WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 583 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs 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: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(zeros) -> zeros encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_uTake1(x_1)) -> uTake1(encArg(x_1)) encArg(cons_uTake2(x_1, x_2, x_3, x_4)) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_uLength(x_1, x_2)) -> uLength(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_uTake1(x_1) -> uTake1(encArg(x_1)) encode_uTake2(x_1, x_2, x_3, x_4) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_uLength(x_1, x_2) -> uLength(encArg(x_1), encArg(x_2)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS 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)) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(zeros) -> zeros encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_uTake1(x_1)) -> uTake1(encArg(x_1)) encArg(cons_uTake2(x_1, x_2, x_3, x_4)) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_uLength(x_1, x_2)) -> uLength(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_uTake1(x_1) -> uTake1(encArg(x_1)) encode_uTake2(x_1, x_2, x_3, x_4) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_uLength(x_1, x_2) -> uLength(encArg(x_1), encArg(x_2)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS 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)) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(zeros) -> zeros encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_uTake1(x_1)) -> uTake1(encArg(x_1)) encArg(cons_uTake2(x_1, x_2, x_3, x_4)) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_uLength(x_1, x_2)) -> uLength(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_uTake1(x_1) -> uTake1(encArg(x_1)) encode_uTake2(x_1, x_2, x_3, x_4) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_uLength(x_1, x_2) -> uLength(encArg(x_1), encArg(x_2)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS 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)) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(zeros) -> zeros encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_uTake1(x_1)) -> uTake1(encArg(x_1)) encArg(cons_uTake2(x_1, x_2, x_3, x_4)) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_uLength(x_1, x_2)) -> uLength(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_uTake1(x_1) -> uTake1(encArg(x_1)) encode_uTake2(x_1, x_2, x_3, x_4) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_uLength(x_1, x_2) -> uLength(encArg(x_1), encArg(x_2)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence uTake2(mark(X1), X2, X3, X4) ->^+ mark(uTake2(X1, X2, X3, X4)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / mark(X1)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS 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)) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(zeros) -> zeros encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_uTake1(x_1)) -> uTake1(encArg(x_1)) encArg(cons_uTake2(x_1, x_2, x_3, x_4)) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_uLength(x_1, x_2)) -> uLength(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_uTake1(x_1) -> uTake1(encArg(x_1)) encode_uTake2(x_1, x_2, x_3, x_4) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_uLength(x_1, x_2) -> uLength(encArg(x_1), encArg(x_2)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS 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)) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(zeros) -> zeros encArg(nil) -> nil encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_uTake1(x_1)) -> uTake1(encArg(x_1)) encArg(cons_uTake2(x_1, x_2, x_3, x_4)) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_uLength(x_1, x_2)) -> uLength(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_uTake1(x_1) -> uTake1(encArg(x_1)) encode_uTake2(x_1, x_2, x_3, x_4) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_uLength(x_1, x_2) -> uLength(encArg(x_1), encArg(x_2)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST