:S) 1.07/1.14 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.14 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.14 a__U52(tt) -> tt 1.07/1.14 a__U52(X:S) -> U52(X:S) 1.07/1.14 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.14 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.14 a__U62(tt) -> tt 1.07/1.14 a__U62(X:S) -> U62(X:S) 1.07/1.14 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.14 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.14 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.14 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.14 a__U81(tt) -> nil 1.07/1.14 a__U81(X:S) -> U81(X:S) 1.07/1.14 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.14 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__isNat(0) -> tt 1.07/1.14 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.14 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.14 a__isNat(X:S) -> isNat(X:S) 1.07/1.14 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatIList(zeros) -> tt 1.07/1.14 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.14 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.14 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(nil) -> tt 1.07/1.14 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.14 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 a__length(nil) -> 0 1.07/1.14 a__length(X:S) -> length(X:S) 1.07/1.14 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.14 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.14 a__zeros -> cons(0,zeros) 1.07/1.14 a__zeros -> zeros 1.07/1.14 mark(0) -> 0 1.07/1.14 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.14 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.14 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.14 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.14 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.14 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.14 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.14 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.14 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.14 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.14 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.14 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.14 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.14 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.14 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.14 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.14 mark(nil) -> nil 1.07/1.14 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.14 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.14 mark(tt) -> tt 1.07/1.14 mark(zeros) -> a__zeros 1.07/1.14 ->Interpretation type: 1.07/1.14 Linear 1.07/1.14 ->Coefficients: 1.07/1.14 Natural Numbers 1.07/1.14 ->Dimension: 1.07/1.14 1 1.07/1.14 ->Bound: 1.07/1.14 2 1.07/1.14 ->Interpretation: 1.07/1.14 1.07/1.14 [a__U11](X) = X 1.07/1.14 [a__U21](X) = X 1.07/1.14 [a__U31](X) = 2 1.07/1.14 [a__U41](X1,X2) = 2 1.07/1.14 [a__U42](X) = X 1.07/1.14 [a__U51](X1,X2) = 2.X2 1.07/1.14 [a__U52](X) = 2.X 1.07/1.14 [a__U61](X1,X2) = X1 + X2 1.07/1.14 [a__U62](X) = X 1.07/1.14 [a__U71](X1,X2,X3) = 2.X2 1.07/1.14 [a__U72](X1,X2) = 2.X2 1.07/1.14 [a__U81](X) = X 1.07/1.14 [a__U91](X1,X2,X3,X4) = 2.X2 + 2.X3 1.07/1.14 [a__U92](X1,X2,X3,X4) = 2.X2 + 2.X3 1.07/1.14 [a__U93](X1,X2,X3,X4) = 2.X2 + 2.X3 1.07/1.14 [a__isNat](X) = X 1.07/1.14 [a__isNatIList](X) = 2 1.07/1.14 [a__isNatList](X) = X 1.07/1.14 [a__length](X) = X 1.07/1.14 [a__take](X1,X2) = X1 + X2 1.07/1.14 [a__zeros] = 0 1.07/1.14 [mark](X) = X 1.07/1.14 [0] = 2 1.07/1.14 [U11](X) = X 1.07/1.14 [U21](X) = X 1.07/1.14 [U31](X) = 2 1.07/1.14 [U41](X1,X2) = 2 1.07/1.14 [U42](X) = X 1.07/1.14 [U51](X1,X2) = 2.X2 1.07/1.14 [U52](X) = 2.X 1.07/1.14 [U61](X1,X2) = X1 + X2 1.07/1.14 [U62](X) = X 1.07/1.14 [U71](X1,X2,X3) = 2.X2 1.07/1.14 [U72](X1,X2) = 2.X2 1.07/1.14 [U81](X) = X 1.07/1.14 [U91](X1,X2,X3,X4) = 2.X2 + 2.X3 1.07/1.14 [U92](X1,X2,X3,X4) = 2.X2 + 2.X3 1.07/1.14 [U93](X1,X2,X3,X4) = 2.X2 + 2.X3 1.07/1.14 [cons](X1,X2) = 2.X2 1.07/1.14 [fSNonEmpty] = 0 1.07/1.14 [isNat](X) = X 1.07/1.14 [isNatIList](X) = 2 1.07/1.14 [isNatList](X) = X 1.07/1.14 [length](X) = X 1.07/1.14 [nil] = 2 1.07/1.14 [s](X) = 2.X 1.07/1.14 [take](X1,X2) = X1 + X2 1.07/1.14 [tt] = 2 1.07/1.14 [zeros] = 0 1.07/1.14 [A__U11](X) = 0 1.07/1.14 [A__U21](X) = 0 1.07/1.14 [A__U31](X) = 0 1.07/1.14 [A__U41](X1,X2) = 0 1.07/1.14 [A__U42](X) = 0 1.07/1.14 [A__U51](X1,X2) = 0 1.07/1.14 [A__U52](X) = 0 1.07/1.14 [A__U61](X1,X2) = 0 1.07/1.14 [A__U62](X) = 0 1.07/1.14 [A__U71](X1,X2,X3) = 2.X1 + 2.X2 1.07/1.14 [A__U72](X1,X2) = 2.X2 + 2 1.07/1.14 [A__U81](X) = 0 1.07/1.14 [A__U91](X1,X2,X3,X4) = 0 1.07/1.14 [A__U92](X1,X2,X3,X4) = 0 1.07/1.14 [A__U93](X1,X2,X3,X4) = 0 1.07/1.14 [A__ISNAT](X) = 0 1.07/1.14 [A__ISNATILIST](X) = 0 1.07/1.14 [A__ISNATLIST](X) = 0 1.07/1.14 [A__LENGTH](X) = 2.X 1.07/1.14 [A__TAKE](X1,X2) = 0 1.07/1.14 [A__ZEROS] = 0 1.07/1.14 [MARK](X) = 0 1.07/1.14 1.07/1.14 Problem 1.2.1: 1.07/1.14 1.07/1.14 SCC Processor: 1.07/1.14 -> Pairs: 1.07/1.14 A__U72(tt,L:S) -> A__LENGTH(mark(L:S)) 1.07/1.14 A__LENGTH(cons(N:S,L:S)) -> A__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 -> Rules: 1.07/1.14 a__U11(tt) -> tt 1.07/1.14 a__U11(X:S) -> U11(X:S) 1.07/1.14 a__U21(tt) -> tt 1.07/1.14 a__U21(X:S) -> U21(X:S) 1.07/1.14 a__U31(tt) -> tt 1.07/1.14 a__U31(X:S) -> U31(X:S) 1.07/1.14 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.14 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.14 a__U42(tt) -> tt 1.07/1.14 a__U42(X:S) -> U42(X:S) 1.07/1.14 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.14 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.14 a__U52(tt) -> tt 1.07/1.14 a__U52(X:S) -> U52(X:S) 1.07/1.14 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.14 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.14 a__U62(tt) -> tt 1.07/1.14 a__U62(X:S) -> U62(X:S) 1.07/1.14 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.14 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.14 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.14 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.14 a__U81(tt) -> nil 1.07/1.14 a__U81(X:S) -> U81(X:S) 1.07/1.14 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.14 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__isNat(0) -> tt 1.07/1.14 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.14 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.14 a__isNat(X:S) -> isNat(X:S) 1.07/1.14 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatIList(zeros) -> tt 1.07/1.14 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.14 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.14 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(nil) -> tt 1.07/1.14 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.14 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 a__length(nil) -> 0 1.07/1.14 a__length(X:S) -> length(X:S) 1.07/1.14 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.14 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.14 a__zeros -> cons(0,zeros) 1.07/1.14 a__zeros -> zeros 1.07/1.14 mark(0) -> 0 1.07/1.14 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.14 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.14 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.14 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.14 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.14 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.14 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.14 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.14 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.14 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.14 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.14 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.14 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.14 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.14 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.14 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.14 mark(nil) -> nil 1.07/1.14 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.14 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.14 mark(tt) -> tt 1.07/1.14 mark(zeros) -> a__zeros 1.07/1.14 ->Strongly Connected Components: 1.07/1.14 There is no strongly connected component 1.07/1.14 1.07/1.14 The problem is finite. 1.07/1.14 1.07/1.14 Problem 1.2.2: 1.07/1.14 1.07/1.14 Reduction Pairs Processor: 1.07/1.14 -> Pairs: 1.07/1.14 A__U91(tt,IL:S,M:S,N:S) -> A__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) 1.07/1.14 A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 MARK(U11(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U21(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U31(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U41(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U42(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U52(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U61(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U62(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.07/1.14 MARK(U72(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U81(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> A__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(length(X:S)) -> MARK(X:S) 1.07/1.14 MARK(s(X:S)) -> MARK(X:S) 1.07/1.14 MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S)) 1.07/1.14 MARK(take(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(take(X1:S,X2:S)) -> MARK(X2:S) 1.07/1.14 -> Rules: 1.07/1.14 a__U11(tt) -> tt 1.07/1.14 a__U11(X:S) -> U11(X:S) 1.07/1.14 a__U21(tt) -> tt 1.07/1.14 a__U21(X:S) -> U21(X:S) 1.07/1.14 a__U31(tt) -> tt 1.07/1.14 a__U31(X:S) -> U31(X:S) 1.07/1.14 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.14 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.14 a__U42(tt) -> tt 1.07/1.14 a__U42(X:S) -> U42(X:S) 1.07/1.14 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.14 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.14 a__U52(tt) -> tt 1.07/1.14 a__U52(X:S) -> U52(X:S) 1.07/1.14 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.14 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.14 a__U62(tt) -> tt 1.07/1.14 a__U62(X:S) -> U62(X:S) 1.07/1.14 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.14 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.14 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.14 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.14 a__U81(tt) -> nil 1.07/1.14 a__U81(X:S) -> U81(X:S) 1.07/1.14 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.14 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__isNat(0) -> tt 1.07/1.14 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.14 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.14 a__isNat(X:S) -> isNat(X:S) 1.07/1.14 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatIList(zeros) -> tt 1.07/1.14 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.14 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.14 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(nil) -> tt 1.07/1.14 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.14 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 a__length(nil) -> 0 1.07/1.14 a__length(X:S) -> length(X:S) 1.07/1.14 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.14 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.14 a__zeros -> cons(0,zeros) 1.07/1.14 a__zeros -> zeros 1.07/1.14 mark(0) -> 0 1.07/1.14 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.14 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.14 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.14 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.14 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.14 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.14 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.14 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.14 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.14 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.14 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.14 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.14 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.14 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.14 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.14 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.14 mark(nil) -> nil 1.07/1.14 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.14 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.14 mark(tt) -> tt 1.07/1.14 mark(zeros) -> a__zeros 1.07/1.14 -> Usable rules: 1.07/1.14 a__U11(tt) -> tt 1.07/1.14 a__U11(X:S) -> U11(X:S) 1.07/1.14 a__U21(tt) -> tt 1.07/1.14 a__U21(X:S) -> U21(X:S) 1.07/1.14 a__U31(tt) -> tt 1.07/1.14 a__U31(X:S) -> U31(X:S) 1.07/1.14 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.14 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.14 a__U42(tt) -> tt 1.07/1.14 a__U42(X:S) -> U42(X:S) 1.07/1.14 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.14 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.14 a__U52(tt) -> tt 1.07/1.14 a__U52(X:S) -> U52(X:S) 1.07/1.14 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.14 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.14 a__U62(tt) -> tt 1.07/1.14 a__U62(X:S) -> U62(X:S) 1.07/1.14 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.14 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.14 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.14 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.14 a__U81(tt) -> nil 1.07/1.14 a__U81(X:S) -> U81(X:S) 1.07/1.14 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.14 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__isNat(0) -> tt 1.07/1.14 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.14 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.14 a__isNat(X:S) -> isNat(X:S) 1.07/1.14 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatIList(zeros) -> tt 1.07/1.14 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.14 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.14 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(nil) -> tt 1.07/1.14 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.14 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 a__length(nil) -> 0 1.07/1.14 a__length(X:S) -> length(X:S) 1.07/1.14 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.14 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.14 a__zeros -> cons(0,zeros) 1.07/1.14 a__zeros -> zeros 1.07/1.14 mark(0) -> 0 1.07/1.14 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.14 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.14 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.14 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.14 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.14 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.14 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.14 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.14 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.14 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.14 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.14 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.14 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.14 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.14 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.14 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.14 mark(nil) -> nil 1.07/1.14 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.14 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.14 mark(tt) -> tt 1.07/1.14 mark(zeros) -> a__zeros 1.07/1.14 ->Interpretation type: 1.07/1.14 Linear 1.07/1.14 ->Coefficients: 1.07/1.14 Natural Numbers 1.07/1.14 ->Dimension: 1.07/1.14 1 1.07/1.14 ->Bound: 1.07/1.14 2 1.07/1.14 ->Interpretation: 1.07/1.14 1.07/1.14 [a__U11](X) = X 1.07/1.14 [a__U21](X) = X 1.07/1.14 [a__U31](X) = X 1.07/1.14 [a__U41](X1,X2) = X1 1.07/1.14 [a__U42](X) = X 1.07/1.14 [a__U51](X1,X2) = X1 1.07/1.14 [a__U52](X) = X 1.07/1.14 [a__U61](X1,X2) = X1 1.07/1.14 [a__U62](X) = X 1.07/1.14 [a__U71](X1,X2,X3) = X1 + 2.X2 + 2.X3 1.07/1.14 [a__U72](X1,X2) = X1 + 2.X2 1.07/1.14 [a__U81](X) = X 1.07/1.14 [a__U91](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 1.07/1.14 [a__U92](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 1.07/1.14 [a__U93](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 1.07/1.14 [a__isNat](X) = 2 1.07/1.14 [a__isNatIList](X) = 2 1.07/1.14 [a__isNatList](X) = 2 1.07/1.14 [a__length](X) = 2.X + 2 1.07/1.14 [a__take](X1,X2) = 2.X1 + 2.X2 + 2 1.07/1.14 [a__zeros] = 2 1.07/1.14 [mark](X) = X 1.07/1.14 [0] = 0 1.07/1.14 [U11](X) = X 1.07/1.14 [U21](X) = X 1.07/1.14 [U31](X) = X 1.07/1.14 [U41](X1,X2) = X1 1.07/1.14 [U42](X) = X 1.07/1.14 [U51](X1,X2) = X1 1.07/1.14 [U52](X) = X 1.07/1.14 [U61](X1,X2) = X1 1.07/1.14 [U62](X) = X 1.07/1.14 [U71](X1,X2,X3) = X1 + 2.X2 + 2.X3 1.07/1.14 [U72](X1,X2) = X1 + 2.X2 1.07/1.14 [U81](X) = X 1.07/1.14 [U91](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 1.07/1.14 [U92](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 1.07/1.14 [U93](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 1.07/1.14 [cons](X1,X2) = X1 + X2 1.07/1.14 [fSNonEmpty] = 0 1.07/1.14 [isNat](X) = 2 1.07/1.14 [isNatIList](X) = 2 1.07/1.14 [isNatList](X) = 2 1.07/1.14 [length](X) = 2.X + 2 1.07/1.14 [nil] = 2 1.07/1.14 [s](X) = X 1.07/1.14 [take](X1,X2) = 2.X1 + 2.X2 + 2 1.07/1.14 [tt] = 2 1.07/1.14 [zeros] = 2 1.07/1.14 [A__U11](X) = 0 1.07/1.14 [A__U21](X) = 0 1.07/1.14 [A__U31](X) = 0 1.07/1.14 [A__U41](X1,X2) = 0 1.07/1.14 [A__U42](X) = 0 1.07/1.14 [A__U51](X1,X2) = 0 1.07/1.14 [A__U52](X) = 0 1.07/1.14 [A__U61](X1,X2) = 0 1.07/1.14 [A__U62](X) = 0 1.07/1.14 [A__U71](X1,X2,X3) = 0 1.07/1.14 [A__U72](X1,X2) = 0 1.07/1.14 [A__U81](X) = 0 1.07/1.14 [A__U91](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 1.07/1.14 [A__U92](X1,X2,X3,X4) = 2.X2 + 2.X3 + 2.X4 + 1 1.07/1.14 [A__U93](X1,X2,X3,X4) = 2.X2 + X3 + 2.X4 + 1 1.07/1.14 [A__ISNAT](X) = 0 1.07/1.14 [A__ISNATILIST](X) = 0 1.07/1.14 [A__ISNATLIST](X) = 0 1.07/1.14 [A__LENGTH](X) = 0 1.07/1.14 [A__TAKE](X1,X2) = 2.X1 + 2.X2 + 2 1.07/1.14 [A__ZEROS] = 0 1.07/1.14 [MARK](X) = 2.X + 1 1.07/1.14 1.07/1.14 Problem 1.2.2: 1.07/1.14 1.07/1.14 SCC Processor: 1.07/1.14 -> Pairs: 1.07/1.14 A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) 1.07/1.14 A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 MARK(U11(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U21(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U31(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U41(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U42(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U52(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U61(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U62(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.07/1.14 MARK(U72(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U81(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> A__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(length(X:S)) -> MARK(X:S) 1.07/1.14 MARK(s(X:S)) -> MARK(X:S) 1.07/1.14 MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S)) 1.07/1.14 MARK(take(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(take(X1:S,X2:S)) -> MARK(X2:S) 1.07/1.14 -> Rules: 1.07/1.14 a__U11(tt) -> tt 1.07/1.14 a__U11(X:S) -> U11(X:S) 1.07/1.14 a__U21(tt) -> tt 1.07/1.14 a__U21(X:S) -> U21(X:S) 1.07/1.14 a__U31(tt) -> tt 1.07/1.14 a__U31(X:S) -> U31(X:S) 1.07/1.14 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.14 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.14 a__U42(tt) -> tt 1.07/1.14 a__U42(X:S) -> U42(X:S) 1.07/1.14 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.14 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.14 a__U52(tt) -> tt 1.07/1.14 a__U52(X:S) -> U52(X:S) 1.07/1.14 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.14 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.14 a__U62(tt) -> tt 1.07/1.14 a__U62(X:S) -> U62(X:S) 1.07/1.14 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.14 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.14 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.14 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.14 a__U81(tt) -> nil 1.07/1.14 a__U81(X:S) -> U81(X:S) 1.07/1.14 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.14 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__isNat(0) -> tt 1.07/1.14 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.14 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.14 a__isNat(X:S) -> isNat(X:S) 1.07/1.14 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatIList(zeros) -> tt 1.07/1.14 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.14 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.14 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(nil) -> tt 1.07/1.14 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.14 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 a__length(nil) -> 0 1.07/1.14 a__length(X:S) -> length(X:S) 1.07/1.14 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.14 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.14 a__zeros -> cons(0,zeros) 1.07/1.14 a__zeros -> zeros 1.07/1.14 mark(0) -> 0 1.07/1.14 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.14 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.14 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.14 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.14 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.14 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.14 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.14 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.14 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.14 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.14 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.14 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.14 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.14 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.14 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.14 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.14 mark(nil) -> nil 1.07/1.14 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.14 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.14 mark(tt) -> tt 1.07/1.14 mark(zeros) -> a__zeros 1.07/1.14 ->Strongly Connected Components: 1.07/1.14 ->->Cycle: 1.07/1.14 ->->-> Pairs: 1.07/1.14 A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) 1.07/1.14 MARK(U11(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U21(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U31(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U41(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U42(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U52(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U61(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U62(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.07/1.14 MARK(U72(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U81(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(length(X:S)) -> MARK(X:S) 1.07/1.14 MARK(s(X:S)) -> MARK(X:S) 1.07/1.14 MARK(take(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(take(X1:S,X2:S)) -> MARK(X2:S) 1.07/1.14 ->->-> Rules: 1.07/1.14 a__U11(tt) -> tt 1.07/1.14 a__U11(X:S) -> U11(X:S) 1.07/1.14 a__U21(tt) -> tt 1.07/1.14 a__U21(X:S) -> U21(X:S) 1.07/1.14 a__U31(tt) -> tt 1.07/1.14 a__U31(X:S) -> U31(X:S) 1.07/1.14 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.14 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.14 a__U42(tt) -> tt 1.07/1.14 a__U42(X:S) -> U42(X:S) 1.07/1.14 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.14 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.14 a__U52(tt) -> tt 1.07/1.14 a__U52(X:S) -> U52(X:S) 1.07/1.14 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.14 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.14 a__U62(tt) -> tt 1.07/1.14 a__U62(X:S) -> U62(X:S) 1.07/1.14 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.14 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.14 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.14 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.14 a__U81(tt) -> nil 1.07/1.14 a__U81(X:S) -> U81(X:S) 1.07/1.14 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.14 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__isNat(0) -> tt 1.07/1.14 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.14 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.14 a__isNat(X:S) -> isNat(X:S) 1.07/1.14 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatIList(zeros) -> tt 1.07/1.14 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.14 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.14 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(nil) -> tt 1.07/1.14 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.14 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 a__length(nil) -> 0 1.07/1.14 a__length(X:S) -> length(X:S) 1.07/1.14 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.14 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.14 a__zeros -> cons(0,zeros) 1.07/1.14 a__zeros -> zeros 1.07/1.14 mark(0) -> 0 1.07/1.14 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.14 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.14 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.14 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.14 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.14 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.14 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.14 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.14 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.14 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.14 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.14 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.14 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.14 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.14 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.14 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.14 mark(nil) -> nil 1.07/1.14 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.14 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.14 mark(tt) -> tt 1.07/1.14 mark(zeros) -> a__zeros 1.07/1.14 1.07/1.14 Problem 1.2.2: 1.07/1.14 1.07/1.14 Subterm Processor: 1.07/1.14 -> Pairs: 1.07/1.14 A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) 1.07/1.14 MARK(U11(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U21(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U31(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U41(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U42(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U52(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U61(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U62(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.07/1.14 MARK(U72(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(U81(X:S)) -> MARK(X:S) 1.07/1.14 MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) 1.07/1.14 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(length(X:S)) -> MARK(X:S) 1.07/1.14 MARK(s(X:S)) -> MARK(X:S) 1.07/1.14 MARK(take(X1:S,X2:S)) -> MARK(X1:S) 1.07/1.14 MARK(take(X1:S,X2:S)) -> MARK(X2:S) 1.07/1.14 -> Rules: 1.07/1.14 a__U11(tt) -> tt 1.07/1.14 a__U11(X:S) -> U11(X:S) 1.07/1.14 a__U21(tt) -> tt 1.07/1.14 a__U21(X:S) -> U21(X:S) 1.07/1.14 a__U31(tt) -> tt 1.07/1.14 a__U31(X:S) -> U31(X:S) 1.07/1.14 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.14 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.14 a__U42(tt) -> tt 1.07/1.14 a__U42(X:S) -> U42(X:S) 1.07/1.14 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.14 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.14 a__U52(tt) -> tt 1.07/1.14 a__U52(X:S) -> U52(X:S) 1.07/1.14 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.14 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.14 a__U62(tt) -> tt 1.07/1.14 a__U62(X:S) -> U62(X:S) 1.07/1.14 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.14 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.14 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.14 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.14 a__U81(tt) -> nil 1.07/1.14 a__U81(X:S) -> U81(X:S) 1.07/1.14 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.14 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__isNat(0) -> tt 1.07/1.14 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.14 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.14 a__isNat(X:S) -> isNat(X:S) 1.07/1.14 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatIList(zeros) -> tt 1.07/1.14 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.14 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.14 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(nil) -> tt 1.07/1.14 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.14 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 a__length(nil) -> 0 1.07/1.14 a__length(X:S) -> length(X:S) 1.07/1.14 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.14 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.14 a__zeros -> cons(0,zeros) 1.07/1.14 a__zeros -> zeros 1.07/1.14 mark(0) -> 0 1.07/1.14 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.14 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.14 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.14 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.14 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.14 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.14 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.14 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.14 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.14 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.14 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.14 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.14 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.14 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.14 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.14 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.14 mark(nil) -> nil 1.07/1.14 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.14 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.14 mark(tt) -> tt 1.07/1.14 mark(zeros) -> a__zeros 1.07/1.14 ->Projection: 1.07/1.14 pi(A__U92) = 4 1.07/1.14 pi(A__U93) = 4 1.07/1.14 pi(MARK) = 1 1.07/1.14 1.07/1.14 Problem 1.2.2: 1.07/1.14 1.07/1.14 SCC Processor: 1.07/1.14 -> Pairs: 1.07/1.14 A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) 1.07/1.14 -> Rules: 1.07/1.14 a__U11(tt) -> tt 1.07/1.14 a__U11(X:S) -> U11(X:S) 1.07/1.14 a__U21(tt) -> tt 1.07/1.14 a__U21(X:S) -> U21(X:S) 1.07/1.14 a__U31(tt) -> tt 1.07/1.14 a__U31(X:S) -> U31(X:S) 1.07/1.14 a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) 1.07/1.14 a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) 1.07/1.14 a__U42(tt) -> tt 1.07/1.14 a__U42(X:S) -> U42(X:S) 1.07/1.14 a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) 1.07/1.14 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.07/1.14 a__U52(tt) -> tt 1.07/1.14 a__U52(X:S) -> U52(X:S) 1.07/1.14 a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) 1.07/1.14 a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) 1.07/1.14 a__U62(tt) -> tt 1.07/1.14 a__U62(X:S) -> U62(X:S) 1.07/1.14 a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) 1.07/1.14 a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) 1.07/1.14 a__U72(tt,L:S) -> s(a__length(mark(L:S))) 1.07/1.14 a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) 1.07/1.14 a__U81(tt) -> nil 1.07/1.14 a__U81(X:S) -> U81(X:S) 1.07/1.14 a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) 1.07/1.14 a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) 1.07/1.14 a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) 1.07/1.14 a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) 1.07/1.14 a__isNat(0) -> tt 1.07/1.14 a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) 1.07/1.14 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 1.07/1.14 a__isNat(X:S) -> isNat(X:S) 1.07/1.14 a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatIList(zeros) -> tt 1.07/1.14 a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) 1.07/1.14 a__isNatIList(X:S) -> isNatIList(X:S) 1.07/1.14 a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(nil) -> tt 1.07/1.14 a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) 1.07/1.14 a__isNatList(X:S) -> isNatList(X:S) 1.07/1.14 a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) 1.07/1.14 a__length(nil) -> 0 1.07/1.14 a__length(X:S) -> length(X:S) 1.07/1.14 a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) 1.07/1.14 a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) 1.07/1.14 a__take(X1:S,X2:S) -> take(X1:S,X2:S) 1.07/1.14 a__zeros -> cons(0,zeros) 1.07/1.14 a__zeros -> zeros 1.07/1.14 mark(0) -> 0 1.07/1.14 mark(U11(X:S)) -> a__U11(mark(X:S)) 1.07/1.14 mark(U21(X:S)) -> a__U21(mark(X:S)) 1.07/1.14 mark(U31(X:S)) -> a__U31(mark(X:S)) 1.07/1.14 mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) 1.07/1.14 mark(U42(X:S)) -> a__U42(mark(X:S)) 1.07/1.14 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.07/1.14 mark(U52(X:S)) -> a__U52(mark(X:S)) 1.07/1.14 mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) 1.07/1.14 mark(U62(X:S)) -> a__U62(mark(X:S)) 1.07/1.14 mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) 1.07/1.14 mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) 1.07/1.14 mark(U81(X:S)) -> a__U81(mark(X:S)) 1.07/1.14 mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) 1.07/1.14 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 1.07/1.14 mark(isNat(X:S)) -> a__isNat(X:S) 1.07/1.14 mark(isNatIList(X:S)) -> a__isNatIList(X:S) 1.07/1.14 mark(isNatList(X:S)) -> a__isNatList(X:S) 1.07/1.14 mark(length(X:S)) -> a__length(mark(X:S)) 1.07/1.14 mark(nil) -> nil 1.07/1.14 mark(s(X:S)) -> s(mark(X:S)) 1.07/1.14 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) 1.07/1.14 mark(tt) -> tt 1.07/1.14 mark(zeros) -> a__zeros 1.07/1.14 ->Strongly Connected Components: 1.07/1.14 There is no strongly connected component 1.07/1.14 1.07/1.14 The problem is finite. 1.07/1.14 EOF