WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).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), 47 ms] (4) CpxTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 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)), 199 ms] (14) CdtProblem (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms] (16) CdtProblem (17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (18) BOUNDS(1, 1) (19) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxTRS (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (22) typed CpxTrs (23) OrderProof [LOWER BOUND(ID), 9 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 448 ms] (26) BEST (27) proven lower bound (28) LowerBoundPropagationProof [FINISHED, 0 ms] (29) BOUNDS(n^1, INF) (30) typed CpxTrs (31) RewriteLemmaProof [LOWER BOUND(ID), 140 ms] (32) typed CpxTrs (33) RewriteLemmaProof [LOWER BOUND(ID), 102 ms] (34) typed CpxTrs (35) RewriteLemmaProof [LOWER BOUND(ID), 125 ms] (36) typed CpxTrs (37) RewriteLemmaProof [LOWER BOUND(ID), 111 ms] (38) typed CpxTrs (39) RewriteLemmaProof [LOWER BOUND(ID), 104 ms] (40) typed CpxTrs (41) RewriteLemmaProof [LOWER BOUND(ID), 90 ms] (42) typed CpxTrs (43) RewriteLemmaProof [LOWER BOUND(ID), 159 ms] (44) typed CpxTrs (45) RewriteLemmaProof [LOWER BOUND(ID), 121 ms] (46) typed CpxTrs (47) RewriteLemmaProof [LOWER BOUND(ID), 167 ms] (48) typed CpxTrs (49) RewriteLemmaProof [LOWER BOUND(ID), 208 ms] (50) typed CpxTrs (51) RewriteLemmaProof [LOWER BOUND(ID), 155 ms] (52) typed CpxTrs (53) RewriteLemmaProof [LOWER BOUND(ID), 131 ms] (54) typed CpxTrs (55) RewriteLemmaProof [LOWER BOUND(ID), 109 ms] (56) typed CpxTrs (57) RewriteLemmaProof [LOWER BOUND(ID), 137 ms] (58) typed CpxTrs (59) RewriteLemmaProof [LOWER BOUND(ID), 136 ms] (60) typed CpxTrs (61) RewriteLemmaProof [LOWER BOUND(ID), 112 ms] (62) typed CpxTrs (63) RewriteLemmaProof [LOWER BOUND(ID), 172 ms] (64) 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(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0)) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0, XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0) -> ok(0) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(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(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0)) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0, XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(sel(X1, X2)) -> sel(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: U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(tt) -> ok(tt) proper(nil) -> ok(nil) proper(0) -> ok(0) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(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: U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(tt) -> ok(tt) proper(nil) -> ok(nil) proper(0) -> ok(0) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(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: U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2)) U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2)) fst(mark(z0)) -> mark(fst(z0)) fst(ok(z0)) -> ok(fst(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)) U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) snd(mark(z0)) -> mark(snd(z0)) snd(ok(z0)) -> ok(snd(z0)) U21(mark(z0), z1) -> mark(U21(z0, z1)) U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) U31(mark(z0), z1) -> mark(U31(z0, z1)) U31(ok(z0), ok(z1)) -> ok(U31(z0, z1)) U41(mark(z0), z1) -> mark(U41(z0, z1)) U41(ok(z0), ok(z1)) -> ok(U41(z0, z1)) cons(mark(z0), z1) -> mark(cons(z0, z1)) cons(ok(z0), ok(z1)) -> ok(cons(z0, z1)) natsFrom(mark(z0)) -> mark(natsFrom(z0)) natsFrom(ok(z0)) -> ok(natsFrom(z0)) s(mark(z0)) -> mark(s(z0)) s(ok(z0)) -> ok(s(z0)) U51(mark(z0), z1, z2) -> mark(U51(z0, z1, z2)) U51(ok(z0), ok(z1), ok(z2)) -> ok(U51(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)) U61(mark(z0), z1) -> mark(U61(z0, z1)) U61(ok(z0), ok(z1)) -> ok(U61(z0, z1)) U71(mark(z0), z1) -> mark(U71(z0, z1)) U71(ok(z0), ok(z1)) -> ok(U71(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)) U81(mark(z0), z1, z2, z3) -> mark(U81(z0, z1, z2, z3)) U81(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U81(z0, z1, z2, z3)) U82(mark(z0), z1) -> mark(U82(z0, z1)) U82(ok(z0), ok(z1)) -> ok(U82(z0, z1)) U91(mark(z0), z1) -> mark(U91(z0, z1)) U91(ok(z0), ok(z1)) -> ok(U91(z0, z1)) and(mark(z0), z1) -> mark(and(z0, z1)) and(ok(z0), ok(z1)) -> ok(and(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)) 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)) proper(tt) -> ok(tt) proper(nil) -> ok(nil) proper(0) -> ok(0) isNatural(ok(z0)) -> ok(isNatural(z0)) isLNat(ok(z0)) -> ok(isLNat(z0)) isPLNat(ok(z0)) -> ok(isPLNat(z0)) top(mark(z0)) -> top(proper(z0)) top(ok(z0)) -> top(active(z0)) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) FST(mark(z0)) -> c2(FST(z0)) FST(ok(z0)) -> c3(FST(z0)) SPLITAT(mark(z0), z1) -> c4(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c5(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c6(SPLITAT(z0, z1)) U11'(mark(z0), z1, z2) -> c7(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c8(U11'(z0, z1, z2)) SND(mark(z0)) -> c9(SND(z0)) SND(ok(z0)) -> c10(SND(z0)) U21'(mark(z0), z1) -> c11(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c12(U21'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U41'(mark(z0), z1) -> c15(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c16(U41'(z0, z1)) CONS(mark(z0), z1) -> c17(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c18(CONS(z0, z1)) NATSFROM(mark(z0)) -> c19(NATSFROM(z0)) NATSFROM(ok(z0)) -> c20(NATSFROM(z0)) S(mark(z0)) -> c21(S(z0)) S(ok(z0)) -> c22(S(z0)) U51'(mark(z0), z1, z2) -> c23(U51'(z0, z1, z2)) U51'(ok(z0), ok(z1), ok(z2)) -> c24(U51'(z0, z1, z2)) HEAD(mark(z0)) -> c25(HEAD(z0)) HEAD(ok(z0)) -> c26(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c27(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c28(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c29(AFTERNTH(z0, z1)) U61'(mark(z0), z1) -> c30(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c31(U61'(z0, z1)) U71'(mark(z0), z1) -> c32(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c33(U71'(z0, z1)) PAIR(mark(z0), z1) -> c34(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c35(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c36(PAIR(z0, z1)) U81'(mark(z0), z1, z2, z3) -> c37(U81'(z0, z1, z2, z3)) U81'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c38(U81'(z0, z1, z2, z3)) U82'(mark(z0), z1) -> c39(U82'(z0, z1)) U82'(ok(z0), ok(z1)) -> c40(U82'(z0, z1)) U91'(mark(z0), z1) -> c41(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c42(U91'(z0, z1)) AND(mark(z0), z1) -> c43(AND(z0, z1)) AND(ok(z0), ok(z1)) -> c44(AND(z0, z1)) TAIL(mark(z0)) -> c45(TAIL(z0)) TAIL(ok(z0)) -> c46(TAIL(z0)) TAKE(mark(z0), z1) -> c47(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c48(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c49(TAKE(z0, z1)) SEL(mark(z0), z1) -> c50(SEL(z0, z1)) SEL(z0, mark(z1)) -> c51(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c52(SEL(z0, z1)) PROPER(tt) -> c53 PROPER(nil) -> c54 PROPER(0) -> c55 ISNATURAL(ok(z0)) -> c56(ISNATURAL(z0)) ISLNAT(ok(z0)) -> c57(ISLNAT(z0)) ISPLNAT(ok(z0)) -> c58(ISPLNAT(z0)) TOP(mark(z0)) -> c59(TOP(proper(z0)), PROPER(z0)) TOP(ok(z0)) -> c60(TOP(active(z0))) S tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) FST(mark(z0)) -> c2(FST(z0)) FST(ok(z0)) -> c3(FST(z0)) SPLITAT(mark(z0), z1) -> c4(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c5(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c6(SPLITAT(z0, z1)) U11'(mark(z0), z1, z2) -> c7(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c8(U11'(z0, z1, z2)) SND(mark(z0)) -> c9(SND(z0)) SND(ok(z0)) -> c10(SND(z0)) U21'(mark(z0), z1) -> c11(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c12(U21'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U41'(mark(z0), z1) -> c15(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c16(U41'(z0, z1)) CONS(mark(z0), z1) -> c17(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c18(CONS(z0, z1)) NATSFROM(mark(z0)) -> c19(NATSFROM(z0)) NATSFROM(ok(z0)) -> c20(NATSFROM(z0)) S(mark(z0)) -> c21(S(z0)) S(ok(z0)) -> c22(S(z0)) U51'(mark(z0), z1, z2) -> c23(U51'(z0, z1, z2)) U51'(ok(z0), ok(z1), ok(z2)) -> c24(U51'(z0, z1, z2)) HEAD(mark(z0)) -> c25(HEAD(z0)) HEAD(ok(z0)) -> c26(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c27(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c28(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c29(AFTERNTH(z0, z1)) U61'(mark(z0), z1) -> c30(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c31(U61'(z0, z1)) U71'(mark(z0), z1) -> c32(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c33(U71'(z0, z1)) PAIR(mark(z0), z1) -> c34(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c35(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c36(PAIR(z0, z1)) U81'(mark(z0), z1, z2, z3) -> c37(U81'(z0, z1, z2, z3)) U81'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c38(U81'(z0, z1, z2, z3)) U82'(mark(z0), z1) -> c39(U82'(z0, z1)) U82'(ok(z0), ok(z1)) -> c40(U82'(z0, z1)) U91'(mark(z0), z1) -> c41(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c42(U91'(z0, z1)) AND(mark(z0), z1) -> c43(AND(z0, z1)) AND(ok(z0), ok(z1)) -> c44(AND(z0, z1)) TAIL(mark(z0)) -> c45(TAIL(z0)) TAIL(ok(z0)) -> c46(TAIL(z0)) TAKE(mark(z0), z1) -> c47(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c48(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c49(TAKE(z0, z1)) SEL(mark(z0), z1) -> c50(SEL(z0, z1)) SEL(z0, mark(z1)) -> c51(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c52(SEL(z0, z1)) PROPER(tt) -> c53 PROPER(nil) -> c54 PROPER(0) -> c55 ISNATURAL(ok(z0)) -> c56(ISNATURAL(z0)) ISLNAT(ok(z0)) -> c57(ISLNAT(z0)) ISPLNAT(ok(z0)) -> c58(ISPLNAT(z0)) TOP(mark(z0)) -> c59(TOP(proper(z0)), PROPER(z0)) TOP(ok(z0)) -> c60(TOP(active(z0))) K tuples:none Defined Rule Symbols: U101_3, fst_1, splitAt_2, U11_3, snd_1, U21_2, U31_2, U41_2, cons_2, natsFrom_1, s_1, U51_3, head_1, afterNth_2, U61_2, U71_2, pair_2, U81_4, U82_2, U91_2, and_2, tail_1, take_2, sel_2, proper_1, isNatural_1, isLNat_1, isPLNat_1, top_1 Defined Pair Symbols: U101'_3, FST_1, SPLITAT_2, U11'_3, SND_1, U21'_2, U31'_2, U41'_2, CONS_2, NATSFROM_1, S_1, U51'_3, HEAD_1, AFTERNTH_2, U61'_2, U71'_2, PAIR_2, U81'_4, U82'_2, U91'_2, AND_2, TAIL_1, TAKE_2, SEL_2, PROPER_1, ISNATURAL_1, ISLNAT_1, ISPLNAT_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, c54, c55, c56_1, c57_1, c58_1, c59_2, c60_1 ---------------------------------------- (7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: TOP(ok(z0)) -> c60(TOP(active(z0))) PROPER(0) -> c55 PROPER(tt) -> c53 PROPER(nil) -> c54 ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2)) U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2)) fst(mark(z0)) -> mark(fst(z0)) fst(ok(z0)) -> ok(fst(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)) U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) snd(mark(z0)) -> mark(snd(z0)) snd(ok(z0)) -> ok(snd(z0)) U21(mark(z0), z1) -> mark(U21(z0, z1)) U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) U31(mark(z0), z1) -> mark(U31(z0, z1)) U31(ok(z0), ok(z1)) -> ok(U31(z0, z1)) U41(mark(z0), z1) -> mark(U41(z0, z1)) U41(ok(z0), ok(z1)) -> ok(U41(z0, z1)) cons(mark(z0), z1) -> mark(cons(z0, z1)) cons(ok(z0), ok(z1)) -> ok(cons(z0, z1)) natsFrom(mark(z0)) -> mark(natsFrom(z0)) natsFrom(ok(z0)) -> ok(natsFrom(z0)) s(mark(z0)) -> mark(s(z0)) s(ok(z0)) -> ok(s(z0)) U51(mark(z0), z1, z2) -> mark(U51(z0, z1, z2)) U51(ok(z0), ok(z1), ok(z2)) -> ok(U51(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)) U61(mark(z0), z1) -> mark(U61(z0, z1)) U61(ok(z0), ok(z1)) -> ok(U61(z0, z1)) U71(mark(z0), z1) -> mark(U71(z0, z1)) U71(ok(z0), ok(z1)) -> ok(U71(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)) U81(mark(z0), z1, z2, z3) -> mark(U81(z0, z1, z2, z3)) U81(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U81(z0, z1, z2, z3)) U82(mark(z0), z1) -> mark(U82(z0, z1)) U82(ok(z0), ok(z1)) -> ok(U82(z0, z1)) U91(mark(z0), z1) -> mark(U91(z0, z1)) U91(ok(z0), ok(z1)) -> ok(U91(z0, z1)) and(mark(z0), z1) -> mark(and(z0, z1)) and(ok(z0), ok(z1)) -> ok(and(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)) 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)) proper(tt) -> ok(tt) proper(nil) -> ok(nil) proper(0) -> ok(0) isNatural(ok(z0)) -> ok(isNatural(z0)) isLNat(ok(z0)) -> ok(isLNat(z0)) isPLNat(ok(z0)) -> ok(isPLNat(z0)) top(mark(z0)) -> top(proper(z0)) top(ok(z0)) -> top(active(z0)) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) FST(mark(z0)) -> c2(FST(z0)) FST(ok(z0)) -> c3(FST(z0)) SPLITAT(mark(z0), z1) -> c4(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c5(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c6(SPLITAT(z0, z1)) U11'(mark(z0), z1, z2) -> c7(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c8(U11'(z0, z1, z2)) SND(mark(z0)) -> c9(SND(z0)) SND(ok(z0)) -> c10(SND(z0)) U21'(mark(z0), z1) -> c11(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c12(U21'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U41'(mark(z0), z1) -> c15(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c16(U41'(z0, z1)) CONS(mark(z0), z1) -> c17(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c18(CONS(z0, z1)) NATSFROM(mark(z0)) -> c19(NATSFROM(z0)) NATSFROM(ok(z0)) -> c20(NATSFROM(z0)) S(mark(z0)) -> c21(S(z0)) S(ok(z0)) -> c22(S(z0)) U51'(mark(z0), z1, z2) -> c23(U51'(z0, z1, z2)) U51'(ok(z0), ok(z1), ok(z2)) -> c24(U51'(z0, z1, z2)) HEAD(mark(z0)) -> c25(HEAD(z0)) HEAD(ok(z0)) -> c26(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c27(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c28(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c29(AFTERNTH(z0, z1)) U61'(mark(z0), z1) -> c30(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c31(U61'(z0, z1)) U71'(mark(z0), z1) -> c32(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c33(U71'(z0, z1)) PAIR(mark(z0), z1) -> c34(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c35(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c36(PAIR(z0, z1)) U81'(mark(z0), z1, z2, z3) -> c37(U81'(z0, z1, z2, z3)) U81'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c38(U81'(z0, z1, z2, z3)) U82'(mark(z0), z1) -> c39(U82'(z0, z1)) U82'(ok(z0), ok(z1)) -> c40(U82'(z0, z1)) U91'(mark(z0), z1) -> c41(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c42(U91'(z0, z1)) AND(mark(z0), z1) -> c43(AND(z0, z1)) AND(ok(z0), ok(z1)) -> c44(AND(z0, z1)) TAIL(mark(z0)) -> c45(TAIL(z0)) TAIL(ok(z0)) -> c46(TAIL(z0)) TAKE(mark(z0), z1) -> c47(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c48(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c49(TAKE(z0, z1)) SEL(mark(z0), z1) -> c50(SEL(z0, z1)) SEL(z0, mark(z1)) -> c51(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c52(SEL(z0, z1)) ISNATURAL(ok(z0)) -> c56(ISNATURAL(z0)) ISLNAT(ok(z0)) -> c57(ISLNAT(z0)) ISPLNAT(ok(z0)) -> c58(ISPLNAT(z0)) TOP(mark(z0)) -> c59(TOP(proper(z0)), PROPER(z0)) S tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) FST(mark(z0)) -> c2(FST(z0)) FST(ok(z0)) -> c3(FST(z0)) SPLITAT(mark(z0), z1) -> c4(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c5(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c6(SPLITAT(z0, z1)) U11'(mark(z0), z1, z2) -> c7(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c8(U11'(z0, z1, z2)) SND(mark(z0)) -> c9(SND(z0)) SND(ok(z0)) -> c10(SND(z0)) U21'(mark(z0), z1) -> c11(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c12(U21'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U41'(mark(z0), z1) -> c15(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c16(U41'(z0, z1)) CONS(mark(z0), z1) -> c17(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c18(CONS(z0, z1)) NATSFROM(mark(z0)) -> c19(NATSFROM(z0)) NATSFROM(ok(z0)) -> c20(NATSFROM(z0)) S(mark(z0)) -> c21(S(z0)) S(ok(z0)) -> c22(S(z0)) U51'(mark(z0), z1, z2) -> c23(U51'(z0, z1, z2)) U51'(ok(z0), ok(z1), ok(z2)) -> c24(U51'(z0, z1, z2)) HEAD(mark(z0)) -> c25(HEAD(z0)) HEAD(ok(z0)) -> c26(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c27(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c28(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c29(AFTERNTH(z0, z1)) U61'(mark(z0), z1) -> c30(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c31(U61'(z0, z1)) U71'(mark(z0), z1) -> c32(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c33(U71'(z0, z1)) PAIR(mark(z0), z1) -> c34(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c35(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c36(PAIR(z0, z1)) U81'(mark(z0), z1, z2, z3) -> c37(U81'(z0, z1, z2, z3)) U81'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c38(U81'(z0, z1, z2, z3)) U82'(mark(z0), z1) -> c39(U82'(z0, z1)) U82'(ok(z0), ok(z1)) -> c40(U82'(z0, z1)) U91'(mark(z0), z1) -> c41(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c42(U91'(z0, z1)) AND(mark(z0), z1) -> c43(AND(z0, z1)) AND(ok(z0), ok(z1)) -> c44(AND(z0, z1)) TAIL(mark(z0)) -> c45(TAIL(z0)) TAIL(ok(z0)) -> c46(TAIL(z0)) TAKE(mark(z0), z1) -> c47(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c48(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c49(TAKE(z0, z1)) SEL(mark(z0), z1) -> c50(SEL(z0, z1)) SEL(z0, mark(z1)) -> c51(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c52(SEL(z0, z1)) ISNATURAL(ok(z0)) -> c56(ISNATURAL(z0)) ISLNAT(ok(z0)) -> c57(ISLNAT(z0)) ISPLNAT(ok(z0)) -> c58(ISPLNAT(z0)) TOP(mark(z0)) -> c59(TOP(proper(z0)), PROPER(z0)) K tuples:none Defined Rule Symbols: U101_3, fst_1, splitAt_2, U11_3, snd_1, U21_2, U31_2, U41_2, cons_2, natsFrom_1, s_1, U51_3, head_1, afterNth_2, U61_2, U71_2, pair_2, U81_4, U82_2, U91_2, and_2, tail_1, take_2, sel_2, proper_1, isNatural_1, isLNat_1, isPLNat_1, top_1 Defined Pair Symbols: U101'_3, FST_1, SPLITAT_2, U11'_3, SND_1, U21'_2, U31'_2, U41'_2, CONS_2, NATSFROM_1, S_1, U51'_3, HEAD_1, AFTERNTH_2, U61'_2, U71'_2, PAIR_2, U81'_4, U82'_2, U91'_2, AND_2, TAIL_1, TAKE_2, SEL_2, ISNATURAL_1, ISLNAT_1, ISPLNAT_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, c56_1, c57_1, c58_1, c59_2 ---------------------------------------- (9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2)) U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2)) fst(mark(z0)) -> mark(fst(z0)) fst(ok(z0)) -> ok(fst(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)) U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) snd(mark(z0)) -> mark(snd(z0)) snd(ok(z0)) -> ok(snd(z0)) U21(mark(z0), z1) -> mark(U21(z0, z1)) U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) U31(mark(z0), z1) -> mark(U31(z0, z1)) U31(ok(z0), ok(z1)) -> ok(U31(z0, z1)) U41(mark(z0), z1) -> mark(U41(z0, z1)) U41(ok(z0), ok(z1)) -> ok(U41(z0, z1)) cons(mark(z0), z1) -> mark(cons(z0, z1)) cons(ok(z0), ok(z1)) -> ok(cons(z0, z1)) natsFrom(mark(z0)) -> mark(natsFrom(z0)) natsFrom(ok(z0)) -> ok(natsFrom(z0)) s(mark(z0)) -> mark(s(z0)) s(ok(z0)) -> ok(s(z0)) U51(mark(z0), z1, z2) -> mark(U51(z0, z1, z2)) U51(ok(z0), ok(z1), ok(z2)) -> ok(U51(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)) U61(mark(z0), z1) -> mark(U61(z0, z1)) U61(ok(z0), ok(z1)) -> ok(U61(z0, z1)) U71(mark(z0), z1) -> mark(U71(z0, z1)) U71(ok(z0), ok(z1)) -> ok(U71(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)) U81(mark(z0), z1, z2, z3) -> mark(U81(z0, z1, z2, z3)) U81(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U81(z0, z1, z2, z3)) U82(mark(z0), z1) -> mark(U82(z0, z1)) U82(ok(z0), ok(z1)) -> ok(U82(z0, z1)) U91(mark(z0), z1) -> mark(U91(z0, z1)) U91(ok(z0), ok(z1)) -> ok(U91(z0, z1)) and(mark(z0), z1) -> mark(and(z0, z1)) and(ok(z0), ok(z1)) -> ok(and(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)) 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)) proper(tt) -> ok(tt) proper(nil) -> ok(nil) proper(0) -> ok(0) isNatural(ok(z0)) -> ok(isNatural(z0)) isLNat(ok(z0)) -> ok(isLNat(z0)) isPLNat(ok(z0)) -> ok(isPLNat(z0)) top(mark(z0)) -> top(proper(z0)) top(ok(z0)) -> top(active(z0)) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) FST(mark(z0)) -> c2(FST(z0)) FST(ok(z0)) -> c3(FST(z0)) SPLITAT(mark(z0), z1) -> c4(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c5(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c6(SPLITAT(z0, z1)) U11'(mark(z0), z1, z2) -> c7(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c8(U11'(z0, z1, z2)) SND(mark(z0)) -> c9(SND(z0)) SND(ok(z0)) -> c10(SND(z0)) U21'(mark(z0), z1) -> c11(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c12(U21'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U41'(mark(z0), z1) -> c15(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c16(U41'(z0, z1)) CONS(mark(z0), z1) -> c17(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c18(CONS(z0, z1)) NATSFROM(mark(z0)) -> c19(NATSFROM(z0)) NATSFROM(ok(z0)) -> c20(NATSFROM(z0)) S(mark(z0)) -> c21(S(z0)) S(ok(z0)) -> c22(S(z0)) U51'(mark(z0), z1, z2) -> c23(U51'(z0, z1, z2)) U51'(ok(z0), ok(z1), ok(z2)) -> c24(U51'(z0, z1, z2)) HEAD(mark(z0)) -> c25(HEAD(z0)) HEAD(ok(z0)) -> c26(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c27(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c28(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c29(AFTERNTH(z0, z1)) U61'(mark(z0), z1) -> c30(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c31(U61'(z0, z1)) U71'(mark(z0), z1) -> c32(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c33(U71'(z0, z1)) PAIR(mark(z0), z1) -> c34(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c35(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c36(PAIR(z0, z1)) U81'(mark(z0), z1, z2, z3) -> c37(U81'(z0, z1, z2, z3)) U81'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c38(U81'(z0, z1, z2, z3)) U82'(mark(z0), z1) -> c39(U82'(z0, z1)) U82'(ok(z0), ok(z1)) -> c40(U82'(z0, z1)) U91'(mark(z0), z1) -> c41(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c42(U91'(z0, z1)) AND(mark(z0), z1) -> c43(AND(z0, z1)) AND(ok(z0), ok(z1)) -> c44(AND(z0, z1)) TAIL(mark(z0)) -> c45(TAIL(z0)) TAIL(ok(z0)) -> c46(TAIL(z0)) TAKE(mark(z0), z1) -> c47(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c48(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c49(TAKE(z0, z1)) SEL(mark(z0), z1) -> c50(SEL(z0, z1)) SEL(z0, mark(z1)) -> c51(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c52(SEL(z0, z1)) ISNATURAL(ok(z0)) -> c56(ISNATURAL(z0)) ISLNAT(ok(z0)) -> c57(ISLNAT(z0)) ISPLNAT(ok(z0)) -> c58(ISPLNAT(z0)) TOP(mark(z0)) -> c59(TOP(proper(z0))) S tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) FST(mark(z0)) -> c2(FST(z0)) FST(ok(z0)) -> c3(FST(z0)) SPLITAT(mark(z0), z1) -> c4(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c5(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c6(SPLITAT(z0, z1)) U11'(mark(z0), z1, z2) -> c7(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c8(U11'(z0, z1, z2)) SND(mark(z0)) -> c9(SND(z0)) SND(ok(z0)) -> c10(SND(z0)) U21'(mark(z0), z1) -> c11(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c12(U21'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U41'(mark(z0), z1) -> c15(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c16(U41'(z0, z1)) CONS(mark(z0), z1) -> c17(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c18(CONS(z0, z1)) NATSFROM(mark(z0)) -> c19(NATSFROM(z0)) NATSFROM(ok(z0)) -> c20(NATSFROM(z0)) S(mark(z0)) -> c21(S(z0)) S(ok(z0)) -> c22(S(z0)) U51'(mark(z0), z1, z2) -> c23(U51'(z0, z1, z2)) U51'(ok(z0), ok(z1), ok(z2)) -> c24(U51'(z0, z1, z2)) HEAD(mark(z0)) -> c25(HEAD(z0)) HEAD(ok(z0)) -> c26(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c27(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c28(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c29(AFTERNTH(z0, z1)) U61'(mark(z0), z1) -> c30(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c31(U61'(z0, z1)) U71'(mark(z0), z1) -> c32(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c33(U71'(z0, z1)) PAIR(mark(z0), z1) -> c34(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c35(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c36(PAIR(z0, z1)) U81'(mark(z0), z1, z2, z3) -> c37(U81'(z0, z1, z2, z3)) U81'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c38(U81'(z0, z1, z2, z3)) U82'(mark(z0), z1) -> c39(U82'(z0, z1)) U82'(ok(z0), ok(z1)) -> c40(U82'(z0, z1)) U91'(mark(z0), z1) -> c41(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c42(U91'(z0, z1)) AND(mark(z0), z1) -> c43(AND(z0, z1)) AND(ok(z0), ok(z1)) -> c44(AND(z0, z1)) TAIL(mark(z0)) -> c45(TAIL(z0)) TAIL(ok(z0)) -> c46(TAIL(z0)) TAKE(mark(z0), z1) -> c47(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c48(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c49(TAKE(z0, z1)) SEL(mark(z0), z1) -> c50(SEL(z0, z1)) SEL(z0, mark(z1)) -> c51(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c52(SEL(z0, z1)) ISNATURAL(ok(z0)) -> c56(ISNATURAL(z0)) ISLNAT(ok(z0)) -> c57(ISLNAT(z0)) ISPLNAT(ok(z0)) -> c58(ISPLNAT(z0)) TOP(mark(z0)) -> c59(TOP(proper(z0))) K tuples:none Defined Rule Symbols: U101_3, fst_1, splitAt_2, U11_3, snd_1, U21_2, U31_2, U41_2, cons_2, natsFrom_1, s_1, U51_3, head_1, afterNth_2, U61_2, U71_2, pair_2, U81_4, U82_2, U91_2, and_2, tail_1, take_2, sel_2, proper_1, isNatural_1, isLNat_1, isPLNat_1, top_1 Defined Pair Symbols: U101'_3, FST_1, SPLITAT_2, U11'_3, SND_1, U21'_2, U31'_2, U41'_2, CONS_2, NATSFROM_1, S_1, U51'_3, HEAD_1, AFTERNTH_2, U61'_2, U71'_2, PAIR_2, U81'_4, U82'_2, U91'_2, AND_2, TAIL_1, TAKE_2, SEL_2, ISNATURAL_1, ISLNAT_1, ISPLNAT_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, c56_1, c57_1, c58_1, c59_1 ---------------------------------------- (11) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2)) U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2)) fst(mark(z0)) -> mark(fst(z0)) fst(ok(z0)) -> ok(fst(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)) U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) snd(mark(z0)) -> mark(snd(z0)) snd(ok(z0)) -> ok(snd(z0)) U21(mark(z0), z1) -> mark(U21(z0, z1)) U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) U31(mark(z0), z1) -> mark(U31(z0, z1)) U31(ok(z0), ok(z1)) -> ok(U31(z0, z1)) U41(mark(z0), z1) -> mark(U41(z0, z1)) U41(ok(z0), ok(z1)) -> ok(U41(z0, z1)) cons(mark(z0), z1) -> mark(cons(z0, z1)) cons(ok(z0), ok(z1)) -> ok(cons(z0, z1)) natsFrom(mark(z0)) -> mark(natsFrom(z0)) natsFrom(ok(z0)) -> ok(natsFrom(z0)) s(mark(z0)) -> mark(s(z0)) s(ok(z0)) -> ok(s(z0)) U51(mark(z0), z1, z2) -> mark(U51(z0, z1, z2)) U51(ok(z0), ok(z1), ok(z2)) -> ok(U51(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)) U61(mark(z0), z1) -> mark(U61(z0, z1)) U61(ok(z0), ok(z1)) -> ok(U61(z0, z1)) U71(mark(z0), z1) -> mark(U71(z0, z1)) U71(ok(z0), ok(z1)) -> ok(U71(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)) U81(mark(z0), z1, z2, z3) -> mark(U81(z0, z1, z2, z3)) U81(ok(z0), ok(z1), ok(z2), ok(z3)) -> ok(U81(z0, z1, z2, z3)) U82(mark(z0), z1) -> mark(U82(z0, z1)) U82(ok(z0), ok(z1)) -> ok(U82(z0, z1)) U91(mark(z0), z1) -> mark(U91(z0, z1)) U91(ok(z0), ok(z1)) -> ok(U91(z0, z1)) and(mark(z0), z1) -> mark(and(z0, z1)) and(ok(z0), ok(z1)) -> ok(and(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)) 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)) isNatural(ok(z0)) -> ok(isNatural(z0)) isLNat(ok(z0)) -> ok(isLNat(z0)) isPLNat(ok(z0)) -> ok(isPLNat(z0)) top(mark(z0)) -> top(proper(z0)) top(ok(z0)) -> top(active(z0)) ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(nil) -> ok(nil) proper(0) -> ok(0) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) FST(mark(z0)) -> c2(FST(z0)) FST(ok(z0)) -> c3(FST(z0)) SPLITAT(mark(z0), z1) -> c4(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c5(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c6(SPLITAT(z0, z1)) U11'(mark(z0), z1, z2) -> c7(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c8(U11'(z0, z1, z2)) SND(mark(z0)) -> c9(SND(z0)) SND(ok(z0)) -> c10(SND(z0)) U21'(mark(z0), z1) -> c11(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c12(U21'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U41'(mark(z0), z1) -> c15(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c16(U41'(z0, z1)) CONS(mark(z0), z1) -> c17(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c18(CONS(z0, z1)) NATSFROM(mark(z0)) -> c19(NATSFROM(z0)) NATSFROM(ok(z0)) -> c20(NATSFROM(z0)) S(mark(z0)) -> c21(S(z0)) S(ok(z0)) -> c22(S(z0)) U51'(mark(z0), z1, z2) -> c23(U51'(z0, z1, z2)) U51'(ok(z0), ok(z1), ok(z2)) -> c24(U51'(z0, z1, z2)) HEAD(mark(z0)) -> c25(HEAD(z0)) HEAD(ok(z0)) -> c26(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c27(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c28(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c29(AFTERNTH(z0, z1)) U61'(mark(z0), z1) -> c30(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c31(U61'(z0, z1)) U71'(mark(z0), z1) -> c32(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c33(U71'(z0, z1)) PAIR(mark(z0), z1) -> c34(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c35(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c36(PAIR(z0, z1)) U81'(mark(z0), z1, z2, z3) -> c37(U81'(z0, z1, z2, z3)) U81'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c38(U81'(z0, z1, z2, z3)) U82'(mark(z0), z1) -> c39(U82'(z0, z1)) U82'(ok(z0), ok(z1)) -> c40(U82'(z0, z1)) U91'(mark(z0), z1) -> c41(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c42(U91'(z0, z1)) AND(mark(z0), z1) -> c43(AND(z0, z1)) AND(ok(z0), ok(z1)) -> c44(AND(z0, z1)) TAIL(mark(z0)) -> c45(TAIL(z0)) TAIL(ok(z0)) -> c46(TAIL(z0)) TAKE(mark(z0), z1) -> c47(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c48(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c49(TAKE(z0, z1)) SEL(mark(z0), z1) -> c50(SEL(z0, z1)) SEL(z0, mark(z1)) -> c51(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c52(SEL(z0, z1)) ISNATURAL(ok(z0)) -> c56(ISNATURAL(z0)) ISLNAT(ok(z0)) -> c57(ISLNAT(z0)) ISPLNAT(ok(z0)) -> c58(ISPLNAT(z0)) TOP(mark(z0)) -> c59(TOP(proper(z0))) S tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) FST(mark(z0)) -> c2(FST(z0)) FST(ok(z0)) -> c3(FST(z0)) SPLITAT(mark(z0), z1) -> c4(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c5(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c6(SPLITAT(z0, z1)) U11'(mark(z0), z1, z2) -> c7(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c8(U11'(z0, z1, z2)) SND(mark(z0)) -> c9(SND(z0)) SND(ok(z0)) -> c10(SND(z0)) U21'(mark(z0), z1) -> c11(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c12(U21'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U41'(mark(z0), z1) -> c15(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c16(U41'(z0, z1)) CONS(mark(z0), z1) -> c17(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c18(CONS(z0, z1)) NATSFROM(mark(z0)) -> c19(NATSFROM(z0)) NATSFROM(ok(z0)) -> c20(NATSFROM(z0)) S(mark(z0)) -> c21(S(z0)) S(ok(z0)) -> c22(S(z0)) U51'(mark(z0), z1, z2) -> c23(U51'(z0, z1, z2)) U51'(ok(z0), ok(z1), ok(z2)) -> c24(U51'(z0, z1, z2)) HEAD(mark(z0)) -> c25(HEAD(z0)) HEAD(ok(z0)) -> c26(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c27(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c28(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c29(AFTERNTH(z0, z1)) U61'(mark(z0), z1) -> c30(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c31(U61'(z0, z1)) U71'(mark(z0), z1) -> c32(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c33(U71'(z0, z1)) PAIR(mark(z0), z1) -> c34(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c35(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c36(PAIR(z0, z1)) U81'(mark(z0), z1, z2, z3) -> c37(U81'(z0, z1, z2, z3)) U81'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c38(U81'(z0, z1, z2, z3)) U82'(mark(z0), z1) -> c39(U82'(z0, z1)) U82'(ok(z0), ok(z1)) -> c40(U82'(z0, z1)) U91'(mark(z0), z1) -> c41(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c42(U91'(z0, z1)) AND(mark(z0), z1) -> c43(AND(z0, z1)) AND(ok(z0), ok(z1)) -> c44(AND(z0, z1)) TAIL(mark(z0)) -> c45(TAIL(z0)) TAIL(ok(z0)) -> c46(TAIL(z0)) TAKE(mark(z0), z1) -> c47(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c48(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c49(TAKE(z0, z1)) SEL(mark(z0), z1) -> c50(SEL(z0, z1)) SEL(z0, mark(z1)) -> c51(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c52(SEL(z0, z1)) ISNATURAL(ok(z0)) -> c56(ISNATURAL(z0)) ISLNAT(ok(z0)) -> c57(ISLNAT(z0)) ISPLNAT(ok(z0)) -> c58(ISPLNAT(z0)) TOP(mark(z0)) -> c59(TOP(proper(z0))) K tuples:none Defined Rule Symbols: proper_1 Defined Pair Symbols: U101'_3, FST_1, SPLITAT_2, U11'_3, SND_1, U21'_2, U31'_2, U41'_2, CONS_2, NATSFROM_1, S_1, U51'_3, HEAD_1, AFTERNTH_2, U61'_2, U71'_2, PAIR_2, U81'_4, U82'_2, U91'_2, AND_2, TAIL_1, TAKE_2, SEL_2, ISNATURAL_1, ISLNAT_1, ISPLNAT_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, c56_1, c57_1, c58_1, c59_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. U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) FST(mark(z0)) -> c2(FST(z0)) FST(ok(z0)) -> c3(FST(z0)) SPLITAT(mark(z0), z1) -> c4(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c5(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c6(SPLITAT(z0, z1)) U11'(mark(z0), z1, z2) -> c7(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c8(U11'(z0, z1, z2)) SND(mark(z0)) -> c9(SND(z0)) SND(ok(z0)) -> c10(SND(z0)) U21'(mark(z0), z1) -> c11(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c12(U21'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U41'(mark(z0), z1) -> c15(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c16(U41'(z0, z1)) CONS(mark(z0), z1) -> c17(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c18(CONS(z0, z1)) NATSFROM(mark(z0)) -> c19(NATSFROM(z0)) NATSFROM(ok(z0)) -> c20(NATSFROM(z0)) S(mark(z0)) -> c21(S(z0)) S(ok(z0)) -> c22(S(z0)) U51'(mark(z0), z1, z2) -> c23(U51'(z0, z1, z2)) U51'(ok(z0), ok(z1), ok(z2)) -> c24(U51'(z0, z1, z2)) HEAD(mark(z0)) -> c25(HEAD(z0)) HEAD(ok(z0)) -> c26(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c27(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c28(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c29(AFTERNTH(z0, z1)) U61'(mark(z0), z1) -> c30(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c31(U61'(z0, z1)) U71'(mark(z0), z1) -> c32(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c33(U71'(z0, z1)) PAIR(mark(z0), z1) -> c34(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c35(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c36(PAIR(z0, z1)) U81'(mark(z0), z1, z2, z3) -> c37(U81'(z0, z1, z2, z3)) U81'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c38(U81'(z0, z1, z2, z3)) U82'(mark(z0), z1) -> c39(U82'(z0, z1)) U82'(ok(z0), ok(z1)) -> c40(U82'(z0, z1)) U91'(mark(z0), z1) -> c41(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c42(U91'(z0, z1)) AND(mark(z0), z1) -> c43(AND(z0, z1)) AND(ok(z0), ok(z1)) -> c44(AND(z0, z1)) TAIL(mark(z0)) -> c45(TAIL(z0)) TAIL(ok(z0)) -> c46(TAIL(z0)) TAKE(mark(z0), z1) -> c47(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c48(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c49(TAKE(z0, z1)) SEL(mark(z0), z1) -> c50(SEL(z0, z1)) SEL(z0, mark(z1)) -> c51(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c52(SEL(z0, z1)) ISNATURAL(ok(z0)) -> c56(ISNATURAL(z0)) ISLNAT(ok(z0)) -> c57(ISLNAT(z0)) ISPLNAT(ok(z0)) -> c58(ISPLNAT(z0)) We considered the (Usable) Rules: proper(tt) -> ok(tt) proper(nil) -> ok(nil) proper(0) -> ok(0) And the Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) FST(mark(z0)) -> c2(FST(z0)) FST(ok(z0)) -> c3(FST(z0)) SPLITAT(mark(z0), z1) -> c4(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c5(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c6(SPLITAT(z0, z1)) U11'(mark(z0), z1, z2) -> c7(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c8(U11'(z0, z1, z2)) SND(mark(z0)) -> c9(SND(z0)) SND(ok(z0)) -> c10(SND(z0)) U21'(mark(z0), z1) -> c11(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c12(U21'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U41'(mark(z0), z1) -> c15(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c16(U41'(z0, z1)) CONS(mark(z0), z1) -> c17(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c18(CONS(z0, z1)) NATSFROM(mark(z0)) -> c19(NATSFROM(z0)) NATSFROM(ok(z0)) -> c20(NATSFROM(z0)) S(mark(z0)) -> c21(S(z0)) S(ok(z0)) -> c22(S(z0)) U51'(mark(z0), z1, z2) -> c23(U51'(z0, z1, z2)) U51'(ok(z0), ok(z1), ok(z2)) -> c24(U51'(z0, z1, z2)) HEAD(mark(z0)) -> c25(HEAD(z0)) HEAD(ok(z0)) -> c26(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c27(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c28(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c29(AFTERNTH(z0, z1)) U61'(mark(z0), z1) -> c30(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c31(U61'(z0, z1)) U71'(mark(z0), z1) -> c32(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c33(U71'(z0, z1)) PAIR(mark(z0), z1) -> c34(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c35(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c36(PAIR(z0, z1)) U81'(mark(z0), z1, z2, z3) -> c37(U81'(z0, z1, z2, z3)) U81'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c38(U81'(z0, z1, z2, z3)) U82'(mark(z0), z1) -> c39(U82'(z0, z1)) U82'(ok(z0), ok(z1)) -> c40(U82'(z0, z1)) U91'(mark(z0), z1) -> c41(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c42(U91'(z0, z1)) AND(mark(z0), z1) -> c43(AND(z0, z1)) AND(ok(z0), ok(z1)) -> c44(AND(z0, z1)) TAIL(mark(z0)) -> c45(TAIL(z0)) TAIL(ok(z0)) -> c46(TAIL(z0)) TAKE(mark(z0), z1) -> c47(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c48(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c49(TAKE(z0, z1)) SEL(mark(z0), z1) -> c50(SEL(z0, z1)) SEL(z0, mark(z1)) -> c51(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c52(SEL(z0, z1)) ISNATURAL(ok(z0)) -> c56(ISNATURAL(z0)) ISLNAT(ok(z0)) -> c57(ISLNAT(z0)) ISPLNAT(ok(z0)) -> c58(ISPLNAT(z0)) TOP(mark(z0)) -> c59(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(AND(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(ISLNAT(x_1)) = x_1 POL(ISNATURAL(x_1)) = x_1 POL(ISPLNAT(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(U101'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U11'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U21'(x_1, x_2)) = x_1 + x_2 POL(U31'(x_1, x_2)) = x_1 + x_2 POL(U41'(x_1, x_2)) = x_1 + x_2 POL(U51'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U61'(x_1, x_2)) = x_1 + x_2 POL(U71'(x_1, x_2)) = x_1 + x_2 POL(U81'(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(U82'(x_1, x_2)) = x_1 + x_2 POL(U91'(x_1, x_2)) = x_1 + x_2 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(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(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] ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(nil) -> ok(nil) proper(0) -> ok(0) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) FST(mark(z0)) -> c2(FST(z0)) FST(ok(z0)) -> c3(FST(z0)) SPLITAT(mark(z0), z1) -> c4(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c5(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c6(SPLITAT(z0, z1)) U11'(mark(z0), z1, z2) -> c7(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c8(U11'(z0, z1, z2)) SND(mark(z0)) -> c9(SND(z0)) SND(ok(z0)) -> c10(SND(z0)) U21'(mark(z0), z1) -> c11(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c12(U21'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U41'(mark(z0), z1) -> c15(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c16(U41'(z0, z1)) CONS(mark(z0), z1) -> c17(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c18(CONS(z0, z1)) NATSFROM(mark(z0)) -> c19(NATSFROM(z0)) NATSFROM(ok(z0)) -> c20(NATSFROM(z0)) S(mark(z0)) -> c21(S(z0)) S(ok(z0)) -> c22(S(z0)) U51'(mark(z0), z1, z2) -> c23(U51'(z0, z1, z2)) U51'(ok(z0), ok(z1), ok(z2)) -> c24(U51'(z0, z1, z2)) HEAD(mark(z0)) -> c25(HEAD(z0)) HEAD(ok(z0)) -> c26(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c27(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c28(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c29(AFTERNTH(z0, z1)) U61'(mark(z0), z1) -> c30(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c31(U61'(z0, z1)) U71'(mark(z0), z1) -> c32(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c33(U71'(z0, z1)) PAIR(mark(z0), z1) -> c34(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c35(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c36(PAIR(z0, z1)) U81'(mark(z0), z1, z2, z3) -> c37(U81'(z0, z1, z2, z3)) U81'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c38(U81'(z0, z1, z2, z3)) U82'(mark(z0), z1) -> c39(U82'(z0, z1)) U82'(ok(z0), ok(z1)) -> c40(U82'(z0, z1)) U91'(mark(z0), z1) -> c41(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c42(U91'(z0, z1)) AND(mark(z0), z1) -> c43(AND(z0, z1)) AND(ok(z0), ok(z1)) -> c44(AND(z0, z1)) TAIL(mark(z0)) -> c45(TAIL(z0)) TAIL(ok(z0)) -> c46(TAIL(z0)) TAKE(mark(z0), z1) -> c47(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c48(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c49(TAKE(z0, z1)) SEL(mark(z0), z1) -> c50(SEL(z0, z1)) SEL(z0, mark(z1)) -> c51(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c52(SEL(z0, z1)) ISNATURAL(ok(z0)) -> c56(ISNATURAL(z0)) ISLNAT(ok(z0)) -> c57(ISLNAT(z0)) ISPLNAT(ok(z0)) -> c58(ISPLNAT(z0)) TOP(mark(z0)) -> c59(TOP(proper(z0))) S tuples: TOP(mark(z0)) -> c59(TOP(proper(z0))) K tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) FST(mark(z0)) -> c2(FST(z0)) FST(ok(z0)) -> c3(FST(z0)) SPLITAT(mark(z0), z1) -> c4(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c5(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c6(SPLITAT(z0, z1)) U11'(mark(z0), z1, z2) -> c7(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c8(U11'(z0, z1, z2)) SND(mark(z0)) -> c9(SND(z0)) SND(ok(z0)) -> c10(SND(z0)) U21'(mark(z0), z1) -> c11(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c12(U21'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U41'(mark(z0), z1) -> c15(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c16(U41'(z0, z1)) CONS(mark(z0), z1) -> c17(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c18(CONS(z0, z1)) NATSFROM(mark(z0)) -> c19(NATSFROM(z0)) NATSFROM(ok(z0)) -> c20(NATSFROM(z0)) S(mark(z0)) -> c21(S(z0)) S(ok(z0)) -> c22(S(z0)) U51'(mark(z0), z1, z2) -> c23(U51'(z0, z1, z2)) U51'(ok(z0), ok(z1), ok(z2)) -> c24(U51'(z0, z1, z2)) HEAD(mark(z0)) -> c25(HEAD(z0)) HEAD(ok(z0)) -> c26(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c27(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c28(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c29(AFTERNTH(z0, z1)) U61'(mark(z0), z1) -> c30(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c31(U61'(z0, z1)) U71'(mark(z0), z1) -> c32(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c33(U71'(z0, z1)) PAIR(mark(z0), z1) -> c34(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c35(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c36(PAIR(z0, z1)) U81'(mark(z0), z1, z2, z3) -> c37(U81'(z0, z1, z2, z3)) U81'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c38(U81'(z0, z1, z2, z3)) U82'(mark(z0), z1) -> c39(U82'(z0, z1)) U82'(ok(z0), ok(z1)) -> c40(U82'(z0, z1)) U91'(mark(z0), z1) -> c41(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c42(U91'(z0, z1)) AND(mark(z0), z1) -> c43(AND(z0, z1)) AND(ok(z0), ok(z1)) -> c44(AND(z0, z1)) TAIL(mark(z0)) -> c45(TAIL(z0)) TAIL(ok(z0)) -> c46(TAIL(z0)) TAKE(mark(z0), z1) -> c47(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c48(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c49(TAKE(z0, z1)) SEL(mark(z0), z1) -> c50(SEL(z0, z1)) SEL(z0, mark(z1)) -> c51(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c52(SEL(z0, z1)) ISNATURAL(ok(z0)) -> c56(ISNATURAL(z0)) ISLNAT(ok(z0)) -> c57(ISLNAT(z0)) ISPLNAT(ok(z0)) -> c58(ISPLNAT(z0)) Defined Rule Symbols: proper_1 Defined Pair Symbols: U101'_3, FST_1, SPLITAT_2, U11'_3, SND_1, U21'_2, U31'_2, U41'_2, CONS_2, NATSFROM_1, S_1, U51'_3, HEAD_1, AFTERNTH_2, U61'_2, U71'_2, PAIR_2, U81'_4, U82'_2, U91'_2, AND_2, TAIL_1, TAKE_2, SEL_2, ISNATURAL_1, ISLNAT_1, ISPLNAT_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, c56_1, c57_1, c58_1, c59_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. TOP(mark(z0)) -> c59(TOP(proper(z0))) We considered the (Usable) Rules: proper(tt) -> ok(tt) proper(nil) -> ok(nil) proper(0) -> ok(0) And the Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) FST(mark(z0)) -> c2(FST(z0)) FST(ok(z0)) -> c3(FST(z0)) SPLITAT(mark(z0), z1) -> c4(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c5(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c6(SPLITAT(z0, z1)) U11'(mark(z0), z1, z2) -> c7(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c8(U11'(z0, z1, z2)) SND(mark(z0)) -> c9(SND(z0)) SND(ok(z0)) -> c10(SND(z0)) U21'(mark(z0), z1) -> c11(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c12(U21'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U41'(mark(z0), z1) -> c15(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c16(U41'(z0, z1)) CONS(mark(z0), z1) -> c17(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c18(CONS(z0, z1)) NATSFROM(mark(z0)) -> c19(NATSFROM(z0)) NATSFROM(ok(z0)) -> c20(NATSFROM(z0)) S(mark(z0)) -> c21(S(z0)) S(ok(z0)) -> c22(S(z0)) U51'(mark(z0), z1, z2) -> c23(U51'(z0, z1, z2)) U51'(ok(z0), ok(z1), ok(z2)) -> c24(U51'(z0, z1, z2)) HEAD(mark(z0)) -> c25(HEAD(z0)) HEAD(ok(z0)) -> c26(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c27(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c28(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c29(AFTERNTH(z0, z1)) U61'(mark(z0), z1) -> c30(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c31(U61'(z0, z1)) U71'(mark(z0), z1) -> c32(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c33(U71'(z0, z1)) PAIR(mark(z0), z1) -> c34(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c35(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c36(PAIR(z0, z1)) U81'(mark(z0), z1, z2, z3) -> c37(U81'(z0, z1, z2, z3)) U81'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c38(U81'(z0, z1, z2, z3)) U82'(mark(z0), z1) -> c39(U82'(z0, z1)) U82'(ok(z0), ok(z1)) -> c40(U82'(z0, z1)) U91'(mark(z0), z1) -> c41(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c42(U91'(z0, z1)) AND(mark(z0), z1) -> c43(AND(z0, z1)) AND(ok(z0), ok(z1)) -> c44(AND(z0, z1)) TAIL(mark(z0)) -> c45(TAIL(z0)) TAIL(ok(z0)) -> c46(TAIL(z0)) TAKE(mark(z0), z1) -> c47(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c48(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c49(TAKE(z0, z1)) SEL(mark(z0), z1) -> c50(SEL(z0, z1)) SEL(z0, mark(z1)) -> c51(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c52(SEL(z0, z1)) ISNATURAL(ok(z0)) -> c56(ISNATURAL(z0)) ISLNAT(ok(z0)) -> c57(ISLNAT(z0)) ISPLNAT(ok(z0)) -> c58(ISPLNAT(z0)) TOP(mark(z0)) -> c59(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(AND(x_1, x_2)) = 0 POL(CONS(x_1, x_2)) = 0 POL(FST(x_1)) = 0 POL(HEAD(x_1)) = 0 POL(ISLNAT(x_1)) = 0 POL(ISNATURAL(x_1)) = 0 POL(ISPLNAT(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)) = x_1 POL(U101'(x_1, x_2, x_3)) = 0 POL(U11'(x_1, x_2, x_3)) = 0 POL(U21'(x_1, x_2)) = 0 POL(U31'(x_1, x_2)) = 0 POL(U41'(x_1, x_2)) = 0 POL(U51'(x_1, x_2, x_3)) = 0 POL(U61'(x_1, x_2)) = 0 POL(U71'(x_1, x_2)) = 0 POL(U81'(x_1, x_2, x_3, x_4)) = 0 POL(U82'(x_1, x_2)) = 0 POL(U91'(x_1, x_2)) = 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(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(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) = 0 POL(ok(x_1)) = 0 POL(proper(x_1)) = x_1 POL(tt) = [2] ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(nil) -> ok(nil) proper(0) -> ok(0) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) FST(mark(z0)) -> c2(FST(z0)) FST(ok(z0)) -> c3(FST(z0)) SPLITAT(mark(z0), z1) -> c4(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c5(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c6(SPLITAT(z0, z1)) U11'(mark(z0), z1, z2) -> c7(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c8(U11'(z0, z1, z2)) SND(mark(z0)) -> c9(SND(z0)) SND(ok(z0)) -> c10(SND(z0)) U21'(mark(z0), z1) -> c11(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c12(U21'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U41'(mark(z0), z1) -> c15(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c16(U41'(z0, z1)) CONS(mark(z0), z1) -> c17(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c18(CONS(z0, z1)) NATSFROM(mark(z0)) -> c19(NATSFROM(z0)) NATSFROM(ok(z0)) -> c20(NATSFROM(z0)) S(mark(z0)) -> c21(S(z0)) S(ok(z0)) -> c22(S(z0)) U51'(mark(z0), z1, z2) -> c23(U51'(z0, z1, z2)) U51'(ok(z0), ok(z1), ok(z2)) -> c24(U51'(z0, z1, z2)) HEAD(mark(z0)) -> c25(HEAD(z0)) HEAD(ok(z0)) -> c26(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c27(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c28(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c29(AFTERNTH(z0, z1)) U61'(mark(z0), z1) -> c30(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c31(U61'(z0, z1)) U71'(mark(z0), z1) -> c32(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c33(U71'(z0, z1)) PAIR(mark(z0), z1) -> c34(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c35(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c36(PAIR(z0, z1)) U81'(mark(z0), z1, z2, z3) -> c37(U81'(z0, z1, z2, z3)) U81'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c38(U81'(z0, z1, z2, z3)) U82'(mark(z0), z1) -> c39(U82'(z0, z1)) U82'(ok(z0), ok(z1)) -> c40(U82'(z0, z1)) U91'(mark(z0), z1) -> c41(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c42(U91'(z0, z1)) AND(mark(z0), z1) -> c43(AND(z0, z1)) AND(ok(z0), ok(z1)) -> c44(AND(z0, z1)) TAIL(mark(z0)) -> c45(TAIL(z0)) TAIL(ok(z0)) -> c46(TAIL(z0)) TAKE(mark(z0), z1) -> c47(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c48(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c49(TAKE(z0, z1)) SEL(mark(z0), z1) -> c50(SEL(z0, z1)) SEL(z0, mark(z1)) -> c51(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c52(SEL(z0, z1)) ISNATURAL(ok(z0)) -> c56(ISNATURAL(z0)) ISLNAT(ok(z0)) -> c57(ISLNAT(z0)) ISPLNAT(ok(z0)) -> c58(ISPLNAT(z0)) TOP(mark(z0)) -> c59(TOP(proper(z0))) S tuples:none K tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) FST(mark(z0)) -> c2(FST(z0)) FST(ok(z0)) -> c3(FST(z0)) SPLITAT(mark(z0), z1) -> c4(SPLITAT(z0, z1)) SPLITAT(z0, mark(z1)) -> c5(SPLITAT(z0, z1)) SPLITAT(ok(z0), ok(z1)) -> c6(SPLITAT(z0, z1)) U11'(mark(z0), z1, z2) -> c7(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c8(U11'(z0, z1, z2)) SND(mark(z0)) -> c9(SND(z0)) SND(ok(z0)) -> c10(SND(z0)) U21'(mark(z0), z1) -> c11(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c12(U21'(z0, z1)) U31'(mark(z0), z1) -> c13(U31'(z0, z1)) U31'(ok(z0), ok(z1)) -> c14(U31'(z0, z1)) U41'(mark(z0), z1) -> c15(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c16(U41'(z0, z1)) CONS(mark(z0), z1) -> c17(CONS(z0, z1)) CONS(ok(z0), ok(z1)) -> c18(CONS(z0, z1)) NATSFROM(mark(z0)) -> c19(NATSFROM(z0)) NATSFROM(ok(z0)) -> c20(NATSFROM(z0)) S(mark(z0)) -> c21(S(z0)) S(ok(z0)) -> c22(S(z0)) U51'(mark(z0), z1, z2) -> c23(U51'(z0, z1, z2)) U51'(ok(z0), ok(z1), ok(z2)) -> c24(U51'(z0, z1, z2)) HEAD(mark(z0)) -> c25(HEAD(z0)) HEAD(ok(z0)) -> c26(HEAD(z0)) AFTERNTH(mark(z0), z1) -> c27(AFTERNTH(z0, z1)) AFTERNTH(z0, mark(z1)) -> c28(AFTERNTH(z0, z1)) AFTERNTH(ok(z0), ok(z1)) -> c29(AFTERNTH(z0, z1)) U61'(mark(z0), z1) -> c30(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c31(U61'(z0, z1)) U71'(mark(z0), z1) -> c32(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c33(U71'(z0, z1)) PAIR(mark(z0), z1) -> c34(PAIR(z0, z1)) PAIR(z0, mark(z1)) -> c35(PAIR(z0, z1)) PAIR(ok(z0), ok(z1)) -> c36(PAIR(z0, z1)) U81'(mark(z0), z1, z2, z3) -> c37(U81'(z0, z1, z2, z3)) U81'(ok(z0), ok(z1), ok(z2), ok(z3)) -> c38(U81'(z0, z1, z2, z3)) U82'(mark(z0), z1) -> c39(U82'(z0, z1)) U82'(ok(z0), ok(z1)) -> c40(U82'(z0, z1)) U91'(mark(z0), z1) -> c41(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c42(U91'(z0, z1)) AND(mark(z0), z1) -> c43(AND(z0, z1)) AND(ok(z0), ok(z1)) -> c44(AND(z0, z1)) TAIL(mark(z0)) -> c45(TAIL(z0)) TAIL(ok(z0)) -> c46(TAIL(z0)) TAKE(mark(z0), z1) -> c47(TAKE(z0, z1)) TAKE(z0, mark(z1)) -> c48(TAKE(z0, z1)) TAKE(ok(z0), ok(z1)) -> c49(TAKE(z0, z1)) SEL(mark(z0), z1) -> c50(SEL(z0, z1)) SEL(z0, mark(z1)) -> c51(SEL(z0, z1)) SEL(ok(z0), ok(z1)) -> c52(SEL(z0, z1)) ISNATURAL(ok(z0)) -> c56(ISNATURAL(z0)) ISLNAT(ok(z0)) -> c57(ISLNAT(z0)) ISPLNAT(ok(z0)) -> c58(ISPLNAT(z0)) TOP(mark(z0)) -> c59(TOP(proper(z0))) Defined Rule Symbols: proper_1 Defined Pair Symbols: U101'_3, FST_1, SPLITAT_2, U11'_3, SND_1, U21'_2, U31'_2, U41'_2, CONS_2, NATSFROM_1, S_1, U51'_3, HEAD_1, AFTERNTH_2, U61'_2, U71'_2, PAIR_2, U81'_4, U82'_2, U91'_2, AND_2, TAIL_1, TAKE_2, SEL_2, ISNATURAL_1, ISLNAT_1, ISPLNAT_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, c56_1, c57_1, c58_1, c59_1 ---------------------------------------- (17) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (18) BOUNDS(1, 1) ---------------------------------------- (19) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (20) 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(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (21) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (22) Obligation: TRS: Rules: active(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok ---------------------------------------- (23) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: active, fst, splitAt, snd, cons, natsFrom, s, head, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: fst < active splitAt < active snd < active cons < active natsFrom < active s < active head < active afterNth < active pair < active U82 < active U11 < active and < active isNatural < active isLNat < active U21 < active U31 < active isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top fst < proper splitAt < proper snd < proper cons < proper natsFrom < proper s < proper head < proper afterNth < proper pair < proper U82 < proper U11 < proper and < proper isNatural < proper isLNat < proper U21 < proper U31 < proper isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (24) Obligation: TRS: Rules: active(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: fst, active, splitAt, snd, cons, natsFrom, s, head, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: fst < active splitAt < active snd < active cons < active natsFrom < active s < active head < active afterNth < active pair < active U82 < active U11 < active and < active isNatural < active isLNat < active U21 < active U31 < active isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top fst < proper splitAt < proper snd < proper cons < proper natsFrom < proper s < proper head < proper afterNth < proper pair < proper U82 < proper U11 < proper and < proper isNatural < proper isLNat < proper U21 < proper U31 < proper isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) Induction Base: fst(gen_tt:mark:nil:0':ok3_0(+(1, 0))) Induction Step: fst(gen_tt:mark:nil:0':ok3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1) mark(fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_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). ---------------------------------------- (26) Complex Obligation (BEST) ---------------------------------------- (27) Obligation: Proved the lower bound n^1 for the following obligation: TRS: Rules: active(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: fst, active, splitAt, snd, cons, natsFrom, s, head, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: fst < active splitAt < active snd < active cons < active natsFrom < active s < active head < active afterNth < active pair < active U82 < active U11 < active and < active isNatural < active isLNat < active U21 < active U31 < active isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top fst < proper splitAt < proper snd < proper cons < proper natsFrom < proper s < proper head < proper afterNth < proper pair < proper U82 < proper U11 < proper and < proper isNatural < proper isLNat < proper U21 < proper U31 < proper isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (28) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (29) BOUNDS(n^1, INF) ---------------------------------------- (30) Obligation: TRS: Rules: active(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: splitAt, active, snd, cons, natsFrom, s, head, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: splitAt < active snd < active cons < active natsFrom < active s < active head < active afterNth < active pair < active U82 < active U11 < active and < active isNatural < active isLNat < active U21 < active U31 < active isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top splitAt < proper snd < proper cons < proper natsFrom < proper s < proper head < proper afterNth < proper pair < proper U82 < proper U11 < proper and < proper isNatural < proper isLNat < proper U21 < proper U31 < proper isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (31) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) Induction Base: splitAt(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b)) Induction Step: splitAt(gen_tt:mark:nil:0':ok3_0(+(1, +(n731_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1) mark(splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':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). ---------------------------------------- (32) Obligation: TRS: Rules: active(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: snd, active, cons, natsFrom, s, head, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: snd < active cons < active natsFrom < active s < active head < active afterNth < active pair < active U82 < active U11 < active and < active isNatural < active isLNat < active U21 < active U31 < active isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top snd < proper cons < proper natsFrom < proper s < proper head < proper afterNth < proper pair < proper U82 < proper U11 < proper and < proper isNatural < proper isLNat < proper U21 < proper U31 < proper isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (33) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) Induction Base: snd(gen_tt:mark:nil:0':ok3_0(+(1, 0))) Induction Step: snd(gen_tt:mark:nil:0':ok3_0(+(1, +(n3715_0, 1)))) ->_R^Omega(1) mark(snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (34) Obligation: TRS: Rules: active(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: cons, active, natsFrom, s, head, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: cons < active natsFrom < active s < active head < active afterNth < active pair < active U82 < active U11 < active and < active isNatural < active isLNat < active U21 < active U31 < active isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top cons < proper natsFrom < proper s < proper head < proper afterNth < proper pair < proper U82 < proper U11 < proper and < proper isNatural < proper isLNat < proper U21 < proper U31 < proper isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (35) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n4692_0) Induction Base: cons(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b)) Induction Step: cons(gen_tt:mark:nil:0':ok3_0(+(1, +(n4692_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1) mark(cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':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). ---------------------------------------- (36) Obligation: TRS: Rules: active(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n4692_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: natsFrom, active, s, head, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: natsFrom < active s < active head < active afterNth < active pair < active U82 < active U11 < active and < active isNatural < active isLNat < active U21 < active U31 < active isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top natsFrom < proper s < proper head < proper afterNth < proper pair < proper U82 < proper U11 < proper and < proper isNatural < proper isLNat < proper U21 < proper U31 < proper isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (37) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n7991_0))) -> *4_0, rt in Omega(n7991_0) Induction Base: natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, 0))) Induction Step: natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, +(n7991_0, 1)))) ->_R^Omega(1) mark(natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n7991_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(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n4692_0) natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n7991_0))) -> *4_0, rt in Omega(n7991_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: s, active, head, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: s < active head < active afterNth < active pair < active U82 < active U11 < active and < active isNatural < active isLNat < active U21 < active U31 < active isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top s < proper head < proper afterNth < proper pair < proper U82 < proper U11 < proper and < proper isNatural < proper isLNat < proper U21 < proper U31 < proper isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (39) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: s(gen_tt:mark:nil:0':ok3_0(+(1, n9219_0))) -> *4_0, rt in Omega(n9219_0) Induction Base: s(gen_tt:mark:nil:0':ok3_0(+(1, 0))) Induction Step: s(gen_tt:mark:nil:0':ok3_0(+(1, +(n9219_0, 1)))) ->_R^Omega(1) mark(s(gen_tt:mark:nil:0':ok3_0(+(1, n9219_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). ---------------------------------------- (40) Obligation: TRS: Rules: active(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n4692_0) natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n7991_0))) -> *4_0, rt in Omega(n7991_0) s(gen_tt:mark:nil:0':ok3_0(+(1, n9219_0))) -> *4_0, rt in Omega(n9219_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: head, active, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: head < active afterNth < active pair < active U82 < active U11 < active and < active isNatural < active isLNat < active U21 < active U31 < active isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top head < proper afterNth < proper pair < proper U82 < proper U11 < proper and < proper isNatural < proper isLNat < proper U21 < proper U31 < proper isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (41) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: head(gen_tt:mark:nil:0':ok3_0(+(1, n10548_0))) -> *4_0, rt in Omega(n10548_0) Induction Base: head(gen_tt:mark:nil:0':ok3_0(+(1, 0))) Induction Step: head(gen_tt:mark:nil:0':ok3_0(+(1, +(n10548_0, 1)))) ->_R^Omega(1) mark(head(gen_tt:mark:nil:0':ok3_0(+(1, n10548_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). ---------------------------------------- (42) Obligation: TRS: Rules: active(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n4692_0) natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n7991_0))) -> *4_0, rt in Omega(n7991_0) s(gen_tt:mark:nil:0':ok3_0(+(1, n9219_0))) -> *4_0, rt in Omega(n9219_0) head(gen_tt:mark:nil:0':ok3_0(+(1, n10548_0))) -> *4_0, rt in Omega(n10548_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: afterNth, active, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: afterNth < active pair < active U82 < active U11 < active and < active isNatural < active isLNat < active U21 < active U31 < active isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top afterNth < proper pair < proper U82 < proper U11 < proper and < proper isNatural < proper isLNat < proper U21 < proper U31 < proper isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (43) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n11978_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n11978_0) Induction Base: afterNth(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b)) Induction Step: afterNth(gen_tt:mark:nil:0':ok3_0(+(1, +(n11978_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1) mark(afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n11978_0)), gen_tt:mark:nil:0':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(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n4692_0) natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n7991_0))) -> *4_0, rt in Omega(n7991_0) s(gen_tt:mark:nil:0':ok3_0(+(1, n9219_0))) -> *4_0, rt in Omega(n9219_0) head(gen_tt:mark:nil:0':ok3_0(+(1, n10548_0))) -> *4_0, rt in Omega(n10548_0) afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n11978_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n11978_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: pair, active, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: pair < active U82 < active U11 < active and < active isNatural < active isLNat < active U21 < active U31 < active isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top pair < proper U82 < proper U11 < proper and < proper isNatural < proper isLNat < proper U21 < proper U31 < proper isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (45) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: pair(gen_tt:mark:nil:0':ok3_0(+(1, n16406_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n16406_0) Induction Base: pair(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b)) Induction Step: pair(gen_tt:mark:nil:0':ok3_0(+(1, +(n16406_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1) mark(pair(gen_tt:mark:nil:0':ok3_0(+(1, n16406_0)), gen_tt:mark:nil:0':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). ---------------------------------------- (46) Obligation: TRS: Rules: active(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n4692_0) natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n7991_0))) -> *4_0, rt in Omega(n7991_0) s(gen_tt:mark:nil:0':ok3_0(+(1, n9219_0))) -> *4_0, rt in Omega(n9219_0) head(gen_tt:mark:nil:0':ok3_0(+(1, n10548_0))) -> *4_0, rt in Omega(n10548_0) afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n11978_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n11978_0) pair(gen_tt:mark:nil:0':ok3_0(+(1, n16406_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n16406_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: U82, active, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: U82 < active U11 < active and < active isNatural < active isLNat < active U21 < active U31 < active isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top U82 < proper U11 < proper and < proper isNatural < proper isLNat < proper U21 < proper U31 < proper isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (47) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U82(gen_tt:mark:nil:0':ok3_0(+(1, n21140_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21140_0) Induction Base: U82(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b)) Induction Step: U82(gen_tt:mark:nil:0':ok3_0(+(1, +(n21140_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1) mark(U82(gen_tt:mark:nil:0':ok3_0(+(1, n21140_0)), gen_tt:mark:nil:0':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). ---------------------------------------- (48) Obligation: TRS: Rules: active(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n4692_0) natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n7991_0))) -> *4_0, rt in Omega(n7991_0) s(gen_tt:mark:nil:0':ok3_0(+(1, n9219_0))) -> *4_0, rt in Omega(n9219_0) head(gen_tt:mark:nil:0':ok3_0(+(1, n10548_0))) -> *4_0, rt in Omega(n10548_0) afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n11978_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n11978_0) pair(gen_tt:mark:nil:0':ok3_0(+(1, n16406_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n16406_0) U82(gen_tt:mark:nil:0':ok3_0(+(1, n21140_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21140_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: U11, active, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: U11 < active and < active isNatural < active isLNat < active U21 < active U31 < active isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top U11 < proper and < proper isNatural < proper isLNat < proper U21 < proper U31 < proper isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (49) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U11(gen_tt:mark:nil:0':ok3_0(+(1, n25981_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n25981_0) Induction Base: U11(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) Induction Step: U11(gen_tt:mark:nil:0':ok3_0(+(1, +(n25981_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) ->_R^Omega(1) mark(U11(gen_tt:mark:nil:0':ok3_0(+(1, n25981_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':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). ---------------------------------------- (50) Obligation: TRS: Rules: active(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n4692_0) natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n7991_0))) -> *4_0, rt in Omega(n7991_0) s(gen_tt:mark:nil:0':ok3_0(+(1, n9219_0))) -> *4_0, rt in Omega(n9219_0) head(gen_tt:mark:nil:0':ok3_0(+(1, n10548_0))) -> *4_0, rt in Omega(n10548_0) afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n11978_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n11978_0) pair(gen_tt:mark:nil:0':ok3_0(+(1, n16406_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n16406_0) U82(gen_tt:mark:nil:0':ok3_0(+(1, n21140_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21140_0) U11(gen_tt:mark:nil:0':ok3_0(+(1, n25981_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n25981_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: and, active, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: and < active isNatural < active isLNat < active U21 < active U31 < active isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top and < proper isNatural < proper isLNat < proper U21 < proper U31 < proper isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (51) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: and(gen_tt:mark:nil:0':ok3_0(+(1, n34294_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n34294_0) Induction Base: and(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b)) Induction Step: and(gen_tt:mark:nil:0':ok3_0(+(1, +(n34294_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1) mark(and(gen_tt:mark:nil:0':ok3_0(+(1, n34294_0)), gen_tt:mark:nil:0':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(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n4692_0) natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n7991_0))) -> *4_0, rt in Omega(n7991_0) s(gen_tt:mark:nil:0':ok3_0(+(1, n9219_0))) -> *4_0, rt in Omega(n9219_0) head(gen_tt:mark:nil:0':ok3_0(+(1, n10548_0))) -> *4_0, rt in Omega(n10548_0) afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n11978_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n11978_0) pair(gen_tt:mark:nil:0':ok3_0(+(1, n16406_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n16406_0) U82(gen_tt:mark:nil:0':ok3_0(+(1, n21140_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21140_0) U11(gen_tt:mark:nil:0':ok3_0(+(1, n25981_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n25981_0) and(gen_tt:mark:nil:0':ok3_0(+(1, n34294_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n34294_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: isNatural, active, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: isNatural < active isLNat < active U21 < active U31 < active isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top isNatural < proper isLNat < proper U21 < proper U31 < proper isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (53) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U21(gen_tt:mark:nil:0':ok3_0(+(1, n39957_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n39957_0) Induction Base: U21(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b)) Induction Step: U21(gen_tt:mark:nil:0':ok3_0(+(1, +(n39957_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1) mark(U21(gen_tt:mark:nil:0':ok3_0(+(1, n39957_0)), gen_tt:mark:nil:0':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). ---------------------------------------- (54) Obligation: TRS: Rules: active(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n4692_0) natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n7991_0))) -> *4_0, rt in Omega(n7991_0) s(gen_tt:mark:nil:0':ok3_0(+(1, n9219_0))) -> *4_0, rt in Omega(n9219_0) head(gen_tt:mark:nil:0':ok3_0(+(1, n10548_0))) -> *4_0, rt in Omega(n10548_0) afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n11978_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n11978_0) pair(gen_tt:mark:nil:0':ok3_0(+(1, n16406_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n16406_0) U82(gen_tt:mark:nil:0':ok3_0(+(1, n21140_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21140_0) U11(gen_tt:mark:nil:0':ok3_0(+(1, n25981_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n25981_0) and(gen_tt:mark:nil:0':ok3_0(+(1, n34294_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n34294_0) U21(gen_tt:mark:nil:0':ok3_0(+(1, n39957_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n39957_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: U31, active, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: U31 < active isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top U31 < proper isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (55) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U31(gen_tt:mark:nil:0':ok3_0(+(1, n45814_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45814_0) Induction Base: U31(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b)) Induction Step: U31(gen_tt:mark:nil:0':ok3_0(+(1, +(n45814_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1) mark(U31(gen_tt:mark:nil:0':ok3_0(+(1, n45814_0)), gen_tt:mark:nil:0':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). ---------------------------------------- (56) Obligation: TRS: Rules: active(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n4692_0) natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n7991_0))) -> *4_0, rt in Omega(n7991_0) s(gen_tt:mark:nil:0':ok3_0(+(1, n9219_0))) -> *4_0, rt in Omega(n9219_0) head(gen_tt:mark:nil:0':ok3_0(+(1, n10548_0))) -> *4_0, rt in Omega(n10548_0) afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n11978_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n11978_0) pair(gen_tt:mark:nil:0':ok3_0(+(1, n16406_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n16406_0) U82(gen_tt:mark:nil:0':ok3_0(+(1, n21140_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21140_0) U11(gen_tt:mark:nil:0':ok3_0(+(1, n25981_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n25981_0) and(gen_tt:mark:nil:0':ok3_0(+(1, n34294_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n34294_0) U21(gen_tt:mark:nil:0':ok3_0(+(1, n39957_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n39957_0) U31(gen_tt:mark:nil:0':ok3_0(+(1, n45814_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45814_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: isPLNat, active, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: isPLNat < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top isPLNat < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (57) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U41(gen_tt:mark:nil:0':ok3_0(+(1, n52043_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n52043_0) Induction Base: U41(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b)) Induction Step: U41(gen_tt:mark:nil:0':ok3_0(+(1, +(n52043_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1) mark(U41(gen_tt:mark:nil:0':ok3_0(+(1, n52043_0)), gen_tt:mark:nil:0':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(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n4692_0) natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n7991_0))) -> *4_0, rt in Omega(n7991_0) s(gen_tt:mark:nil:0':ok3_0(+(1, n9219_0))) -> *4_0, rt in Omega(n9219_0) head(gen_tt:mark:nil:0':ok3_0(+(1, n10548_0))) -> *4_0, rt in Omega(n10548_0) afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n11978_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n11978_0) pair(gen_tt:mark:nil:0':ok3_0(+(1, n16406_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n16406_0) U82(gen_tt:mark:nil:0':ok3_0(+(1, n21140_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21140_0) U11(gen_tt:mark:nil:0':ok3_0(+(1, n25981_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n25981_0) and(gen_tt:mark:nil:0':ok3_0(+(1, n34294_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n34294_0) U21(gen_tt:mark:nil:0':ok3_0(+(1, n39957_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n39957_0) U31(gen_tt:mark:nil:0':ok3_0(+(1, n45814_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45814_0) U41(gen_tt:mark:nil:0':ok3_0(+(1, n52043_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n52043_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: U51, active, U61, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (59) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U51(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n58512_0) Induction Base: U51(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) Induction Step: U51(gen_tt:mark:nil:0':ok3_0(+(1, +(n58512_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) ->_R^Omega(1) mark(U51(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':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). ---------------------------------------- (60) Obligation: TRS: Rules: active(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n4692_0) natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n7991_0))) -> *4_0, rt in Omega(n7991_0) s(gen_tt:mark:nil:0':ok3_0(+(1, n9219_0))) -> *4_0, rt in Omega(n9219_0) head(gen_tt:mark:nil:0':ok3_0(+(1, n10548_0))) -> *4_0, rt in Omega(n10548_0) afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n11978_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n11978_0) pair(gen_tt:mark:nil:0':ok3_0(+(1, n16406_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n16406_0) U82(gen_tt:mark:nil:0':ok3_0(+(1, n21140_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21140_0) U11(gen_tt:mark:nil:0':ok3_0(+(1, n25981_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n25981_0) and(gen_tt:mark:nil:0':ok3_0(+(1, n34294_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n34294_0) U21(gen_tt:mark:nil:0':ok3_0(+(1, n39957_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n39957_0) U31(gen_tt:mark:nil:0':ok3_0(+(1, n45814_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45814_0) U41(gen_tt:mark:nil:0':ok3_0(+(1, n52043_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n52043_0) U51(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n58512_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: U61, active, U71, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: U61 < active U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (61) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U61(gen_tt:mark:nil:0':ok3_0(+(1, n69282_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n69282_0) Induction Base: U61(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b)) Induction Step: U61(gen_tt:mark:nil:0':ok3_0(+(1, +(n69282_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1) mark(U61(gen_tt:mark:nil:0':ok3_0(+(1, n69282_0)), gen_tt:mark:nil:0':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(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n4692_0) natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n7991_0))) -> *4_0, rt in Omega(n7991_0) s(gen_tt:mark:nil:0':ok3_0(+(1, n9219_0))) -> *4_0, rt in Omega(n9219_0) head(gen_tt:mark:nil:0':ok3_0(+(1, n10548_0))) -> *4_0, rt in Omega(n10548_0) afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n11978_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n11978_0) pair(gen_tt:mark:nil:0':ok3_0(+(1, n16406_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n16406_0) U82(gen_tt:mark:nil:0':ok3_0(+(1, n21140_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21140_0) U11(gen_tt:mark:nil:0':ok3_0(+(1, n25981_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n25981_0) and(gen_tt:mark:nil:0':ok3_0(+(1, n34294_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n34294_0) U21(gen_tt:mark:nil:0':ok3_0(+(1, n39957_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n39957_0) U31(gen_tt:mark:nil:0':ok3_0(+(1, n45814_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45814_0) U41(gen_tt:mark:nil:0':ok3_0(+(1, n52043_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n52043_0) U51(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n58512_0) U61(gen_tt:mark:nil:0':ok3_0(+(1, n69282_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n69282_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: U71, active, U81, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: U71 < active U81 < active U91 < active U101 < active tail < active take < active sel < active active < top U71 < proper U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top ---------------------------------------- (63) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U71(gen_tt:mark:nil:0':ok3_0(+(1, n76461_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n76461_0) Induction Base: U71(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b)) Induction Step: U71(gen_tt:mark:nil:0':ok3_0(+(1, +(n76461_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) ->_R^Omega(1) mark(U71(gen_tt:mark:nil:0':ok3_0(+(1, n76461_0)), gen_tt:mark:nil:0':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(U101(tt, N, XS)) -> mark(fst(splitAt(N, XS))) active(U11(tt, N, XS)) -> mark(snd(splitAt(N, XS))) active(U21(tt, X)) -> mark(X) active(U31(tt, N)) -> mark(N) active(U41(tt, N)) -> mark(cons(N, natsFrom(s(N)))) active(U51(tt, N, XS)) -> mark(head(afterNth(N, XS))) active(U61(tt, Y)) -> mark(Y) active(U71(tt, XS)) -> mark(pair(nil, XS)) active(U81(tt, N, X, XS)) -> mark(U82(splitAt(N, XS), X)) active(U82(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) active(U91(tt, XS)) -> mark(XS) active(afterNth(N, XS)) -> mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) active(and(tt, X)) -> mark(X) active(fst(pair(X, Y))) -> mark(U21(and(isLNat(X), isLNat(Y)), X)) active(head(cons(N, XS))) -> mark(U31(and(isNatural(N), isLNat(XS)), N)) active(isLNat(nil)) -> mark(tt) active(isLNat(afterNth(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(cons(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isNatural(0')) -> mark(tt) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(isPLNat(pair(V1, V2))) -> mark(and(isLNat(V1), isLNat(V2))) active(isPLNat(splitAt(V1, V2))) -> mark(and(isNatural(V1), isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N), N)) active(sel(N, XS)) -> mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) active(snd(pair(X, Y))) -> mark(U61(and(isLNat(X), isLNat(Y)), Y)) active(splitAt(0', XS)) -> mark(U71(isLNat(XS), XS)) active(splitAt(s(N), cons(X, XS))) -> mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) active(tail(cons(N, XS))) -> mark(U91(and(isNatural(N), isLNat(XS)), XS)) active(take(N, XS)) -> mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1, X2)) -> splitAt(active(X1), X2) active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(snd(X)) -> snd(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2)) -> U41(active(X1), X2) active(cons(X1, X2)) -> cons(active(X1), X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1, X2, X3)) -> U51(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(U61(X1, X2)) -> U61(active(X1), X2) active(U71(X1, X2)) -> U71(active(X1), X2) active(pair(X1, X2)) -> pair(active(X1), X2) active(pair(X1, X2)) -> pair(X1, active(X2)) active(U81(X1, X2, X3, X4)) -> U81(active(X1), X2, X3, X4) active(U82(X1, X2)) -> U82(active(X1), X2) active(U91(X1, X2)) -> U91(active(X1), X2) active(and(X1, X2)) -> and(active(X1), X2) active(tail(X)) -> tail(active(X)) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2) -> mark(U41(X1, X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1), X2, X3) -> mark(U51(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)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U71(mark(X1), X2) -> mark(U71(X1, X2)) pair(mark(X1), X2) -> mark(pair(X1, X2)) pair(X1, mark(X2)) -> mark(pair(X1, X2)) U81(mark(X1), X2, X3, X4) -> mark(U81(X1, X2, X3, X4)) U82(mark(X1), X2) -> mark(U82(X1, X2)) U91(mark(X1), X2) -> mark(U91(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U81(X1, X2, X3, X4)) -> U81(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(0') -> ok(0') proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) U81(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U81(X1, X2, X3, X4)) U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U101 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok tt :: tt:mark:nil:0':ok mark :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok fst :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok splitAt :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U11 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok snd :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U21 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U31 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U41 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok cons :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok natsFrom :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok s :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok U51 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok head :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok afterNth :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U61 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U71 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok pair :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok nil :: tt:mark:nil:0':ok U81 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U82 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok U91 :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok and :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok isNatural :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok isPLNat :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok tail :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok take :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok 0' :: tt:mark:nil:0':ok sel :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok -> tt:mark:nil:0':ok proper :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok ok :: tt:mark:nil:0':ok -> tt:mark:nil:0':ok top :: tt:mark:nil:0':ok -> top hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok hole_top2_0 :: top gen_tt:mark:nil:0':ok3_0 :: Nat -> tt:mark:nil:0':ok Lemmas: fst(gen_tt:mark:nil:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n731_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n731_0) snd(gen_tt:mark:nil:0':ok3_0(+(1, n3715_0))) -> *4_0, rt in Omega(n3715_0) cons(gen_tt:mark:nil:0':ok3_0(+(1, n4692_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n4692_0) natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n7991_0))) -> *4_0, rt in Omega(n7991_0) s(gen_tt:mark:nil:0':ok3_0(+(1, n9219_0))) -> *4_0, rt in Omega(n9219_0) head(gen_tt:mark:nil:0':ok3_0(+(1, n10548_0))) -> *4_0, rt in Omega(n10548_0) afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n11978_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n11978_0) pair(gen_tt:mark:nil:0':ok3_0(+(1, n16406_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n16406_0) U82(gen_tt:mark:nil:0':ok3_0(+(1, n21140_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n21140_0) U11(gen_tt:mark:nil:0':ok3_0(+(1, n25981_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n25981_0) and(gen_tt:mark:nil:0':ok3_0(+(1, n34294_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n34294_0) U21(gen_tt:mark:nil:0':ok3_0(+(1, n39957_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n39957_0) U31(gen_tt:mark:nil:0':ok3_0(+(1, n45814_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n45814_0) U41(gen_tt:mark:nil:0':ok3_0(+(1, n52043_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n52043_0) U51(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) -> *4_0, rt in Omega(n58512_0) U61(gen_tt:mark:nil:0':ok3_0(+(1, n69282_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n69282_0) U71(gen_tt:mark:nil:0':ok3_0(+(1, n76461_0)), gen_tt:mark:nil:0':ok3_0(b)) -> *4_0, rt in Omega(n76461_0) Generator Equations: gen_tt:mark:nil:0':ok3_0(0) <=> tt gen_tt:mark:nil:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:nil:0':ok3_0(x)) The following defined symbols remain to be analysed: U81, active, U91, U101, tail, take, sel, proper, top They will be analysed ascendingly in the following order: U81 < active U91 < active U101 < active tail < active take < active sel < active active < top U81 < proper U91 < proper U101 < proper tail < proper take < proper sel < proper proper < top