/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), O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 14 ms] (6) CdtProblem (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CdtProblem (9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CdtProblem (11) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CdtProblem (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 473 ms] (14) CdtProblem (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 78 ms] (16) CdtProblem (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 75 ms] (18) CdtProblem (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 67 ms] (20) CdtProblem (21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 69 ms] (22) CdtProblem (23) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (24) BOUNDS(1, 1) (25) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CpxTRS (27) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (28) typed CpxTrs (29) OrderProof [LOWER BOUND(ID), 0 ms] (30) typed CpxTrs (31) RewriteLemmaProof [LOWER BOUND(ID), 508 ms] (32) BEST (33) proven lower bound (34) LowerBoundPropagationProof [FINISHED, 0 ms] (35) BOUNDS(n^1, INF) (36) typed CpxTrs (37) RewriteLemmaProof [LOWER BOUND(ID), 106 ms] (38) typed CpxTrs (39) RewriteLemmaProof [LOWER BOUND(ID), 134 ms] (40) typed CpxTrs (41) RewriteLemmaProof [LOWER BOUND(ID), 120 ms] (42) typed CpxTrs (43) RewriteLemmaProof [LOWER BOUND(ID), 112 ms] (44) typed CpxTrs (45) RewriteLemmaProof [LOWER BOUND(ID), 176 ms] (46) typed CpxTrs (47) RewriteLemmaProof [LOWER BOUND(ID), 77 ms] (48) typed CpxTrs (49) RewriteLemmaProof [LOWER BOUND(ID), 152 ms] (50) typed CpxTrs (51) RewriteLemmaProof [LOWER BOUND(ID), 175 ms] (52) typed CpxTrs (53) RewriteLemmaProof [LOWER BOUND(ID), 267 ms] (54) typed CpxTrs (55) RewriteLemmaProof [LOWER BOUND(ID), 241 ms] (56) typed CpxTrs (57) RewriteLemmaProof [LOWER BOUND(ID), 155 ms] (58) typed CpxTrs (59) RewriteLemmaProof [LOWER BOUND(ID), 175 ms] (60) typed CpxTrs (61) RewriteLemmaProof [LOWER BOUND(ID), 161 ms] (62) typed CpxTrs (63) RewriteLemmaProof [LOWER BOUND(ID), 101 ms] (64) typed CpxTrs (65) RewriteLemmaProof [LOWER BOUND(ID), 151 ms] (66) typed CpxTrs (67) RewriteLemmaProof [LOWER BOUND(ID), 96 ms] (68) typed CpxTrs (69) RewriteLemmaProof [LOWER BOUND(ID), 210 ms] (70) typed CpxTrs (71) RewriteLemmaProof [LOWER BOUND(ID), 135 ms] (72) typed CpxTrs (73) RewriteLemmaProof [LOWER BOUND(ID), 136 ms] (74) typed CpxTrs (75) RewriteLemmaProof [LOWER BOUND(ID), 118 ms] (76) typed CpxTrs (77) RewriteLemmaProof [LOWER BOUND(ID), 78 ms] (78) typed CpxTrs (79) RewriteLemmaProof [LOWER BOUND(ID), 262 ms] (80) typed CpxTrs (81) RewriteLemmaProof [LOWER BOUND(ID), 165 ms] (82) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0, XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0) -> ok(0) proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) 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) NestedDefinedSymbolProof (UPPER BOUND(ID)) The following defined symbols can occur below the 0th argument of top: proper, active The following defined symbols can occur below the 0th argument of proper: proper, active The following defined symbols can occur below the 0th argument of active: proper, active Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0, XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(tt) -> ok(tt) proper(0) -> ok(0) proper(nil) -> ok(nil) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) 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) RcToIrcProof (BOTH BOUNDS(ID, ID)) Converted rc-obligation to irc-obligation. As the TRS is a non-duplicating overlay system, we have rc = irc. ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(tt) -> ok(tt) proper(0) -> ok(0) proper(nil) -> ok(nil) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) 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: INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2)) U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2)) snd(mark(z0)) -> mark(snd(z0)) snd(ok(z0)) -> ok(snd(z0)) splitAt(mark(z0), z1) -> mark(splitAt(z0, z1)) splitAt(z0, mark(z1)) -> mark(splitAt(z0, z1)) splitAt(ok(z0), ok(z1)) -> ok(splitAt(z0, z1)) U21(mark(z0), z1) -> mark(U21(z0, z1)) U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) U22(mark(z0), z1) -> mark(U22(z0, z1)) U22(ok(z0), ok(z1)) -> ok(U22(z0, z1)) U31(mark(z0), z1) -> mark(U31(z0, z1)) U31(ok(z0), ok(z1)) -> ok(U31(z0, z1)) U32(mark(z0), z1) -> mark(U32(z0, z1)) U32(ok(z0), ok(z1)) -> ok(U32(z0, z1)) U41(mark(z0), z1, z2) -> mark(U41(z0, z1, z2)) U41(ok(z0), ok(z1), ok(z2)) -> ok(U41(z0, z1, z2)) U42(mark(z0), z1, z2) -> mark(U42(z0, z1, z2)) U42(ok(z0), ok(z1), ok(z2)) -> ok(U42(z0, z1, z2)) head(mark(z0)) -> mark(head(z0)) head(ok(z0)) -> ok(head(z0)) afterNth(mark(z0), z1) -> mark(afterNth(z0, z1)) afterNth(z0, mark(z1)) -> mark(afterNth(z0, z1)) afterNth(ok(z0), ok(z1)) -> ok(afterNth(z0, z1)) U51(mark(z0), z1) -> mark(U51(z0, z1)) U51(ok(z0), ok(z1)) -> ok(U51(z0, z1)) U52(mark(z0), z1) -> mark(U52(z0, z1)) U52(ok(z0), ok(z1)) -> ok(U52(z0, z1)) U61(mark(z0), z1, z2, z3) -> mark(U61(z0, z1, z2, z3)) U61(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U61(z0, z1, z2, z3)) U62(mark(z0), z1, z2, z3) -> mark(U62(z0, z1, z2, z3)) U62(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U62(z0, z1, z2, z3)) U63(mark(z0), z1, z2, z3) -> mark(U63(z0, z1, z2, z3)) U63(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U63(z0, z1, z2, z3)) U64(mark(z0), z1) -> mark(U64(z0, z1)) U64(ok(z0), ok(z1)) -> ok(U64(z0, z1)) pair(mark(z0), z1) -> mark(pair(z0, z1)) pair(z0, mark(z1)) -> mark(pair(z0, z1)) pair(ok(z0), ok(z1)) -> ok(pair(z0, z1)) cons(mark(z0), z1) -> mark(cons(z0, z1)) cons(ok(z0), ok(z1)) -> ok(cons(z0, z1)) U71(mark(z0), z1) -> mark(U71(z0, z1)) U71(ok(z0), ok(z1)) -> ok(U71(z0, z1)) U72(mark(z0), z1) -> mark(U72(z0, z1)) U72(ok(z0), ok(z1)) -> ok(U72(z0, z1)) U81(mark(z0), z1, z2) -> mark(U81(z0, z1, z2)) U81(ok(z0), ok(z1), ok(z2)) -> ok(U81(z0, z1, z2)) U82(mark(z0), z1, z2) -> mark(U82(z0, z1, z2)) U82(ok(z0), ok(z1), ok(z2)) -> ok(U82(z0, z1, z2)) fst(mark(z0)) -> mark(fst(z0)) fst(ok(z0)) -> ok(fst(z0)) natsFrom(mark(z0)) -> mark(natsFrom(z0)) natsFrom(ok(z0)) -> ok(natsFrom(z0)) s(mark(z0)) -> mark(s(z0)) s(ok(z0)) -> ok(s(z0)) sel(mark(z0), z1) -> mark(sel(z0, z1)) sel(z0, mark(z1)) -> mark(sel(z0, z1)) sel(ok(z0), ok(z1)) -> ok(sel(z0, z1)) tail(mark(z0)) -> mark(tail(z0)) tail(ok(z0)) -> ok(tail(z0)) take(mark(z0), z1) -> mark(take(z0, z1)) take(z0, mark(z1)) -> mark(take(z0, z1)) take(ok(z0), ok(z1)) -> ok(take(z0, z1)) proper(tt) -> ok(tt) proper(0) -> ok(0) proper(nil) -> ok(nil) top(mark(z0)) -> top(proper(z0)) top(ok(z0)) -> top(active(z0)) Tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) PROPER(tt) -> c65 PROPER(0) -> c66 PROPER(nil) -> c67 TOP(mark(z0)) -> c68(TOP(proper(z0)), PROPER(z0)) TOP(ok(z0)) -> c69(TOP(active(z0))) S tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) PROPER(tt) -> c65 PROPER(0) -> c66 PROPER(nil) -> c67 TOP(mark(z0)) -> c68(TOP(proper(z0)), PROPER(z0)) TOP(ok(z0)) -> c69(TOP(active(z0))) K tuples:none Defined Rule Symbols: U11_3, U12_3, snd_1, splitAt_2, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, head_1, afterNth_2, U51_2, U52_2, U61_4, U62_4, U63_4, U64_2, pair_2, cons_2, U71_2, U72_2, U81_3, U82_3, fst_1, natsFrom_1, s_1, sel_2, tail_1, take_2, proper_1, top_1 Defined Pair Symbols: U11'_3, U12'_3, SND_1, SPLITAT_2, U21'_2, U22'_2, U31'_2, U32'_2, U41'_3, U42'_3, HEAD_1, AFTERNTH_2, U51'_2, U52'_2, U61'_4, U62'_4, U63'_4, U64'_2, PAIR_2, CONS_2, U71'_2, U72'_2, U81'_3, U82'_3, FST_1, NATSFROM_1, S_1, SEL_2, TAIL_1, TAKE_2, PROPER_1, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65, c66, c67, c68_2, c69_1 ---------------------------------------- (7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: PROPER(0) -> c66 PROPER(tt) -> c65 PROPER(nil) -> c67 TOP(ok(z0)) -> c69(TOP(active(z0))) ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2)) U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2)) snd(mark(z0)) -> mark(snd(z0)) snd(ok(z0)) -> ok(snd(z0)) splitAt(mark(z0), z1) -> mark(splitAt(z0, z1)) splitAt(z0, mark(z1)) -> mark(splitAt(z0, z1)) splitAt(ok(z0), ok(z1)) -> ok(splitAt(z0, z1)) U21(mark(z0), z1) -> mark(U21(z0, z1)) U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) U22(mark(z0), z1) -> mark(U22(z0, z1)) U22(ok(z0), ok(z1)) -> ok(U22(z0, z1)) U31(mark(z0), z1) -> mark(U31(z0, z1)) U31(ok(z0), ok(z1)) -> ok(U31(z0, z1)) U32(mark(z0), z1) -> mark(U32(z0, z1)) U32(ok(z0), ok(z1)) -> ok(U32(z0, z1)) U41(mark(z0), z1, z2) -> mark(U41(z0, z1, z2)) U41(ok(z0), ok(z1), ok(z2)) -> ok(U41(z0, z1, z2)) U42(mark(z0), z1, z2) -> mark(U42(z0, z1, z2)) U42(ok(z0), ok(z1), ok(z2)) -> ok(U42(z0, z1, z2)) head(mark(z0)) -> mark(head(z0)) head(ok(z0)) -> ok(head(z0)) afterNth(mark(z0), z1) -> mark(afterNth(z0, z1)) afterNth(z0, mark(z1)) -> mark(afterNth(z0, z1)) afterNth(ok(z0), ok(z1)) -> ok(afterNth(z0, z1)) U51(mark(z0), z1) -> mark(U51(z0, z1)) U51(ok(z0), ok(z1)) -> ok(U51(z0, z1)) U52(mark(z0), z1) -> mark(U52(z0, z1)) U52(ok(z0), ok(z1)) -> ok(U52(z0, z1)) U61(mark(z0), z1, z2, z3) -> mark(U61(z0, z1, z2, z3)) U61(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U61(z0, z1, z2, z3)) U62(mark(z0), z1, z2, z3) -> mark(U62(z0, z1, z2, z3)) U62(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U62(z0, z1, z2, z3)) U63(mark(z0), z1, z2, z3) -> mark(U63(z0, z1, z2, z3)) U63(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U63(z0, z1, z2, z3)) U64(mark(z0), z1) -> mark(U64(z0, z1)) U64(ok(z0), ok(z1)) -> ok(U64(z0, z1)) pair(mark(z0), z1) -> mark(pair(z0, z1)) pair(z0, mark(z1)) -> mark(pair(z0, z1)) pair(ok(z0), ok(z1)) -> ok(pair(z0, z1)) cons(mark(z0), z1) -> mark(cons(z0, z1)) cons(ok(z0), ok(z1)) -> ok(cons(z0, z1)) U71(mark(z0), z1) -> mark(U71(z0, z1)) U71(ok(z0), ok(z1)) -> ok(U71(z0, z1)) U72(mark(z0), z1) -> mark(U72(z0, z1)) U72(ok(z0), ok(z1)) -> ok(U72(z0, z1)) U81(mark(z0), z1, z2) -> mark(U81(z0, z1, z2)) U81(ok(z0), ok(z1), ok(z2)) -> ok(U81(z0, z1, z2)) U82(mark(z0), z1, z2) -> mark(U82(z0, z1, z2)) U82(ok(z0), ok(z1), ok(z2)) -> ok(U82(z0, z1, z2)) fst(mark(z0)) -> mark(fst(z0)) fst(ok(z0)) -> ok(fst(z0)) natsFrom(mark(z0)) -> mark(natsFrom(z0)) natsFrom(ok(z0)) -> ok(natsFrom(z0)) s(mark(z0)) -> mark(s(z0)) s(ok(z0)) -> ok(s(z0)) sel(mark(z0), z1) -> mark(sel(z0, z1)) sel(z0, mark(z1)) -> mark(sel(z0, z1)) sel(ok(z0), ok(z1)) -> ok(sel(z0, z1)) tail(mark(z0)) -> mark(tail(z0)) tail(ok(z0)) -> ok(tail(z0)) take(mark(z0), z1) -> mark(take(z0, z1)) take(z0, mark(z1)) -> mark(take(z0, z1)) take(ok(z0), ok(z1)) -> ok(take(z0, z1)) proper(tt) -> ok(tt) proper(0) -> ok(0) proper(nil) -> ok(nil) top(mark(z0)) -> top(proper(z0)) top(ok(z0)) -> top(active(z0)) Tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) TOP(mark(z0)) -> c68(TOP(proper(z0)), PROPER(z0)) S tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) TOP(mark(z0)) -> c68(TOP(proper(z0)), PROPER(z0)) K tuples:none Defined Rule Symbols: U11_3, U12_3, snd_1, splitAt_2, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, head_1, afterNth_2, U51_2, U52_2, U61_4, U62_4, U63_4, U64_2, pair_2, cons_2, U71_2, U72_2, U81_3, U82_3, fst_1, natsFrom_1, s_1, sel_2, tail_1, take_2, proper_1, top_1 Defined Pair Symbols: U11'_3, U12'_3, SND_1, SPLITAT_2, U21'_2, U22'_2, U31'_2, U32'_2, U41'_3, U42'_3, HEAD_1, AFTERNTH_2, U51'_2, U52'_2, U61'_4, U62'_4, U63'_4, U64'_2, PAIR_2, CONS_2, U71'_2, U72'_2, U81'_3, U82'_3, FST_1, NATSFROM_1, S_1, SEL_2, TAIL_1, TAKE_2, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c68_2 ---------------------------------------- (9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2)) U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2)) snd(mark(z0)) -> mark(snd(z0)) snd(ok(z0)) -> ok(snd(z0)) splitAt(mark(z0), z1) -> mark(splitAt(z0, z1)) splitAt(z0, mark(z1)) -> mark(splitAt(z0, z1)) splitAt(ok(z0), ok(z1)) -> ok(splitAt(z0, z1)) U21(mark(z0), z1) -> mark(U21(z0, z1)) U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) U22(mark(z0), z1) -> mark(U22(z0, z1)) U22(ok(z0), ok(z1)) -> ok(U22(z0, z1)) U31(mark(z0), z1) -> mark(U31(z0, z1)) U31(ok(z0), ok(z1)) -> ok(U31(z0, z1)) U32(mark(z0), z1) -> mark(U32(z0, z1)) U32(ok(z0), ok(z1)) -> ok(U32(z0, z1)) U41(mark(z0), z1, z2) -> mark(U41(z0, z1, z2)) U41(ok(z0), ok(z1), ok(z2)) -> ok(U41(z0, z1, z2)) U42(mark(z0), z1, z2) -> mark(U42(z0, z1, z2)) U42(ok(z0), ok(z1), ok(z2)) -> ok(U42(z0, z1, z2)) head(mark(z0)) -> mark(head(z0)) head(ok(z0)) -> ok(head(z0)) afterNth(mark(z0), z1) -> mark(afterNth(z0, z1)) afterNth(z0, mark(z1)) -> mark(afterNth(z0, z1)) afterNth(ok(z0), ok(z1)) -> ok(afterNth(z0, z1)) U51(mark(z0), z1) -> mark(U51(z0, z1)) U51(ok(z0), ok(z1)) -> ok(U51(z0, z1)) U52(mark(z0), z1) -> mark(U52(z0, z1)) U52(ok(z0), ok(z1)) -> ok(U52(z0, z1)) U61(mark(z0), z1, z2, z3) -> mark(U61(z0, z1, z2, z3)) U61(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U61(z0, z1, z2, z3)) U62(mark(z0), z1, z2, z3) -> mark(U62(z0, z1, z2, z3)) U62(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U62(z0, z1, z2, z3)) U63(mark(z0), z1, z2, z3) -> mark(U63(z0, z1, z2, z3)) U63(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U63(z0, z1, z2, z3)) U64(mark(z0), z1) -> mark(U64(z0, z1)) U64(ok(z0), ok(z1)) -> ok(U64(z0, z1)) pair(mark(z0), z1) -> mark(pair(z0, z1)) pair(z0, mark(z1)) -> mark(pair(z0, z1)) pair(ok(z0), ok(z1)) -> ok(pair(z0, z1)) cons(mark(z0), z1) -> mark(cons(z0, z1)) cons(ok(z0), ok(z1)) -> ok(cons(z0, z1)) U71(mark(z0), z1) -> mark(U71(z0, z1)) U71(ok(z0), ok(z1)) -> ok(U71(z0, z1)) U72(mark(z0), z1) -> mark(U72(z0, z1)) U72(ok(z0), ok(z1)) -> ok(U72(z0, z1)) U81(mark(z0), z1, z2) -> mark(U81(z0, z1, z2)) U81(ok(z0), ok(z1), ok(z2)) -> ok(U81(z0, z1, z2)) U82(mark(z0), z1, z2) -> mark(U82(z0, z1, z2)) U82(ok(z0), ok(z1), ok(z2)) -> ok(U82(z0, z1, z2)) fst(mark(z0)) -> mark(fst(z0)) fst(ok(z0)) -> ok(fst(z0)) natsFrom(mark(z0)) -> mark(natsFrom(z0)) natsFrom(ok(z0)) -> ok(natsFrom(z0)) s(mark(z0)) -> mark(s(z0)) s(ok(z0)) -> ok(s(z0)) sel(mark(z0), z1) -> mark(sel(z0, z1)) sel(z0, mark(z1)) -> mark(sel(z0, z1)) sel(ok(z0), ok(z1)) -> ok(sel(z0, z1)) tail(mark(z0)) -> mark(tail(z0)) tail(ok(z0)) -> ok(tail(z0)) take(mark(z0), z1) -> mark(take(z0, z1)) take(z0, mark(z1)) -> mark(take(z0, z1)) take(ok(z0), ok(z1)) -> ok(take(z0, z1)) proper(tt) -> ok(tt) proper(0) -> ok(0) proper(nil) -> ok(nil) top(mark(z0)) -> top(proper(z0)) top(ok(z0)) -> top(active(z0)) Tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) TOP(mark(z0)) -> c68(TOP(proper(z0))) S tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) TOP(mark(z0)) -> c68(TOP(proper(z0))) K tuples:none Defined Rule Symbols: U11_3, U12_3, snd_1, splitAt_2, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, head_1, afterNth_2, U51_2, U52_2, U61_4, U62_4, U63_4, U64_2, pair_2, cons_2, U71_2, U72_2, U81_3, U82_3, fst_1, natsFrom_1, s_1, sel_2, tail_1, take_2, proper_1, top_1 Defined Pair Symbols: U11'_3, U12'_3, SND_1, SPLITAT_2, U21'_2, U22'_2, U31'_2, U32'_2, U41'_3, U42'_3, HEAD_1, AFTERNTH_2, U51'_2, U52'_2, U61'_4, U62'_4, U63'_4, U64'_2, PAIR_2, CONS_2, U71'_2, U72'_2, U81'_3, U82'_3, FST_1, NATSFROM_1, S_1, SEL_2, TAIL_1, TAKE_2, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c68_1 ---------------------------------------- (11) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2)) U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2)) snd(mark(z0)) -> mark(snd(z0)) snd(ok(z0)) -> ok(snd(z0)) splitAt(mark(z0), z1) -> mark(splitAt(z0, z1)) splitAt(z0, mark(z1)) -> mark(splitAt(z0, z1)) splitAt(ok(z0), ok(z1)) -> ok(splitAt(z0, z1)) U21(mark(z0), z1) -> mark(U21(z0, z1)) U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) U22(mark(z0), z1) -> mark(U22(z0, z1)) U22(ok(z0), ok(z1)) -> ok(U22(z0, z1)) U31(mark(z0), z1) -> mark(U31(z0, z1)) U31(ok(z0), ok(z1)) -> ok(U31(z0, z1)) U32(mark(z0), z1) -> mark(U32(z0, z1)) U32(ok(z0), ok(z1)) -> ok(U32(z0, z1)) U41(mark(z0), z1, z2) -> mark(U41(z0, z1, z2)) U41(ok(z0), ok(z1), ok(z2)) -> ok(U41(z0, z1, z2)) U42(mark(z0), z1, z2) -> mark(U42(z0, z1, z2)) U42(ok(z0), ok(z1), ok(z2)) -> ok(U42(z0, z1, z2)) head(mark(z0)) -> mark(head(z0)) head(ok(z0)) -> ok(head(z0)) afterNth(mark(z0), z1) -> mark(afterNth(z0, z1)) afterNth(z0, mark(z1)) -> mark(afterNth(z0, z1)) afterNth(ok(z0), ok(z1)) -> ok(afterNth(z0, z1)) U51(mark(z0), z1) -> mark(U51(z0, z1)) U51(ok(z0), ok(z1)) -> ok(U51(z0, z1)) U52(mark(z0), z1) -> mark(U52(z0, z1)) U52(ok(z0), ok(z1)) -> ok(U52(z0, z1)) U61(mark(z0), z1, z2, z3) -> mark(U61(z0, z1, z2, z3)) U61(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U61(z0, z1, z2, z3)) U62(mark(z0), z1, z2, z3) -> mark(U62(z0, z1, z2, z3)) U62(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U62(z0, z1, z2, z3)) U63(mark(z0), z1, z2, z3) -> mark(U63(z0, z1, z2, z3)) U63(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U63(z0, z1, z2, z3)) U64(mark(z0), z1) -> mark(U64(z0, z1)) U64(ok(z0), ok(z1)) -> ok(U64(z0, z1)) pair(mark(z0), z1) -> mark(pair(z0, z1)) pair(z0, mark(z1)) -> mark(pair(z0, z1)) pair(ok(z0), ok(z1)) -> ok(pair(z0, z1)) cons(mark(z0), z1) -> mark(cons(z0, z1)) cons(ok(z0), ok(z1)) -> ok(cons(z0, z1)) U71(mark(z0), z1) -> mark(U71(z0, z1)) U71(ok(z0), ok(z1)) -> ok(U71(z0, z1)) U72(mark(z0), z1) -> mark(U72(z0, z1)) U72(ok(z0), ok(z1)) -> ok(U72(z0, z1)) U81(mark(z0), z1, z2) -> mark(U81(z0, z1, z2)) U81(ok(z0), ok(z1), ok(z2)) -> ok(U81(z0, z1, z2)) U82(mark(z0), z1, z2) -> mark(U82(z0, z1, z2)) U82(ok(z0), ok(z1), ok(z2)) -> ok(U82(z0, z1, z2)) fst(mark(z0)) -> mark(fst(z0)) fst(ok(z0)) -> ok(fst(z0)) natsFrom(mark(z0)) -> mark(natsFrom(z0)) natsFrom(ok(z0)) -> ok(natsFrom(z0)) s(mark(z0)) -> mark(s(z0)) s(ok(z0)) -> ok(s(z0)) sel(mark(z0), z1) -> mark(sel(z0, z1)) sel(z0, mark(z1)) -> mark(sel(z0, z1)) sel(ok(z0), ok(z1)) -> ok(sel(z0, z1)) tail(mark(z0)) -> mark(tail(z0)) tail(ok(z0)) -> ok(tail(z0)) take(mark(z0), z1) -> mark(take(z0, z1)) take(z0, mark(z1)) -> mark(take(z0, z1)) take(ok(z0), ok(z1)) -> ok(take(z0, z1)) top(mark(z0)) -> top(proper(z0)) top(ok(z0)) -> top(active(z0)) ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) proper(nil) -> ok(nil) Tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) TOP(mark(z0)) -> c68(TOP(proper(z0))) S tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) TOP(mark(z0)) -> c68(TOP(proper(z0))) K tuples:none Defined Rule Symbols: proper_1 Defined Pair Symbols: U11'_3, U12'_3, SND_1, SPLITAT_2, U21'_2, U22'_2, U31'_2, U32'_2, U41'_3, U42'_3, HEAD_1, AFTERNTH_2, U51'_2, U52'_2, U61'_4, U62'_4, U63'_4, U64'_2, PAIR_2, CONS_2, U71'_2, U72'_2, U81'_3, U82'_3, FST_1, NATSFROM_1, S_1, SEL_2, TAIL_1, TAKE_2, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c68_1 ---------------------------------------- (13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. TOP(mark(z0)) -> c68(TOP(proper(z0))) We considered the (Usable) Rules: proper(tt) -> ok(tt) proper(nil) -> ok(nil) proper(0) -> ok(0) And the Tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) TOP(mark(z0)) -> c68(TOP(proper(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(AFTERNTH(x_1, x_2)) = 0 POL(CONS(x_1, x_2)) = 0 POL(FST(x_1)) = 0 POL(HEAD(x_1)) = 0 POL(NATSFROM(x_1)) = 0 POL(PAIR(x_1, x_2)) = 0 POL(S(x_1)) = 0 POL(SEL(x_1, x_2)) = 0 POL(SND(x_1)) = 0 POL(SPLITAT(x_1, x_2)) = 0 POL(TAIL(x_1)) = 0 POL(TAKE(x_1, x_2)) = 0 POL(TOP(x_1)) = [2]x_1 POL(U11'(x_1, x_2, x_3)) = 0 POL(U12'(x_1, x_2, x_3)) = 0 POL(U21'(x_1, x_2)) = 0 POL(U22'(x_1, x_2)) = 0 POL(U31'(x_1, x_2)) = 0 POL(U32'(x_1, x_2)) = 0 POL(U41'(x_1, x_2, x_3)) = 0 POL(U42'(x_1, x_2, x_3)) = 0 POL(U51'(x_1, x_2)) = 0 POL(U52'(x_1, x_2)) = 0 POL(U61'(x_1, x_2, x_3, x_4)) = 0 POL(U62'(x_1, x_2, x_3, x_4)) = 0 POL(U63'(x_1, x_2, x_3, x_4)) = 0 POL(U64'(x_1, x_2)) = 0 POL(U71'(x_1, x_2)) = 0 POL(U72'(x_1, x_2)) = 0 POL(U81'(x_1, x_2, x_3)) = 0 POL(U82'(x_1, x_2, x_3)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c33(x_1)) = x_1 POL(c34(x_1)) = x_1 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c37(x_1)) = x_1 POL(c38(x_1)) = x_1 POL(c39(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c40(x_1)) = x_1 POL(c41(x_1)) = x_1 POL(c42(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c44(x_1)) = x_1 POL(c45(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(c47(x_1)) = x_1 POL(c48(x_1)) = x_1 POL(c49(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c50(x_1)) = x_1 POL(c51(x_1)) = x_1 POL(c52(x_1)) = x_1 POL(c53(x_1)) = x_1 POL(c54(x_1)) = x_1 POL(c55(x_1)) = x_1 POL(c56(x_1)) = x_1 POL(c57(x_1)) = x_1 POL(c58(x_1)) = x_1 POL(c59(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c60(x_1)) = x_1 POL(c61(x_1)) = x_1 POL(c62(x_1)) = x_1 POL(c63(x_1)) = x_1 POL(c64(x_1)) = x_1 POL(c68(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(mark(x_1)) = [2] + x_1 POL(nil) = 0 POL(ok(x_1)) = 0 POL(proper(x_1)) = x_1 POL(tt) = 0 ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) proper(nil) -> ok(nil) Tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) TOP(mark(z0)) -> c68(TOP(proper(z0))) S tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) K tuples: TOP(mark(z0)) -> c68(TOP(proper(z0))) Defined Rule Symbols: proper_1 Defined Pair Symbols: U11'_3, U12'_3, SND_1, SPLITAT_2, U21'_2, U22'_2, U31'_2, U32'_2, U41'_3, U42'_3, HEAD_1, AFTERNTH_2, U51'_2, U52'_2, U61'_4, U62'_4, U63'_4, U64'_2, PAIR_2, CONS_2, U71'_2, U72'_2, U81'_3, U82'_3, FST_1, NATSFROM_1, S_1, SEL_2, TAIL_1, TAKE_2, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c68_1 ---------------------------------------- (15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) We considered the (Usable) Rules:none And the Tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) TOP(mark(z0)) -> c68(TOP(proper(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [3] POL(AFTERNTH(x_1, x_2)) = 0 POL(CONS(x_1, x_2)) = 0 POL(FST(x_1)) = [3]x_1 POL(HEAD(x_1)) = [3]x_1 POL(NATSFROM(x_1)) = [3]x_1 POL(PAIR(x_1, x_2)) = 0 POL(S(x_1)) = [3]x_1 POL(SEL(x_1, x_2)) = 0 POL(SND(x_1)) = [3]x_1 POL(SPLITAT(x_1, x_2)) = 0 POL(TAIL(x_1)) = [3]x_1 POL(TAKE(x_1, x_2)) = [2]x_1 POL(TOP(x_1)) = 0 POL(U11'(x_1, x_2, x_3)) = 0 POL(U12'(x_1, x_2, x_3)) = 0 POL(U21'(x_1, x_2)) = 0 POL(U22'(x_1, x_2)) = 0 POL(U31'(x_1, x_2)) = 0 POL(U32'(x_1, x_2)) = 0 POL(U41'(x_1, x_2, x_3)) = 0 POL(U42'(x_1, x_2, x_3)) = 0 POL(U51'(x_1, x_2)) = 0 POL(U52'(x_1, x_2)) = 0 POL(U61'(x_1, x_2, x_3, x_4)) = 0 POL(U62'(x_1, x_2, x_3, x_4)) = 0 POL(U63'(x_1, x_2, x_3, x_4)) = 0 POL(U64'(x_1, x_2)) = 0 POL(U71'(x_1, x_2)) = 0 POL(U72'(x_1, x_2)) = 0 POL(U81'(x_1, x_2, x_3)) = 0 POL(U82'(x_1, x_2, x_3)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c33(x_1)) = x_1 POL(c34(x_1)) = x_1 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c37(x_1)) = x_1 POL(c38(x_1)) = x_1 POL(c39(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c40(x_1)) = x_1 POL(c41(x_1)) = x_1 POL(c42(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c44(x_1)) = x_1 POL(c45(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(c47(x_1)) = x_1 POL(c48(x_1)) = x_1 POL(c49(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c50(x_1)) = x_1 POL(c51(x_1)) = x_1 POL(c52(x_1)) = x_1 POL(c53(x_1)) = x_1 POL(c54(x_1)) = x_1 POL(c55(x_1)) = x_1 POL(c56(x_1)) = x_1 POL(c57(x_1)) = x_1 POL(c58(x_1)) = x_1 POL(c59(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c60(x_1)) = x_1 POL(c61(x_1)) = x_1 POL(c62(x_1)) = x_1 POL(c63(x_1)) = x_1 POL(c64(x_1)) = x_1 POL(c68(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(mark(x_1)) = [1] + x_1 POL(nil) = [3] POL(ok(x_1)) = [1] + x_1 POL(proper(x_1)) = [1] + [3]x_1 POL(tt) = [1] ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) proper(nil) -> ok(nil) Tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) TOP(mark(z0)) -> c68(TOP(proper(z0))) S tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) K tuples: TOP(mark(z0)) -> c68(TOP(proper(z0))) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) Defined Rule Symbols: proper_1 Defined Pair Symbols: U11'_3, U12'_3, SND_1, SPLITAT_2, U21'_2, U22'_2, U31'_2, U32'_2, U41'_3, U42'_3, HEAD_1, AFTERNTH_2, U51'_2, U52'_2, U61'_4, U62'_4, U63'_4, U64'_2, PAIR_2, CONS_2, U71'_2, U72'_2, U81'_3, U82'_3, FST_1, NATSFROM_1, S_1, SEL_2, TAIL_1, TAKE_2, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c68_1 ---------------------------------------- (17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) We considered the (Usable) Rules:none And the Tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) TOP(mark(z0)) -> c68(TOP(proper(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(AFTERNTH(x_1, x_2)) = 0 POL(CONS(x_1, x_2)) = 0 POL(FST(x_1)) = [3]x_1 POL(HEAD(x_1)) = [3]x_1 POL(NATSFROM(x_1)) = [3]x_1 POL(PAIR(x_1, x_2)) = 0 POL(S(x_1)) = [3]x_1 POL(SEL(x_1, x_2)) = x_2 POL(SND(x_1)) = [3]x_1 POL(SPLITAT(x_1, x_2)) = 0 POL(TAIL(x_1)) = [3]x_1 POL(TAKE(x_1, x_2)) = [3]x_1 + [3]x_2 POL(TOP(x_1)) = 0 POL(U11'(x_1, x_2, x_3)) = 0 POL(U12'(x_1, x_2, x_3)) = 0 POL(U21'(x_1, x_2)) = 0 POL(U22'(x_1, x_2)) = 0 POL(U31'(x_1, x_2)) = 0 POL(U32'(x_1, x_2)) = 0 POL(U41'(x_1, x_2, x_3)) = 0 POL(U42'(x_1, x_2, x_3)) = 0 POL(U51'(x_1, x_2)) = 0 POL(U52'(x_1, x_2)) = 0 POL(U61'(x_1, x_2, x_3, x_4)) = 0 POL(U62'(x_1, x_2, x_3, x_4)) = 0 POL(U63'(x_1, x_2, x_3, x_4)) = 0 POL(U64'(x_1, x_2)) = 0 POL(U71'(x_1, x_2)) = 0 POL(U72'(x_1, x_2)) = 0 POL(U81'(x_1, x_2, x_3)) = 0 POL(U82'(x_1, x_2, x_3)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c33(x_1)) = x_1 POL(c34(x_1)) = x_1 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c37(x_1)) = x_1 POL(c38(x_1)) = x_1 POL(c39(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c40(x_1)) = x_1 POL(c41(x_1)) = x_1 POL(c42(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c44(x_1)) = x_1 POL(c45(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(c47(x_1)) = x_1 POL(c48(x_1)) = x_1 POL(c49(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c50(x_1)) = x_1 POL(c51(x_1)) = x_1 POL(c52(x_1)) = x_1 POL(c53(x_1)) = x_1 POL(c54(x_1)) = x_1 POL(c55(x_1)) = x_1 POL(c56(x_1)) = x_1 POL(c57(x_1)) = x_1 POL(c58(x_1)) = x_1 POL(c59(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c60(x_1)) = x_1 POL(c61(x_1)) = x_1 POL(c62(x_1)) = x_1 POL(c63(x_1)) = x_1 POL(c64(x_1)) = x_1 POL(c68(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(mark(x_1)) = [1] + x_1 POL(nil) = [3] POL(ok(x_1)) = [1] + x_1 POL(proper(x_1)) = [1] + [3]x_1 POL(tt) = 0 ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) proper(nil) -> ok(nil) Tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) TOP(mark(z0)) -> c68(TOP(proper(z0))) S tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) K tuples: TOP(mark(z0)) -> c68(TOP(proper(z0))) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) Defined Rule Symbols: proper_1 Defined Pair Symbols: U11'_3, U12'_3, SND_1, SPLITAT_2, U21'_2, U22'_2, U31'_2, U32'_2, U41'_3, U42'_3, HEAD_1, AFTERNTH_2, U51'_2, U52'_2, U61'_4, U62'_4, U63'_4, U64'_2, PAIR_2, CONS_2, U71'_2, U72'_2, U81'_3, U82'_3, FST_1, NATSFROM_1, S_1, SEL_2, TAIL_1, TAKE_2, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c68_1 ---------------------------------------- (19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) We considered the (Usable) Rules:none And the Tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) TOP(mark(z0)) -> c68(TOP(proper(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(AFTERNTH(x_1, x_2)) = 0 POL(CONS(x_1, x_2)) = 0 POL(FST(x_1)) = [3]x_1 POL(HEAD(x_1)) = [3]x_1 POL(NATSFROM(x_1)) = [3]x_1 POL(PAIR(x_1, x_2)) = 0 POL(S(x_1)) = [3]x_1 POL(SEL(x_1, x_2)) = [3]x_1 + [3]x_2 POL(SND(x_1)) = [3]x_1 POL(SPLITAT(x_1, x_2)) = 0 POL(TAIL(x_1)) = [3]x_1 POL(TAKE(x_1, x_2)) = [3]x_1 + [3]x_2 POL(TOP(x_1)) = 0 POL(U11'(x_1, x_2, x_3)) = 0 POL(U12'(x_1, x_2, x_3)) = 0 POL(U21'(x_1, x_2)) = 0 POL(U22'(x_1, x_2)) = 0 POL(U31'(x_1, x_2)) = 0 POL(U32'(x_1, x_2)) = 0 POL(U41'(x_1, x_2, x_3)) = 0 POL(U42'(x_1, x_2, x_3)) = 0 POL(U51'(x_1, x_2)) = 0 POL(U52'(x_1, x_2)) = 0 POL(U61'(x_1, x_2, x_3, x_4)) = 0 POL(U62'(x_1, x_2, x_3, x_4)) = 0 POL(U63'(x_1, x_2, x_3, x_4)) = 0 POL(U64'(x_1, x_2)) = x_1 POL(U71'(x_1, x_2)) = 0 POL(U72'(x_1, x_2)) = 0 POL(U81'(x_1, x_2, x_3)) = 0 POL(U82'(x_1, x_2, x_3)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c33(x_1)) = x_1 POL(c34(x_1)) = x_1 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c37(x_1)) = x_1 POL(c38(x_1)) = x_1 POL(c39(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c40(x_1)) = x_1 POL(c41(x_1)) = x_1 POL(c42(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c44(x_1)) = x_1 POL(c45(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(c47(x_1)) = x_1 POL(c48(x_1)) = x_1 POL(c49(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c50(x_1)) = x_1 POL(c51(x_1)) = x_1 POL(c52(x_1)) = x_1 POL(c53(x_1)) = x_1 POL(c54(x_1)) = x_1 POL(c55(x_1)) = x_1 POL(c56(x_1)) = x_1 POL(c57(x_1)) = x_1 POL(c58(x_1)) = x_1 POL(c59(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c60(x_1)) = x_1 POL(c61(x_1)) = x_1 POL(c62(x_1)) = x_1 POL(c63(x_1)) = x_1 POL(c64(x_1)) = x_1 POL(c68(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(mark(x_1)) = [1] + x_1 POL(nil) = [3] POL(ok(x_1)) = [1] + x_1 POL(proper(x_1)) = [1] + [3]x_1 POL(tt) = 0 ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) proper(nil) -> ok(nil) Tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) TOP(mark(z0)) -> c68(TOP(proper(z0))) S tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) K tuples: TOP(mark(z0)) -> c68(TOP(proper(z0))) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) Defined Rule Symbols: proper_1 Defined Pair Symbols: U11'_3, U12'_3, SND_1, SPLITAT_2, U21'_2, U22'_2, U31'_2, U32'_2, U41'_3, U42'_3, HEAD_1, AFTERNTH_2, U51'_2, U52'_2, U61'_4, U62'_4, U63'_4, U64'_2, PAIR_2, CONS_2, U71'_2, U72'_2, U81'_3, U82'_3, FST_1, NATSFROM_1, S_1, SEL_2, TAIL_1, TAKE_2, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c68_1 ---------------------------------------- (21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) We considered the (Usable) Rules: proper(tt) -> ok(tt) proper(nil) -> ok(nil) proper(0) -> ok(0) And the Tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) TOP(mark(z0)) -> c68(TOP(proper(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(AFTERNTH(x_1, x_2)) = x_1 + x_2 POL(CONS(x_1, x_2)) = x_1 + x_2 POL(FST(x_1)) = x_1 POL(HEAD(x_1)) = x_1 POL(NATSFROM(x_1)) = x_1 POL(PAIR(x_1, x_2)) = x_1 + x_2 POL(S(x_1)) = x_1 POL(SEL(x_1, x_2)) = x_1 + x_2 POL(SND(x_1)) = x_1 POL(SPLITAT(x_1, x_2)) = x_1 + x_2 POL(TAIL(x_1)) = x_1 POL(TAKE(x_1, x_2)) = x_1 + x_2 POL(TOP(x_1)) = x_1 POL(U11'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U12'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U21'(x_1, x_2)) = x_1 + x_2 POL(U22'(x_1, x_2)) = x_1 + x_2 POL(U31'(x_1, x_2)) = x_1 + x_2 POL(U32'(x_1, x_2)) = x_1 + x_2 POL(U41'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U42'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U51'(x_1, x_2)) = x_1 + x_2 POL(U52'(x_1, x_2)) = x_1 + x_2 POL(U61'(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(U62'(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(U63'(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(U64'(x_1, x_2)) = x_1 + x_2 POL(U71'(x_1, x_2)) = x_1 + x_2 POL(U72'(x_1, x_2)) = x_1 + x_2 POL(U81'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U82'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c33(x_1)) = x_1 POL(c34(x_1)) = x_1 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c37(x_1)) = x_1 POL(c38(x_1)) = x_1 POL(c39(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c40(x_1)) = x_1 POL(c41(x_1)) = x_1 POL(c42(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c44(x_1)) = x_1 POL(c45(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(c47(x_1)) = x_1 POL(c48(x_1)) = x_1 POL(c49(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c50(x_1)) = x_1 POL(c51(x_1)) = x_1 POL(c52(x_1)) = x_1 POL(c53(x_1)) = x_1 POL(c54(x_1)) = x_1 POL(c55(x_1)) = x_1 POL(c56(x_1)) = x_1 POL(c57(x_1)) = x_1 POL(c58(x_1)) = x_1 POL(c59(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c60(x_1)) = x_1 POL(c61(x_1)) = x_1 POL(c62(x_1)) = x_1 POL(c63(x_1)) = x_1 POL(c64(x_1)) = x_1 POL(c68(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(mark(x_1)) = [1] + x_1 POL(nil) = [1] POL(ok(x_1)) = [1] + x_1 POL(proper(x_1)) = [1] + x_1 POL(tt) = [1] ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) proper(nil) -> ok(nil) Tuples: U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) TOP(mark(z0)) -> c68(TOP(proper(z0))) S tuples:none K tuples: TOP(mark(z0)) -> c68(TOP(proper(z0))) SND(mark(z0)) -> c4(SND(z0)) SND(ok(z0)) -> c5(SND(z0)) HEAD(mark(z0)) -> c21(HEAD(z0)) HEAD(ok(z0)) -> c22(HEAD(z0)) FST(mark(z0)) -> c51(FST(z0)) FST(ok(z0)) -> c52(FST(z0)) NATSFROM(mark(z0)) -> c53(NATSFROM(z0)) NATSFROM(ok(z0)) -> c54(NATSFROM(z0)) S(mark(z0)) -> c55(S(z0)) S(ok(z0)) -> c56(S(z0)) TAIL(mark(z0)) -> c60(TAIL(z0)) TAIL(ok(z0)) -> c61(TAIL(z0)) TAKE(mark(z0), z1) -> c62(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c64(TAKE(z0, z1)) SEL(z0, mark(z1)) -> c58(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c59(SEL(z0, z1)) TAKE(z0, mark(z1)) -> c63(TAKE(z0, z1)) U64'(mark(z0), z1) -> c36(U64'(z0, z1)) U64'(ok(z0), ok(z1)) -> c37(U64'(z0, z1)) SEL(mark(z0), z1) -> c57(SEL(z0, z1)) U11'(mark(z0), z1, z2) -> c(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c1(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c2(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c3(U12'(z0, z1, z2)) SPLITAT(mark(z0), z1) -> c6(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c7(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c8(SPLITAT(z0, z1)) U21'(mark(z0), z1) -> c9(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c10(U21'(z0, z1)) U22'(mark(z0), z1) -> c11(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c12(U22'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U32'(mark(z0), z1) -> c15(U32'(z0, z1)) U32'(ok(z0), ok(z1)) -> c16(U32'(z0, z1)) U41'(mark(z0), z1, z2) -> c17(U41'(z0, z1, z2)) U41'(ok(z0), ok(z1), ok(z2)) -> c18(U41'(z0, z1, z2)) U42'(mark(z0), z1, z2) -> c19(U42'(z0, z1, z2)) U42'(ok(z0), ok(z1), ok(z2)) -> c20(U42'(z0, z1, z2)) AFTERNTH(mark(z0), z1) -> c23(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c24(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c25(AFTERNTH(z0, z1)) U51'(mark(z0), z1) -> c26(U51'(z0, z1)) U51'(ok(z0), ok(z1)) -> c27(U51'(z0, z1)) U52'(mark(z0), z1) -> c28(U52'(z0, z1)) U52'(ok(z0), ok(z1)) -> c29(U52'(z0, z1)) U61'(mark(z0), z1, z2, z3) -> c30(U61'(z0, z1, z2, z3)) U61'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c31(U61'(z0, z1, z2, z3)) U62'(mark(z0), z1, z2, z3) -> c32(U62'(z0, z1, z2, z3)) U62'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c33(U62'(z0, z1, z2, z3)) U63'(mark(z0), z1, z2, z3) -> c34(U63'(z0, z1, z2, z3)) U63'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c35(U63'(z0, z1, z2, z3)) PAIR(mark(z0), z1) -> c38(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c39(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c40(PAIR(z0, z1)) CONS(mark(z0), z1) -> c41(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c42(CONS(z0, z1)) U71'(mark(z0), z1) -> c43(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c44(U71'(z0, z1)) U72'(mark(z0), z1) -> c45(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c46(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c47(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c48(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c49(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c50(U82'(z0, z1, z2)) Defined Rule Symbols: proper_1 Defined Pair Symbols: U11'_3, U12'_3, SND_1, SPLITAT_2, U21'_2, U22'_2, U31'_2, U32'_2, U41'_3, U42'_3, HEAD_1, AFTERNTH_2, U51'_2, U52'_2, U61'_4, U62'_4, U63'_4, U64'_2, PAIR_2, CONS_2, U71'_2, U72'_2, U81'_3, U82'_3, FST_1, NATSFROM_1, S_1, SEL_2, TAIL_1, TAKE_2, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c68_1 ---------------------------------------- (23) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (24) BOUNDS(1, 1) ---------------------------------------- (25) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (26) 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(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) 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 ---------------------------------------- (27) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (28) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok ---------------------------------------- (29) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: active, U12, snd, splitAt, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U12 < active snd < active splitAt < active U22 < active U32 < active U42 < active head < active afterNth < active U52 < active U62 < active U63 < active U64 < active pair < active cons < active U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U12 < proper snd < proper splitAt < proper U22 < proper U32 < proper U42 < proper head < proper afterNth < proper U52 < proper U62 < proper U63 < proper U64 < proper pair < proper cons < proper U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (30) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U12, active, snd, splitAt, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U12 < active snd < active splitAt < active U22 < active U32 < active U42 < active head < active afterNth < active U52 < active U62 < active U63 < active U64 < active pair < active cons < active U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U12 < proper snd < proper splitAt < proper U22 < proper U32 < proper U42 < proper head < proper afterNth < proper U52 < proper U62 < proper U63 < proper U64 < proper pair < proper cons < proper U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (31) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) Induction Base: U12(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) Induction Step: U12(gen_tt:mark:0':nil:ok3_0(+(1, +(n5_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) ->_R^Omega(1) mark(U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':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). ---------------------------------------- (32) Complex Obligation (BEST) ---------------------------------------- (33) Obligation: Proved the lower bound n^1 for the following obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U12, active, snd, splitAt, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U12 < active snd < active splitAt < active U22 < active U32 < active U42 < active head < active afterNth < active U52 < active U62 < active U63 < active U64 < active pair < active cons < active U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U12 < proper snd < proper splitAt < proper U22 < proper U32 < proper U42 < proper head < proper afterNth < proper U52 < proper U62 < proper U63 < proper U64 < proper pair < proper cons < proper U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (34) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (35) BOUNDS(n^1, INF) ---------------------------------------- (36) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: snd, active, splitAt, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: snd < active splitAt < active U22 < active U32 < active U42 < active head < active afterNth < active U52 < active U62 < active U63 < active U64 < active pair < active cons < active U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top snd < proper splitAt < proper U22 < proper U32 < proper U42 < proper head < proper afterNth < proper U52 < proper U62 < proper U63 < proper U64 < proper pair < proper cons < proper U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (37) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) Induction Base: snd(gen_tt:mark:0':nil:ok3_0(+(1, 0))) Induction Step: snd(gen_tt:mark:0':nil:ok3_0(+(1, +(n4683_0, 1)))) ->_R^Omega(1) mark(snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_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). ---------------------------------------- (38) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: splitAt, active, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: splitAt < active U22 < active U32 < active U42 < active head < active afterNth < active U52 < active U62 < active U63 < active U64 < active pair < active cons < active U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top splitAt < proper U22 < proper U32 < proper U42 < proper head < proper afterNth < proper U52 < proper U62 < proper U63 < proper U64 < proper pair < proper cons < proper U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (39) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) Induction Base: splitAt(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b)) Induction Step: splitAt(gen_tt:mark:0':nil:ok3_0(+(1, +(n5638_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1) mark(splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':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(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U22, active, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U22 < active U32 < active U42 < active head < active afterNth < active U52 < active U62 < active U63 < active U64 < active pair < active cons < active U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U22 < proper U32 < proper U42 < proper head < proper afterNth < proper U52 < proper U62 < proper U63 < proper U64 < proper pair < proper cons < proper U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (41) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) Induction Base: U22(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b)) Induction Step: U22(gen_tt:mark:0':nil:ok3_0(+(1, +(n9166_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1) mark(U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':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(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U32, active, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U32 < active U42 < active head < active afterNth < active U52 < active U62 < active U63 < active U64 < active pair < active cons < active U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U32 < proper U42 < proper head < proper afterNth < proper U52 < proper U62 < proper U63 < proper U64 < proper pair < proper cons < proper U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (43) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) Induction Base: U32(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b)) Induction Step: U32(gen_tt:mark:0':nil:ok3_0(+(1, +(n12801_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1) mark(U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':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). ---------------------------------------- (44) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U42, active, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U42 < active head < active afterNth < active U52 < active U62 < active U63 < active U64 < active pair < active cons < active U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U42 < proper head < proper afterNth < proper U52 < proper U62 < proper U63 < proper U64 < proper pair < proper cons < proper U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (45) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) Induction Base: U42(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) Induction Step: U42(gen_tt:mark:0':nil:ok3_0(+(1, +(n16742_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) ->_R^Omega(1) mark(U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':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). ---------------------------------------- (46) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: head, active, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: head < active afterNth < active U52 < active U62 < active U63 < active U64 < active pair < active cons < active U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top head < proper afterNth < proper U52 < proper U62 < proper U63 < proper U64 < proper pair < proper cons < proper U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (47) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) Induction Base: head(gen_tt:mark:0':nil:ok3_0(+(1, 0))) Induction Step: head(gen_tt:mark:0':nil:ok3_0(+(1, +(n23730_0, 1)))) ->_R^Omega(1) mark(head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_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). ---------------------------------------- (48) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: afterNth, active, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: afterNth < active U52 < active U62 < active U63 < active U64 < active pair < active cons < active U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top afterNth < proper U52 < proper U62 < proper U63 < proper U64 < proper pair < proper cons < proper U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (49) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) Induction Base: afterNth(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b)) Induction Step: afterNth(gen_tt:mark:0':nil:ok3_0(+(1, +(n25435_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1) mark(afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':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). ---------------------------------------- (50) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U52, active, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U52 < active U62 < active U63 < active U64 < active pair < active cons < active U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U52 < proper U62 < proper U63 < proper U64 < proper pair < proper cons < proper U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (51) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) Induction Base: U52(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b)) Induction Step: U52(gen_tt:mark:0':nil:ok3_0(+(1, +(n30493_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1) mark(U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':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). ---------------------------------------- (52) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U62, active, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U62 < active U63 < active U64 < active pair < active cons < active U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U62 < proper U63 < proper U64 < proper pair < proper cons < proper U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (53) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n35658_0) Induction Base: U62(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) Induction Step: U62(gen_tt:mark:0':nil:ok3_0(+(1, +(n35658_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) ->_R^Omega(1) mark(U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':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). ---------------------------------------- (54) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n35658_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U63, active, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U63 < active U64 < active pair < active cons < active U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U63 < proper U64 < proper pair < proper cons < proper U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (55) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U63(gen_tt:mark:0':nil:ok3_0(+(1, n47859_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n47859_0) Induction Base: U63(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) Induction Step: U63(gen_tt:mark:0':nil:ok3_0(+(1, +(n47859_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) ->_R^Omega(1) mark(U63(gen_tt:mark:0':nil:ok3_0(+(1, n47859_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':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). ---------------------------------------- (56) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n35658_0) U63(gen_tt:mark:0':nil:ok3_0(+(1, n47859_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n47859_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U64, active, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U64 < active pair < active cons < active U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U64 < proper pair < proper cons < proper U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (57) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U64(gen_tt:mark:0':nil:ok3_0(+(1, n61070_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n61070_0) Induction Base: U64(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b)) Induction Step: U64(gen_tt:mark:0':nil:ok3_0(+(1, +(n61070_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1) mark(U64(gen_tt:mark:0':nil:ok3_0(+(1, n61070_0)), gen_tt:mark:0':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). ---------------------------------------- (58) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n35658_0) U63(gen_tt:mark:0':nil:ok3_0(+(1, n47859_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n47859_0) U64(gen_tt:mark:0':nil:ok3_0(+(1, n61070_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n61070_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: pair, active, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: pair < active cons < active U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top pair < proper cons < proper U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (59) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: pair(gen_tt:mark:0':nil:ok3_0(+(1, n67545_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n67545_0) Induction Base: pair(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b)) Induction Step: pair(gen_tt:mark:0':nil:ok3_0(+(1, +(n67545_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1) mark(pair(gen_tt:mark:0':nil:ok3_0(+(1, n67545_0)), gen_tt:mark:0':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). ---------------------------------------- (60) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n35658_0) U63(gen_tt:mark:0':nil:ok3_0(+(1, n47859_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n47859_0) U64(gen_tt:mark:0':nil:ok3_0(+(1, n61070_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n61070_0) pair(gen_tt:mark:0':nil:ok3_0(+(1, n67545_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n67545_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: cons, active, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: cons < active U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top cons < proper U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (61) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: cons(gen_tt:mark:0':nil:ok3_0(+(1, n74525_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n74525_0) Induction Base: cons(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b)) Induction Step: cons(gen_tt:mark:0':nil:ok3_0(+(1, +(n74525_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1) mark(cons(gen_tt:mark:0':nil:ok3_0(+(1, n74525_0)), gen_tt:mark:0':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). ---------------------------------------- (62) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n35658_0) U63(gen_tt:mark:0':nil:ok3_0(+(1, n47859_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n47859_0) U64(gen_tt:mark:0':nil:ok3_0(+(1, n61070_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n61070_0) pair(gen_tt:mark:0':nil:ok3_0(+(1, n67545_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n67545_0) cons(gen_tt:mark:0':nil:ok3_0(+(1, n74525_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n74525_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U72, active, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U72 < active U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U72 < proper U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (63) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U72(gen_tt:mark:0':nil:ok3_0(+(1, n81612_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n81612_0) Induction Base: U72(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b)) Induction Step: U72(gen_tt:mark:0':nil:ok3_0(+(1, +(n81612_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1) mark(U72(gen_tt:mark:0':nil:ok3_0(+(1, n81612_0)), gen_tt:mark:0':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). ---------------------------------------- (64) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n35658_0) U63(gen_tt:mark:0':nil:ok3_0(+(1, n47859_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n47859_0) U64(gen_tt:mark:0':nil:ok3_0(+(1, n61070_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n61070_0) pair(gen_tt:mark:0':nil:ok3_0(+(1, n67545_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n67545_0) cons(gen_tt:mark:0':nil:ok3_0(+(1, n74525_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n74525_0) U72(gen_tt:mark:0':nil:ok3_0(+(1, n81612_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n81612_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U82, active, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U82 < active fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U82 < proper fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (65) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U82(gen_tt:mark:0':nil:ok3_0(+(1, n89005_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n89005_0) Induction Base: U82(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) Induction Step: U82(gen_tt:mark:0':nil:ok3_0(+(1, +(n89005_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) ->_R^Omega(1) mark(U82(gen_tt:mark:0':nil:ok3_0(+(1, n89005_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':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). ---------------------------------------- (66) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n35658_0) U63(gen_tt:mark:0':nil:ok3_0(+(1, n47859_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n47859_0) U64(gen_tt:mark:0':nil:ok3_0(+(1, n61070_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n61070_0) pair(gen_tt:mark:0':nil:ok3_0(+(1, n67545_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n67545_0) cons(gen_tt:mark:0':nil:ok3_0(+(1, n74525_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n74525_0) U72(gen_tt:mark:0':nil:ok3_0(+(1, n81612_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n81612_0) U82(gen_tt:mark:0':nil:ok3_0(+(1, n89005_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n89005_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: fst, active, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: fst < active U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top fst < proper U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (67) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: fst(gen_tt:mark:0':nil:ok3_0(+(1, n101201_0))) -> *4_0, rt in Omega(n101201_0) Induction Base: fst(gen_tt:mark:0':nil:ok3_0(+(1, 0))) Induction Step: fst(gen_tt:mark:0':nil:ok3_0(+(1, +(n101201_0, 1)))) ->_R^Omega(1) mark(fst(gen_tt:mark:0':nil:ok3_0(+(1, n101201_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). ---------------------------------------- (68) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n35658_0) U63(gen_tt:mark:0':nil:ok3_0(+(1, n47859_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n47859_0) U64(gen_tt:mark:0':nil:ok3_0(+(1, n61070_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n61070_0) pair(gen_tt:mark:0':nil:ok3_0(+(1, n67545_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n67545_0) cons(gen_tt:mark:0':nil:ok3_0(+(1, n74525_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n74525_0) U72(gen_tt:mark:0':nil:ok3_0(+(1, n81612_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n81612_0) U82(gen_tt:mark:0':nil:ok3_0(+(1, n89005_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n89005_0) fst(gen_tt:mark:0':nil:ok3_0(+(1, n101201_0))) -> *4_0, rt in Omega(n101201_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U11, active, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U11 < active U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U11 < proper U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (69) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U11(gen_tt:mark:0':nil:ok3_0(+(1, n104602_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n104602_0) Induction Base: U11(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) Induction Step: U11(gen_tt:mark:0':nil:ok3_0(+(1, +(n104602_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) ->_R^Omega(1) mark(U11(gen_tt:mark:0':nil:ok3_0(+(1, n104602_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':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). ---------------------------------------- (70) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n35658_0) U63(gen_tt:mark:0':nil:ok3_0(+(1, n47859_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n47859_0) U64(gen_tt:mark:0':nil:ok3_0(+(1, n61070_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n61070_0) pair(gen_tt:mark:0':nil:ok3_0(+(1, n67545_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n67545_0) cons(gen_tt:mark:0':nil:ok3_0(+(1, n74525_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n74525_0) U72(gen_tt:mark:0':nil:ok3_0(+(1, n81612_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n81612_0) U82(gen_tt:mark:0':nil:ok3_0(+(1, n89005_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n89005_0) fst(gen_tt:mark:0':nil:ok3_0(+(1, n101201_0))) -> *4_0, rt in Omega(n101201_0) U11(gen_tt:mark:0':nil:ok3_0(+(1, n104602_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n104602_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U21, active, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U21 < active U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U21 < proper U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (71) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U21(gen_tt:mark:0':nil:ok3_0(+(1, n117722_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n117722_0) Induction Base: U21(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b)) Induction Step: U21(gen_tt:mark:0':nil:ok3_0(+(1, +(n117722_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1) mark(U21(gen_tt:mark:0':nil:ok3_0(+(1, n117722_0)), gen_tt:mark:0':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). ---------------------------------------- (72) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n35658_0) U63(gen_tt:mark:0':nil:ok3_0(+(1, n47859_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n47859_0) U64(gen_tt:mark:0':nil:ok3_0(+(1, n61070_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n61070_0) pair(gen_tt:mark:0':nil:ok3_0(+(1, n67545_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n67545_0) cons(gen_tt:mark:0':nil:ok3_0(+(1, n74525_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n74525_0) U72(gen_tt:mark:0':nil:ok3_0(+(1, n81612_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n81612_0) U82(gen_tt:mark:0':nil:ok3_0(+(1, n89005_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n89005_0) fst(gen_tt:mark:0':nil:ok3_0(+(1, n101201_0))) -> *4_0, rt in Omega(n101201_0) U11(gen_tt:mark:0':nil:ok3_0(+(1, n104602_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n104602_0) U21(gen_tt:mark:0':nil:ok3_0(+(1, n117722_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n117722_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U31, active, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U31 < active natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U31 < proper natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (73) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U31(gen_tt:mark:0':nil:ok3_0(+(1, n126437_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n126437_0) Induction Base: U31(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b)) Induction Step: U31(gen_tt:mark:0':nil:ok3_0(+(1, +(n126437_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1) mark(U31(gen_tt:mark:0':nil:ok3_0(+(1, n126437_0)), gen_tt:mark:0':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). ---------------------------------------- (74) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n35658_0) U63(gen_tt:mark:0':nil:ok3_0(+(1, n47859_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n47859_0) U64(gen_tt:mark:0':nil:ok3_0(+(1, n61070_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n61070_0) pair(gen_tt:mark:0':nil:ok3_0(+(1, n67545_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n67545_0) cons(gen_tt:mark:0':nil:ok3_0(+(1, n74525_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n74525_0) U72(gen_tt:mark:0':nil:ok3_0(+(1, n81612_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n81612_0) U82(gen_tt:mark:0':nil:ok3_0(+(1, n89005_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n89005_0) fst(gen_tt:mark:0':nil:ok3_0(+(1, n101201_0))) -> *4_0, rt in Omega(n101201_0) U11(gen_tt:mark:0':nil:ok3_0(+(1, n104602_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n104602_0) U21(gen_tt:mark:0':nil:ok3_0(+(1, n117722_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n117722_0) U31(gen_tt:mark:0':nil:ok3_0(+(1, n126437_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n126437_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: natsFrom, active, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: natsFrom < active s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top natsFrom < proper s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (75) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n135458_0))) -> *4_0, rt in Omega(n135458_0) Induction Base: natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, 0))) Induction Step: natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, +(n135458_0, 1)))) ->_R^Omega(1) mark(natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n135458_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). ---------------------------------------- (76) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n35658_0) U63(gen_tt:mark:0':nil:ok3_0(+(1, n47859_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n47859_0) U64(gen_tt:mark:0':nil:ok3_0(+(1, n61070_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n61070_0) pair(gen_tt:mark:0':nil:ok3_0(+(1, n67545_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n67545_0) cons(gen_tt:mark:0':nil:ok3_0(+(1, n74525_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n74525_0) U72(gen_tt:mark:0':nil:ok3_0(+(1, n81612_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n81612_0) U82(gen_tt:mark:0':nil:ok3_0(+(1, n89005_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n89005_0) fst(gen_tt:mark:0':nil:ok3_0(+(1, n101201_0))) -> *4_0, rt in Omega(n101201_0) U11(gen_tt:mark:0':nil:ok3_0(+(1, n104602_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n104602_0) U21(gen_tt:mark:0':nil:ok3_0(+(1, n117722_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n117722_0) U31(gen_tt:mark:0':nil:ok3_0(+(1, n126437_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n126437_0) natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n135458_0))) -> *4_0, rt in Omega(n135458_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: s, active, U41, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: s < active U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top s < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (77) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: s(gen_tt:mark:0':nil:ok3_0(+(1, n139459_0))) -> *4_0, rt in Omega(n139459_0) Induction Base: s(gen_tt:mark:0':nil:ok3_0(+(1, 0))) Induction Step: s(gen_tt:mark:0':nil:ok3_0(+(1, +(n139459_0, 1)))) ->_R^Omega(1) mark(s(gen_tt:mark:0':nil:ok3_0(+(1, n139459_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). ---------------------------------------- (78) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n35658_0) U63(gen_tt:mark:0':nil:ok3_0(+(1, n47859_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n47859_0) U64(gen_tt:mark:0':nil:ok3_0(+(1, n61070_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n61070_0) pair(gen_tt:mark:0':nil:ok3_0(+(1, n67545_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n67545_0) cons(gen_tt:mark:0':nil:ok3_0(+(1, n74525_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n74525_0) U72(gen_tt:mark:0':nil:ok3_0(+(1, n81612_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n81612_0) U82(gen_tt:mark:0':nil:ok3_0(+(1, n89005_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n89005_0) fst(gen_tt:mark:0':nil:ok3_0(+(1, n101201_0))) -> *4_0, rt in Omega(n101201_0) U11(gen_tt:mark:0':nil:ok3_0(+(1, n104602_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n104602_0) U21(gen_tt:mark:0':nil:ok3_0(+(1, n117722_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n117722_0) U31(gen_tt:mark:0':nil:ok3_0(+(1, n126437_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n126437_0) natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n135458_0))) -> *4_0, rt in Omega(n135458_0) s(gen_tt:mark:0':nil:ok3_0(+(1, n139459_0))) -> *4_0, rt in Omega(n139459_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U41, active, U51, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U41 < active U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (79) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U41(gen_tt:mark:0':nil:ok3_0(+(1, n143561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n143561_0) Induction Base: U41(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) Induction Step: U41(gen_tt:mark:0':nil:ok3_0(+(1, +(n143561_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) ->_R^Omega(1) mark(U41(gen_tt:mark:0':nil:ok3_0(+(1, n143561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':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). ---------------------------------------- (80) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n35658_0) U63(gen_tt:mark:0':nil:ok3_0(+(1, n47859_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n47859_0) U64(gen_tt:mark:0':nil:ok3_0(+(1, n61070_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n61070_0) pair(gen_tt:mark:0':nil:ok3_0(+(1, n67545_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n67545_0) cons(gen_tt:mark:0':nil:ok3_0(+(1, n74525_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n74525_0) U72(gen_tt:mark:0':nil:ok3_0(+(1, n81612_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n81612_0) U82(gen_tt:mark:0':nil:ok3_0(+(1, n89005_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n89005_0) fst(gen_tt:mark:0':nil:ok3_0(+(1, n101201_0))) -> *4_0, rt in Omega(n101201_0) U11(gen_tt:mark:0':nil:ok3_0(+(1, n104602_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n104602_0) U21(gen_tt:mark:0':nil:ok3_0(+(1, n117722_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n117722_0) U31(gen_tt:mark:0':nil:ok3_0(+(1, n126437_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n126437_0) natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n135458_0))) -> *4_0, rt in Omega(n135458_0) s(gen_tt:mark:0':nil:ok3_0(+(1, n139459_0))) -> *4_0, rt in Omega(n139459_0) U41(gen_tt:mark:0':nil:ok3_0(+(1, n143561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n143561_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U51, active, U61, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U51 < active U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U51 < proper U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top ---------------------------------------- (81) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U51(gen_tt:mark:0':nil:ok3_0(+(1, n158844_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n158844_0) Induction Base: U51(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b)) Induction Step: U51(gen_tt:mark:0':nil:ok3_0(+(1, +(n158844_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1) mark(U51(gen_tt:mark:0':nil:ok3_0(+(1, n158844_0)), gen_tt:mark:0':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). ---------------------------------------- (82) Obligation: TRS: Rules: active(U11(tt, N, XS)) -> mark(U12(tt, N, XS)) active(U12(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(U22(tt, X)) active(U22(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(U32(tt, N)) active(U32(tt, N)) -> mark(N) active(U41(tt, N, XS)) -> mark(U42(tt, N, XS)) active(U42(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U51(tt, Y)) -> mark(U52(tt, Y)) active(U52(tt, Y)) -> mark(Y) active(U61(tt, N, X, XS)) -> mark(U62(tt, N, X, XS)) active(U62(tt, N, X, XS)) -> mark(U63(tt, N, X, XS)) active(U63(tt, N, X, XS)) -> mark(U64(splitAt(N, XS), X)) active(U64(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U71(tt, XS)) -> mark(U72(tt, XS)) active(U72(tt, XS)) -> mark(XS) active(U81(tt, N, XS)) -> mark(U82(tt, N, XS)) active(U82(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(afterNth(N, XS)) -> mark(U11(tt, N, XS)) active(fst(pair(X, Y))) -> mark(U21(tt, X)) active(head(cons(N, XS))) -> mark(U31(tt, N)) active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N)))) active(sel(N, XS)) -> mark(U41(tt, N, XS)) active(snd(pair(X, Y))) -> mark(U51(tt, Y)) active(splitAt(0', XS)) -> mark(pair(nil, XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U61(tt, N, X, XS)) active(tail(cons(N, XS))) -> mark(U71(tt, XS)) active(take(N, XS)) -> mark(U81(tt, N, XS)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X1, X2)) -> U32(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) active(head(X)) -> head(active(X)) active(afterNth(X1, X2)) -> afterNth(active(X1), X2) active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X1, X2)) -> U52(active(X1), X2) active(U61(X1, X2, X3, X4)) -> U61(active(X1), X2, X3, X4) active(U62(X1, X2, X3, X4)) -> U62(active(X1), X2, X3, X4) active(U63(X1, X2, X3, X4)) -> U63(active(X1), X2, X3, X4) active(U64(X1, X2)) -> U64(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U61(mark(X1), X2, X3, X4) -> mark(U61(X1, X2, X3, X4)) U62(mark(X1), X2, X3, X4) -> mark(U62(X1, X2, X3, X4)) U63(mark(X1), X2, X3, X4) -> mark(U63(X1, X2, X3, X4)) U64(mark(X1), X2) -> mark(U64(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U61(X1, X2, X3, X4)) -> U61(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U62(X1, X2, X3, X4)) -> U62(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U63(X1, X2, X3, X4)) -> U63(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U64(X1, X2)) -> U64(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(0') -> ok(0') proper(nil) -> ok(nil) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U61(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U61(X1, X2, X3, X4)) U62(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U62(X1, X2, X3, X4)) U63(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U63(X1, X2, X3, X4)) U64(ok(X1), ok(X2)) -> ok(U64(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U11 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok tt :: tt:mark:0':nil:ok mark :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok U12 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok snd :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok splitAt :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U21 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U22 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U31 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U32 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U41 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U42 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok head :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok afterNth :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U51 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U52 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U61 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U62 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U63 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U64 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok pair :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok cons :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U71 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U72 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U81 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok U82 :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok fst :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok natsFrom :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok s :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok sel :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok 0' :: tt:mark:0':nil:ok nil :: tt:mark:0':nil:ok tail :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok take :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok -> tt:mark:0':nil:ok proper :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok ok :: tt:mark:0':nil:ok -> tt:mark:0':nil:ok top :: tt:mark:0':nil:ok -> top hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok hole_top2_0 :: top gen_tt:mark:0':nil:ok3_0 :: Nat -> tt:mark:0':nil:ok Lemmas: U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n5_0) snd(gen_tt:mark:0':nil:ok3_0(+(1, n4683_0))) -> *4_0, rt in Omega(n4683_0) splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5638_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5638_0) U22(gen_tt:mark:0':nil:ok3_0(+(1, n9166_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n9166_0) U32(gen_tt:mark:0':nil:ok3_0(+(1, n12801_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n12801_0) U42(gen_tt:mark:0':nil:ok3_0(+(1, n16742_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n16742_0) head(gen_tt:mark:0':nil:ok3_0(+(1, n23730_0))) -> *4_0, rt in Omega(n23730_0) afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n25435_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n25435_0) U52(gen_tt:mark:0':nil:ok3_0(+(1, n30493_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n30493_0) U62(gen_tt:mark:0':nil:ok3_0(+(1, n35658_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n35658_0) U63(gen_tt:mark:0':nil:ok3_0(+(1, n47859_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n47859_0) U64(gen_tt:mark:0':nil:ok3_0(+(1, n61070_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n61070_0) pair(gen_tt:mark:0':nil:ok3_0(+(1, n67545_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n67545_0) cons(gen_tt:mark:0':nil:ok3_0(+(1, n74525_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n74525_0) U72(gen_tt:mark:0':nil:ok3_0(+(1, n81612_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n81612_0) U82(gen_tt:mark:0':nil:ok3_0(+(1, n89005_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n89005_0) fst(gen_tt:mark:0':nil:ok3_0(+(1, n101201_0))) -> *4_0, rt in Omega(n101201_0) U11(gen_tt:mark:0':nil:ok3_0(+(1, n104602_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n104602_0) U21(gen_tt:mark:0':nil:ok3_0(+(1, n117722_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n117722_0) U31(gen_tt:mark:0':nil:ok3_0(+(1, n126437_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n126437_0) natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n135458_0))) -> *4_0, rt in Omega(n135458_0) s(gen_tt:mark:0':nil:ok3_0(+(1, n139459_0))) -> *4_0, rt in Omega(n139459_0) U41(gen_tt:mark:0':nil:ok3_0(+(1, n143561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) -> *4_0, rt in Omega(n143561_0) U51(gen_tt:mark:0':nil:ok3_0(+(1, n158844_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n158844_0) Generator Equations: gen_tt:mark:0':nil:ok3_0(0) <=> tt gen_tt:mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':nil:ok3_0(x)) The following defined symbols remain to be analysed: U61, active, U71, U81, sel, tail, take, proper, top They will be analysed ascendingly in the following order: U61 < active U71 < active U81 < active sel < active tail < active take < active active < top U61 < proper U71 < proper U81 < proper sel < proper tail < proper take < proper proper < top