/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) typed CpxTrs (5) OrderProof [LOWER BOUND(ID), 0 ms] (6) typed CpxTrs (7) RewriteLemmaProof [LOWER BOUND(ID), 437 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 115 ms] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 85 ms] (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 164 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 105 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 49 ms] (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 86 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 111 ms] (26) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) active(and(X1, X2)) -> and(active(X1), X2) active(and(X1, X2)) -> and(X1, active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4) active(uLength(X1, X2)) -> uLength(active(X1), X2) and(mark(X1), X2) -> mark(and(X1, X2)) and(X1, mark(X2)) -> mark(and(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4)) uLength(mark(X1), X2) -> mark(uLength(X1, X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0) -> ok(0) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1, X2, X3, X4)) -> uTake2(proper(X1), proper(X2), proper(X3), proper(X4)) proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(uTake2(X1, X2, X3, X4)) uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0')) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0', zeros)) active(take(0', IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) active(and(X1, X2)) -> and(active(X1), X2) active(and(X1, X2)) -> and(X1, active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4) active(uLength(X1, X2)) -> uLength(active(X1), X2) and(mark(X1), X2) -> mark(and(X1, X2)) and(X1, mark(X2)) -> mark(and(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4)) uLength(mark(X1), X2) -> mark(uLength(X1, X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1, X2, X3, X4)) -> uTake2(proper(X1), proper(X2), proper(X3), proper(X4)) proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(uTake2(X1, X2, X3, X4)) uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: TRS: Rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0')) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0', zeros)) active(take(0', IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) active(and(X1, X2)) -> and(active(X1), X2) active(and(X1, X2)) -> and(X1, active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4) active(uLength(X1, X2)) -> uLength(active(X1), X2) and(mark(X1), X2) -> mark(and(X1, X2)) and(X1, mark(X2)) -> mark(and(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4)) uLength(mark(X1), X2) -> mark(uLength(X1, X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1, X2, X3, X4)) -> uTake2(proper(X1), proper(X2), proper(X3), proper(X4)) proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(uTake2(X1, X2, X3, X4)) uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok and :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok tt :: tt:mark:0':zeros:nil:ok mark :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatIList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNat :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok 0' :: tt:mark:0':zeros:nil:ok s :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok length :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok zeros :: tt:mark:0':zeros:nil:ok cons :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok nil :: tt:mark:0':zeros:nil:ok take :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake1 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake2 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uLength :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok proper :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok ok :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok top :: tt:mark:0':zeros:nil:ok -> top hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok hole_top2_0 :: top gen_tt:mark:0':zeros:nil:ok3_0 :: Nat -> tt:mark:0':zeros:nil:ok ---------------------------------------- (5) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: active, isNatList, isNat, and, isNatIList, cons, uTake1, uTake2, take, uLength, s, length, proper, top They will be analysed ascendingly in the following order: isNatList < active isNat < active and < active isNatIList < active cons < active uTake1 < active uTake2 < active take < active uLength < active s < active length < active active < top isNatList < proper isNat < proper and < proper isNatIList < proper cons < proper uTake1 < proper uTake2 < proper take < proper uLength < proper s < proper length < proper proper < top ---------------------------------------- (6) Obligation: TRS: Rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0')) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0', zeros)) active(take(0', IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) active(and(X1, X2)) -> and(active(X1), X2) active(and(X1, X2)) -> and(X1, active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4) active(uLength(X1, X2)) -> uLength(active(X1), X2) and(mark(X1), X2) -> mark(and(X1, X2)) and(X1, mark(X2)) -> mark(and(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4)) uLength(mark(X1), X2) -> mark(uLength(X1, X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1, X2, X3, X4)) -> uTake2(proper(X1), proper(X2), proper(X3), proper(X4)) proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(uTake2(X1, X2, X3, X4)) uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok and :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok tt :: tt:mark:0':zeros:nil:ok mark :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatIList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNat :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok 0' :: tt:mark:0':zeros:nil:ok s :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok length :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok zeros :: tt:mark:0':zeros:nil:ok cons :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok nil :: tt:mark:0':zeros:nil:ok take :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake1 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake2 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uLength :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok proper :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok ok :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok top :: tt:mark:0':zeros:nil:ok -> top hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok hole_top2_0 :: top gen_tt:mark:0':zeros:nil:ok3_0 :: Nat -> tt:mark:0':zeros:nil:ok Generator Equations: gen_tt:mark:0':zeros:nil:ok3_0(0) <=> tt gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':zeros:nil:ok3_0(x)) The following defined symbols remain to be analysed: isNatList, active, isNat, and, isNatIList, cons, uTake1, uTake2, take, uLength, s, length, proper, top They will be analysed ascendingly in the following order: isNatList < active isNat < active and < active isNatIList < active cons < active uTake1 < active uTake2 < active take < active uLength < active s < active length < active active < top isNatList < proper isNat < proper and < proper isNatIList < proper cons < proper uTake1 < proper uTake2 < proper take < proper uLength < proper s < proper length < proper proper < top ---------------------------------------- (7) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n13_0) Induction Base: and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, 0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) Induction Step: and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, +(n13_0, 1))), gen_tt:mark:0':zeros:nil:ok3_0(b)) ->_R^Omega(1) mark(and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: TRS: Rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0')) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0', zeros)) active(take(0', IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) active(and(X1, X2)) -> and(active(X1), X2) active(and(X1, X2)) -> and(X1, active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4) active(uLength(X1, X2)) -> uLength(active(X1), X2) and(mark(X1), X2) -> mark(and(X1, X2)) and(X1, mark(X2)) -> mark(and(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4)) uLength(mark(X1), X2) -> mark(uLength(X1, X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1, X2, X3, X4)) -> uTake2(proper(X1), proper(X2), proper(X3), proper(X4)) proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(uTake2(X1, X2, X3, X4)) uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok and :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok tt :: tt:mark:0':zeros:nil:ok mark :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatIList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNat :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok 0' :: tt:mark:0':zeros:nil:ok s :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok length :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok zeros :: tt:mark:0':zeros:nil:ok cons :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok nil :: tt:mark:0':zeros:nil:ok take :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake1 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake2 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uLength :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok proper :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok ok :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok top :: tt:mark:0':zeros:nil:ok -> top hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok hole_top2_0 :: top gen_tt:mark:0':zeros:nil:ok3_0 :: Nat -> tt:mark:0':zeros:nil:ok Generator Equations: gen_tt:mark:0':zeros:nil:ok3_0(0) <=> tt gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':zeros:nil:ok3_0(x)) The following defined symbols remain to be analysed: and, active, isNatIList, cons, uTake1, uTake2, take, uLength, s, length, proper, top They will be analysed ascendingly in the following order: and < active isNatIList < active cons < active uTake1 < active uTake2 < active take < active uLength < active s < active length < active active < top and < proper isNatIList < proper cons < proper uTake1 < proper uTake2 < proper take < proper uLength < proper s < proper length < proper proper < top ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: TRS: Rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0')) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0', zeros)) active(take(0', IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) active(and(X1, X2)) -> and(active(X1), X2) active(and(X1, X2)) -> and(X1, active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4) active(uLength(X1, X2)) -> uLength(active(X1), X2) and(mark(X1), X2) -> mark(and(X1, X2)) and(X1, mark(X2)) -> mark(and(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4)) uLength(mark(X1), X2) -> mark(uLength(X1, X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1, X2, X3, X4)) -> uTake2(proper(X1), proper(X2), proper(X3), proper(X4)) proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(uTake2(X1, X2, X3, X4)) uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok and :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok tt :: tt:mark:0':zeros:nil:ok mark :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatIList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNat :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok 0' :: tt:mark:0':zeros:nil:ok s :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok length :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok zeros :: tt:mark:0':zeros:nil:ok cons :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok nil :: tt:mark:0':zeros:nil:ok take :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake1 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake2 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uLength :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok proper :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok ok :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok top :: tt:mark:0':zeros:nil:ok -> top hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok hole_top2_0 :: top gen_tt:mark:0':zeros:nil:ok3_0 :: Nat -> tt:mark:0':zeros:nil:ok Lemmas: and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n13_0) Generator Equations: gen_tt:mark:0':zeros:nil:ok3_0(0) <=> tt gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':zeros:nil:ok3_0(x)) The following defined symbols remain to be analysed: isNatIList, active, cons, uTake1, uTake2, take, uLength, s, length, proper, top They will be analysed ascendingly in the following order: isNatIList < active cons < active uTake1 < active uTake2 < active take < active uLength < active s < active length < active active < top isNatIList < proper cons < proper uTake1 < proper uTake2 < proper take < proper uLength < proper s < proper length < proper proper < top ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1566_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n1566_0) Induction Base: cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, 0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) Induction Step: cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, +(n1566_0, 1))), gen_tt:mark:0':zeros:nil:ok3_0(b)) ->_R^Omega(1) mark(cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1566_0)), gen_tt:mark:0':zeros:nil:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (14) Obligation: TRS: Rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0')) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0', zeros)) active(take(0', IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) active(and(X1, X2)) -> and(active(X1), X2) active(and(X1, X2)) -> and(X1, active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4) active(uLength(X1, X2)) -> uLength(active(X1), X2) and(mark(X1), X2) -> mark(and(X1, X2)) and(X1, mark(X2)) -> mark(and(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4)) uLength(mark(X1), X2) -> mark(uLength(X1, X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1, X2, X3, X4)) -> uTake2(proper(X1), proper(X2), proper(X3), proper(X4)) proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(uTake2(X1, X2, X3, X4)) uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok and :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok tt :: tt:mark:0':zeros:nil:ok mark :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatIList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNat :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok 0' :: tt:mark:0':zeros:nil:ok s :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok length :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok zeros :: tt:mark:0':zeros:nil:ok cons :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok nil :: tt:mark:0':zeros:nil:ok take :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake1 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake2 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uLength :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok proper :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok ok :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok top :: tt:mark:0':zeros:nil:ok -> top hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok hole_top2_0 :: top gen_tt:mark:0':zeros:nil:ok3_0 :: Nat -> tt:mark:0':zeros:nil:ok Lemmas: and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n13_0) cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1566_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n1566_0) Generator Equations: gen_tt:mark:0':zeros:nil:ok3_0(0) <=> tt gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':zeros:nil:ok3_0(x)) The following defined symbols remain to be analysed: uTake1, active, uTake2, take, uLength, s, length, proper, top They will be analysed ascendingly in the following order: uTake1 < active uTake2 < active take < active uLength < active s < active length < active active < top uTake1 < proper uTake2 < proper take < proper uLength < proper s < proper length < proper proper < top ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3217_0))) -> *4_0, rt in Omega(n3217_0) Induction Base: uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, 0))) Induction Step: uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, +(n3217_0, 1)))) ->_R^Omega(1) mark(uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3217_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (16) Obligation: TRS: Rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0')) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0', zeros)) active(take(0', IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) active(and(X1, X2)) -> and(active(X1), X2) active(and(X1, X2)) -> and(X1, active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4) active(uLength(X1, X2)) -> uLength(active(X1), X2) and(mark(X1), X2) -> mark(and(X1, X2)) and(X1, mark(X2)) -> mark(and(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4)) uLength(mark(X1), X2) -> mark(uLength(X1, X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1, X2, X3, X4)) -> uTake2(proper(X1), proper(X2), proper(X3), proper(X4)) proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(uTake2(X1, X2, X3, X4)) uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok and :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok tt :: tt:mark:0':zeros:nil:ok mark :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatIList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNat :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok 0' :: tt:mark:0':zeros:nil:ok s :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok length :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok zeros :: tt:mark:0':zeros:nil:ok cons :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok nil :: tt:mark:0':zeros:nil:ok take :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake1 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake2 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uLength :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok proper :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok ok :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok top :: tt:mark:0':zeros:nil:ok -> top hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok hole_top2_0 :: top gen_tt:mark:0':zeros:nil:ok3_0 :: Nat -> tt:mark:0':zeros:nil:ok Lemmas: and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n13_0) cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1566_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n1566_0) uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3217_0))) -> *4_0, rt in Omega(n3217_0) Generator Equations: gen_tt:mark:0':zeros:nil:ok3_0(0) <=> tt gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':zeros:nil:ok3_0(x)) The following defined symbols remain to be analysed: uTake2, active, take, uLength, s, length, proper, top They will be analysed ascendingly in the following order: uTake2 < active take < active uLength < active s < active length < active active < top uTake2 < proper take < proper uLength < proper s < proper length < proper proper < top ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3979_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) -> *4_0, rt in Omega(n3979_0) Induction Base: uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, 0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) Induction Step: uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, +(n3979_0, 1))), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) ->_R^Omega(1) mark(uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3979_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: TRS: Rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0')) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0', zeros)) active(take(0', IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) active(and(X1, X2)) -> and(active(X1), X2) active(and(X1, X2)) -> and(X1, active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4) active(uLength(X1, X2)) -> uLength(active(X1), X2) and(mark(X1), X2) -> mark(and(X1, X2)) and(X1, mark(X2)) -> mark(and(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4)) uLength(mark(X1), X2) -> mark(uLength(X1, X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1, X2, X3, X4)) -> uTake2(proper(X1), proper(X2), proper(X3), proper(X4)) proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(uTake2(X1, X2, X3, X4)) uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok and :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok tt :: tt:mark:0':zeros:nil:ok mark :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatIList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNat :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok 0' :: tt:mark:0':zeros:nil:ok s :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok length :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok zeros :: tt:mark:0':zeros:nil:ok cons :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok nil :: tt:mark:0':zeros:nil:ok take :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake1 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake2 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uLength :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok proper :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok ok :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok top :: tt:mark:0':zeros:nil:ok -> top hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok hole_top2_0 :: top gen_tt:mark:0':zeros:nil:ok3_0 :: Nat -> tt:mark:0':zeros:nil:ok Lemmas: and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n13_0) cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1566_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n1566_0) uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3217_0))) -> *4_0, rt in Omega(n3217_0) uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3979_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) -> *4_0, rt in Omega(n3979_0) Generator Equations: gen_tt:mark:0':zeros:nil:ok3_0(0) <=> tt gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':zeros:nil:ok3_0(x)) The following defined symbols remain to be analysed: take, active, uLength, s, length, proper, top They will be analysed ascendingly in the following order: take < active uLength < active s < active length < active active < top take < proper uLength < proper s < proper length < proper proper < top ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n8748_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n8748_0) Induction Base: take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, 0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) Induction Step: take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, +(n8748_0, 1))), gen_tt:mark:0':zeros:nil:ok3_0(b)) ->_R^Omega(1) mark(take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n8748_0)), gen_tt:mark:0':zeros:nil:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (20) Obligation: TRS: Rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0')) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0', zeros)) active(take(0', IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) active(and(X1, X2)) -> and(active(X1), X2) active(and(X1, X2)) -> and(X1, active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4) active(uLength(X1, X2)) -> uLength(active(X1), X2) and(mark(X1), X2) -> mark(and(X1, X2)) and(X1, mark(X2)) -> mark(and(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4)) uLength(mark(X1), X2) -> mark(uLength(X1, X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1, X2, X3, X4)) -> uTake2(proper(X1), proper(X2), proper(X3), proper(X4)) proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(uTake2(X1, X2, X3, X4)) uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok and :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok tt :: tt:mark:0':zeros:nil:ok mark :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatIList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNat :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok 0' :: tt:mark:0':zeros:nil:ok s :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok length :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok zeros :: tt:mark:0':zeros:nil:ok cons :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok nil :: tt:mark:0':zeros:nil:ok take :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake1 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake2 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uLength :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok proper :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok ok :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok top :: tt:mark:0':zeros:nil:ok -> top hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok hole_top2_0 :: top gen_tt:mark:0':zeros:nil:ok3_0 :: Nat -> tt:mark:0':zeros:nil:ok Lemmas: and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n13_0) cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1566_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n1566_0) uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3217_0))) -> *4_0, rt in Omega(n3217_0) uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3979_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) -> *4_0, rt in Omega(n3979_0) take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n8748_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n8748_0) Generator Equations: gen_tt:mark:0':zeros:nil:ok3_0(0) <=> tt gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':zeros:nil:ok3_0(x)) The following defined symbols remain to be analysed: uLength, active, s, length, proper, top They will be analysed ascendingly in the following order: uLength < active s < active length < active active < top uLength < proper s < proper length < proper proper < top ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n11614_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n11614_0) Induction Base: uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, 0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) Induction Step: uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, +(n11614_0, 1))), gen_tt:mark:0':zeros:nil:ok3_0(b)) ->_R^Omega(1) mark(uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n11614_0)), gen_tt:mark:0':zeros:nil:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (22) Obligation: TRS: Rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0')) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0', zeros)) active(take(0', IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) active(and(X1, X2)) -> and(active(X1), X2) active(and(X1, X2)) -> and(X1, active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4) active(uLength(X1, X2)) -> uLength(active(X1), X2) and(mark(X1), X2) -> mark(and(X1, X2)) and(X1, mark(X2)) -> mark(and(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4)) uLength(mark(X1), X2) -> mark(uLength(X1, X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1, X2, X3, X4)) -> uTake2(proper(X1), proper(X2), proper(X3), proper(X4)) proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(uTake2(X1, X2, X3, X4)) uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok and :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok tt :: tt:mark:0':zeros:nil:ok mark :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatIList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNat :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok 0' :: tt:mark:0':zeros:nil:ok s :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok length :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok zeros :: tt:mark:0':zeros:nil:ok cons :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok nil :: tt:mark:0':zeros:nil:ok take :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake1 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake2 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uLength :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok proper :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok ok :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok top :: tt:mark:0':zeros:nil:ok -> top hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok hole_top2_0 :: top gen_tt:mark:0':zeros:nil:ok3_0 :: Nat -> tt:mark:0':zeros:nil:ok Lemmas: and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n13_0) cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1566_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n1566_0) uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3217_0))) -> *4_0, rt in Omega(n3217_0) uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3979_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) -> *4_0, rt in Omega(n3979_0) take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n8748_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n8748_0) uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n11614_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n11614_0) Generator Equations: gen_tt:mark:0':zeros:nil:ok3_0(0) <=> tt gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':zeros:nil:ok3_0(x)) The following defined symbols remain to be analysed: s, active, length, proper, top They will be analysed ascendingly in the following order: s < active length < active active < top s < proper length < proper proper < top ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: s(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n14587_0))) -> *4_0, rt in Omega(n14587_0) Induction Base: s(gen_tt:mark:0':zeros:nil:ok3_0(+(1, 0))) Induction Step: s(gen_tt:mark:0':zeros:nil:ok3_0(+(1, +(n14587_0, 1)))) ->_R^Omega(1) mark(s(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n14587_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (24) Obligation: TRS: Rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0')) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0', zeros)) active(take(0', IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) active(and(X1, X2)) -> and(active(X1), X2) active(and(X1, X2)) -> and(X1, active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4) active(uLength(X1, X2)) -> uLength(active(X1), X2) and(mark(X1), X2) -> mark(and(X1, X2)) and(X1, mark(X2)) -> mark(and(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4)) uLength(mark(X1), X2) -> mark(uLength(X1, X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1, X2, X3, X4)) -> uTake2(proper(X1), proper(X2), proper(X3), proper(X4)) proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(uTake2(X1, X2, X3, X4)) uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok and :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok tt :: tt:mark:0':zeros:nil:ok mark :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatIList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNat :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok 0' :: tt:mark:0':zeros:nil:ok s :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok length :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok zeros :: tt:mark:0':zeros:nil:ok cons :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok nil :: tt:mark:0':zeros:nil:ok take :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake1 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake2 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uLength :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok proper :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok ok :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok top :: tt:mark:0':zeros:nil:ok -> top hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok hole_top2_0 :: top gen_tt:mark:0':zeros:nil:ok3_0 :: Nat -> tt:mark:0':zeros:nil:ok Lemmas: and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n13_0) cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1566_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n1566_0) uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3217_0))) -> *4_0, rt in Omega(n3217_0) uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3979_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) -> *4_0, rt in Omega(n3979_0) take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n8748_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n8748_0) uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n11614_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n11614_0) s(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n14587_0))) -> *4_0, rt in Omega(n14587_0) Generator Equations: gen_tt:mark:0':zeros:nil:ok3_0(0) <=> tt gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':zeros:nil:ok3_0(x)) The following defined symbols remain to be analysed: length, active, proper, top They will be analysed ascendingly in the following order: length < active active < top length < proper proper < top ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: length(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n15998_0))) -> *4_0, rt in Omega(n15998_0) Induction Base: length(gen_tt:mark:0':zeros:nil:ok3_0(+(1, 0))) Induction Step: length(gen_tt:mark:0':zeros:nil:ok3_0(+(1, +(n15998_0, 1)))) ->_R^Omega(1) mark(length(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n15998_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) Obligation: TRS: Rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0')) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0', zeros)) active(take(0', IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) active(and(X1, X2)) -> and(active(X1), X2) active(and(X1, X2)) -> and(X1, active(X2)) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(uTake1(X)) -> uTake1(active(X)) active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4) active(uLength(X1, X2)) -> uLength(active(X1), X2) and(mark(X1), X2) -> mark(and(X1, X2)) and(X1, mark(X2)) -> mark(and(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) uTake1(mark(X)) -> mark(uTake1(X)) uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4)) uLength(mark(X1), X2) -> mark(uLength(X1, X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(uTake1(X)) -> uTake1(proper(X)) proper(uTake2(X1, X2, X3, X4)) -> uTake2(proper(X1), proper(X2), proper(X3), proper(X4)) proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) uTake1(ok(X)) -> ok(uTake1(X)) uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(uTake2(X1, X2, X3, X4)) uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok and :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok tt :: tt:mark:0':zeros:nil:ok mark :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatIList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNatList :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok isNat :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok 0' :: tt:mark:0':zeros:nil:ok s :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok length :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok zeros :: tt:mark:0':zeros:nil:ok cons :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok nil :: tt:mark:0':zeros:nil:ok take :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake1 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uTake2 :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok uLength :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok proper :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok ok :: tt:mark:0':zeros:nil:ok -> tt:mark:0':zeros:nil:ok top :: tt:mark:0':zeros:nil:ok -> top hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok hole_top2_0 :: top gen_tt:mark:0':zeros:nil:ok3_0 :: Nat -> tt:mark:0':zeros:nil:ok Lemmas: and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n13_0) cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1566_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n1566_0) uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3217_0))) -> *4_0, rt in Omega(n3217_0) uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3979_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) -> *4_0, rt in Omega(n3979_0) take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n8748_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n8748_0) uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n11614_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) -> *4_0, rt in Omega(n11614_0) s(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n14587_0))) -> *4_0, rt in Omega(n14587_0) length(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n15998_0))) -> *4_0, rt in Omega(n15998_0) Generator Equations: gen_tt:mark:0':zeros:nil:ok3_0(0) <=> tt gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':zeros:nil:ok3_0(x)) The following defined symbols remain to be analysed: active, proper, top They will be analysed ascendingly in the following order: active < top proper < top