/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) typed CpxTrs (5) OrderProof [LOWER BOUND(ID), 0 ms] (6) typed CpxTrs (7) RewriteLemmaProof [LOWER BOUND(ID), 485 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 124 ms] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 127 ms] (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 93 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 86 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 107 ms] (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 117 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 228 ms] (26) typed CpxTrs (27) RewriteLemmaProof [LOWER BOUND(ID), 204 ms] (28) typed CpxTrs (29) RewriteLemmaProof [LOWER BOUND(ID), 141 ms] (30) typed CpxTrs (31) RewriteLemmaProof [LOWER BOUND(ID), 77 ms] (32) typed CpxTrs (33) RewriteLemmaProof [LOWER BOUND(ID), 114 ms] (34) typed CpxTrs (35) RewriteLemmaProof [LOWER BOUND(ID), 42 ms] (36) typed CpxTrs (37) RewriteLemmaProof [LOWER BOUND(ID), 122 ms] (38) typed CpxTrs (39) RewriteLemmaProof [LOWER BOUND(ID), 118 ms] (40) typed CpxTrs (41) RewriteLemmaProof [LOWER BOUND(ID), 145 ms] (42) typed CpxTrs (43) RewriteLemmaProof [LOWER BOUND(ID), 165 ms] (44) typed CpxTrs (45) RewriteLemmaProof [LOWER BOUND(ID), 55 ms] (46) typed CpxTrs (47) RewriteLemmaProof [LOWER BOUND(ID), 282 ms] (48) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok ---------------------------------------- (5) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: active, cons, U42, isNatIList, U52, isNatList, U62, U72, isNat, s, length, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: cons < active U42 < active isNatIList < active U52 < active isNatList < active U62 < active U72 < active isNat < active s < active length < active U92 < active U93 < active take < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active active < top cons < proper U42 < proper isNatIList < proper U52 < proper isNatList < proper U62 < proper U72 < proper isNat < proper s < proper length < proper U92 < proper U93 < proper take < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (6) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: cons, active, U42, isNatIList, U52, isNatList, U62, U72, isNat, s, length, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: cons < active U42 < active isNatIList < active U52 < active isNatList < active U62 < active U72 < active isNat < active s < active length < active U92 < active U93 < active take < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active active < top cons < proper U42 < proper isNatIList < proper U52 < proper isNatList < proper U62 < proper U72 < proper isNat < proper s < proper length < proper U92 < proper U93 < proper take < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (7) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) Induction Base: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) Induction Step: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n5_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1) mark(cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: cons, active, U42, isNatIList, U52, isNatList, U62, U72, isNat, s, length, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: cons < active U42 < active isNatIList < active U52 < active isNatList < active U62 < active U72 < active isNat < active s < active length < active U92 < active U93 < active take < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active active < top cons < proper U42 < proper isNatIList < proper U52 < proper isNatList < proper U62 < proper U72 < proper isNat < proper s < proper length < proper U92 < proper U93 < proper take < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: U42, active, isNatIList, U52, isNatList, U62, U72, isNat, s, length, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: U42 < active isNatIList < active U52 < active isNatList < active U62 < active U72 < active isNat < active s < active length < active U92 < active U93 < active take < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active active < top U42 < proper isNatIList < proper U52 < proper isNatList < proper U62 < proper U72 < proper isNat < proper s < proper length < proper U92 < proper U93 < proper take < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) Induction Base: U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0))) Induction Step: U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n2106_0, 1)))) ->_R^Omega(1) mark(U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (14) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: isNatIList, active, U52, isNatList, U62, U72, isNat, s, length, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: isNatIList < active U52 < active isNatList < active U62 < active U72 < active isNat < active s < active length < active U92 < active U93 < active take < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active active < top isNatIList < proper U52 < proper isNatList < proper U62 < proper U72 < proper isNat < proper s < proper length < proper U92 < proper U93 < proper take < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) Induction Base: U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0))) Induction Step: U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n2892_0, 1)))) ->_R^Omega(1) mark(U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (16) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: isNatList, active, U62, U72, isNat, s, length, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: isNatList < active U62 < active U72 < active isNat < active s < active length < active U92 < active U93 < active take < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active active < top isNatList < proper U62 < proper U72 < proper isNat < proper s < proper length < proper U92 < proper U93 < proper take < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) Induction Base: U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0))) Induction Step: U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n3782_0, 1)))) ->_R^Omega(1) mark(U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: U72, active, isNat, s, length, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: U72 < active isNat < active s < active length < active U92 < active U93 < active take < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active active < top U72 < proper isNat < proper s < proper length < proper U92 < proper U93 < proper take < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0) Induction Base: U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) Induction Step: U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n4758_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1) mark(U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (20) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: isNat, active, s, length, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: isNat < active s < active length < active U92 < active U93 < active take < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active active < top isNat < proper s < proper length < proper U92 < proper U93 < proper take < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0) Induction Base: s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0))) Induction Step: s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n7812_0, 1)))) ->_R^Omega(1) mark(s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (22) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: length, active, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: length < active U92 < active U93 < active take < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active active < top length < proper U92 < proper U93 < proper take < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0) Induction Base: length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0))) Induction Step: length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n9039_0, 1)))) ->_R^Omega(1) mark(length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (24) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: U92, active, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: U92 < active U93 < active take < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active active < top U92 < proper U93 < proper take < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0) Induction Base: U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) Induction Step: U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n10367_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) ->_R^Omega(1) mark(U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (26) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0) U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: U93, active, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: U93 < active take < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active active < top U93 < proper take < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (27) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0) Induction Base: U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) Induction Step: U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n18768_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) ->_R^Omega(1) mark(U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (28) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0) U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0) U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: take, active, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: take < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active active < top take < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (29) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0) Induction Base: take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) Induction Step: take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n28179_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1) mark(take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (30) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0) U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0) U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0) take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: U11, active, U21, U31, U41, U51, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active active < top U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (31) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0) Induction Base: U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0))) Induction Step: U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n33135_0, 1)))) ->_R^Omega(1) mark(U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (32) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0) U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0) U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0) take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0) U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: U21, active, U31, U41, U51, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active active < top U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (33) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0) Induction Base: U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0))) Induction Step: U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n35210_0, 1)))) ->_R^Omega(1) mark(U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (34) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0) U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0) U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0) take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0) U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0) U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: U31, active, U41, U51, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active active < top U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (35) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0))) -> *4_0, rt in Omega(n37386_0) Induction Base: U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0))) Induction Step: U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n37386_0, 1)))) ->_R^Omega(1) mark(U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (36) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0) U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0) U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0) take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0) U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0) U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0) U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0))) -> *4_0, rt in Omega(n37386_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: U41, active, U51, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active active < top U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (37) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n39663_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n39663_0) Induction Base: U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) Induction Step: U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n39663_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1) mark(U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n39663_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (38) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0) U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0) U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0) take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0) U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0) U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0) U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0))) -> *4_0, rt in Omega(n37386_0) U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n39663_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n39663_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: U51, active, U61, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: U51 < active U61 < active U71 < active U81 < active U91 < active active < top U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (39) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n45350_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n45350_0) Induction Base: U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) Induction Step: U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n45350_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1) mark(U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n45350_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (40) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0) U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0) U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0) take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0) U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0) U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0) U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0))) -> *4_0, rt in Omega(n37386_0) U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n39663_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n39663_0) U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n45350_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n45350_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: U61, active, U71, U81, U91, proper, top They will be analysed ascendingly in the following order: U61 < active U71 < active U81 < active U91 < active active < top U61 < proper U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (41) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n51343_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n51343_0) Induction Base: U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) Induction Step: U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n51343_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1) mark(U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n51343_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (42) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0) U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0) U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0) take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0) U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0) U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0) U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0))) -> *4_0, rt in Omega(n37386_0) U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n39663_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n39663_0) U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n45350_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n45350_0) U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n51343_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n51343_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: U71, active, U81, U91, proper, top They will be analysed ascendingly in the following order: U71 < active U81 < active U91 < active active < top U71 < proper U81 < proper U91 < proper proper < top ---------------------------------------- (43) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n57642_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n57642_0) Induction Base: U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) Induction Step: U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n57642_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1) mark(U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n57642_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (44) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0) U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0) U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0) take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0) U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0) U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0) U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0))) -> *4_0, rt in Omega(n37386_0) U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n39663_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n39663_0) U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n45350_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n45350_0) U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n51343_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n51343_0) U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n57642_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n57642_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: U81, active, U91, proper, top They will be analysed ascendingly in the following order: U81 < active U91 < active active < top U81 < proper U91 < proper proper < top ---------------------------------------- (45) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n68024_0))) -> *4_0, rt in Omega(n68024_0) Induction Base: U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0))) Induction Step: U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n68024_0, 1)))) ->_R^Omega(1) mark(U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n68024_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (46) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0) U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0) U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0) take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0) U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0) U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0) U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0))) -> *4_0, rt in Omega(n37386_0) U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n39663_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n39663_0) U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n45350_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n45350_0) U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n51343_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n51343_0) U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n57642_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n57642_0) U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n68024_0))) -> *4_0, rt in Omega(n68024_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: U91, active, proper, top They will be analysed ascendingly in the following order: U91 < active active < top U91 < proper proper < top ---------------------------------------- (47) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n71051_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n71051_0) Induction Base: U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) Induction Step: U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n71051_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) ->_R^Omega(1) mark(U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n71051_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (48) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0', IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(U72(X1, X2)) -> U72(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4) active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) U72(mark(X1), X2) -> mark(U72(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4)) U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil) -> ok(nil) proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4)) U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0) U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0) U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0) U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0) U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0) U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0) take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0) U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0) U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0) U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0))) -> *4_0, rt in Omega(n37386_0) U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n39663_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n39663_0) U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n45350_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n45350_0) U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n51343_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n51343_0) U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n57642_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n57642_0) U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n68024_0))) -> *4_0, rt in Omega(n68024_0) U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n71051_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n71051_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: active, proper, top They will be analysed ascendingly in the following order: active < top proper < top