/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 753 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(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) active(__(X, nil)) -> mark(X) active(__(nil, X)) -> mark(X) active(U11(tt, V)) -> mark(U12(isNeList(V))) active(U12(tt)) -> mark(tt) active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2)) active(U22(tt, V2)) -> mark(U23(isList(V2))) active(U23(tt)) -> mark(tt) active(U31(tt, V)) -> mark(U32(isQid(V))) active(U32(tt)) -> mark(tt) active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2)) active(U42(tt, V2)) -> mark(U43(isNeList(V2))) active(U43(tt)) -> mark(tt) active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2)) active(U52(tt, V2)) -> mark(U53(isList(V2))) active(U53(tt)) -> mark(tt) active(U61(tt, V)) -> mark(U62(isQid(V))) active(U62(tt)) -> mark(tt) active(U71(tt, V)) -> mark(U72(isNePal(V))) active(U72(tt)) -> mark(tt) active(and(tt, X)) -> mark(X) active(isList(V)) -> mark(U11(isPalListKind(V), V)) active(isList(nil)) -> mark(tt) active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNeList(V)) -> mark(U31(isPalListKind(V), V)) active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNePal(V)) -> mark(U61(isPalListKind(V), V)) active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))) active(isPal(V)) -> mark(U71(isPalListKind(V), V)) active(isPal(nil)) -> mark(tt) active(isPalListKind(a)) -> mark(tt) active(isPalListKind(e)) -> mark(tt) active(isPalListKind(i)) -> mark(tt) active(isPalListKind(nil)) -> mark(tt) active(isPalListKind(o)) -> mark(tt) active(isPalListKind(u)) -> mark(tt) active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2))) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) active(__(X1, X2)) -> __(active(X1), X2) active(__(X1, X2)) -> __(X1, active(X2)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X)) -> U32(active(X)) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2)) -> U42(active(X1), X2) active(U43(X)) -> U43(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(U52(X1, X2)) -> U52(active(X1), X2) active(U53(X)) -> U53(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X)) -> U72(active(X)) active(and(X1, X2)) -> and(active(X1), X2) __(mark(X1), X2) -> mark(__(X1, X2)) __(X1, mark(X2)) -> mark(__(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2) -> mark(U42(X1, X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X)) -> mark(U72(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(__(X1, X2)) -> __(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X)) -> U12(proper(X)) proper(isNeList(X)) -> isNeList(proper(X)) proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(isList(X)) -> isList(proper(X)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(isQid(X)) -> isQid(proper(X)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X)) -> U72(proper(X)) proper(isNePal(X)) -> isNePal(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isPalListKind(X)) -> isPalListKind(proper(X)) proper(isPal(X)) -> isPal(proper(X)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(o) -> ok(o) proper(u) -> ok(u) __(ok(X1), ok(X2)) -> ok(__(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNeList(ok(X)) -> ok(isNeList(X)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) isList(ok(X)) -> ok(isList(X)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X)) -> ok(U32(X)) isQid(ok(X)) -> ok(isQid(X)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) U43(ok(X)) -> ok(U43(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X)) -> ok(U72(X)) isNePal(ok(X)) -> ok(isNePal(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isPalListKind(ok(X)) -> ok(isPalListKind(X)) isPal(ok(X)) -> ok(isPal(X)) 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(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(tt) -> tt encArg(a) -> a encArg(e) -> e encArg(i) -> i encArg(o) -> o encArg(u) -> u encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1)) -> U12(encArg(x_1)) encArg(cons_U21(x_1, x_2, x_3)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U22(x_1, x_2)) -> U22(encArg(x_1), encArg(x_2)) encArg(cons_U23(x_1)) -> U23(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1)) -> U32(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2)) -> U42(encArg(x_1), encArg(x_2)) encArg(cons_U43(x_1)) -> U43(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U52(x_1, x_2)) -> U52(encArg(x_1), encArg(x_2)) encArg(cons_U53(x_1)) -> U53(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U62(x_1)) -> U62(encArg(x_1)) encArg(cons_U71(x_1, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_U72(x_1)) -> U72(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNeList(x_1)) -> isNeList(encArg(x_1)) encArg(cons_isList(x_1)) -> isList(encArg(x_1)) encArg(cons_isQid(x_1)) -> isQid(encArg(x_1)) encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1)) encArg(cons_isPalListKind(x_1)) -> isPalListKind(encArg(x_1)) encArg(cons_isPal(x_1)) -> isPal(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_nil -> nil encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_U12(x_1) -> U12(encArg(x_1)) encode_isNeList(x_1) -> isNeList(encArg(x_1)) encode_U21(x_1, x_2, x_3) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U22(x_1, x_2) -> U22(encArg(x_1), encArg(x_2)) encode_isList(x_1) -> isList(encArg(x_1)) encode_U23(x_1) -> U23(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1) -> U32(encArg(x_1)) encode_isQid(x_1) -> isQid(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2) -> U42(encArg(x_1), encArg(x_2)) encode_U43(x_1) -> U43(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U52(x_1, x_2) -> U52(encArg(x_1), encArg(x_2)) encode_U53(x_1) -> U53(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U62(x_1) -> U62(encArg(x_1)) encode_U71(x_1, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_U72(x_1) -> U72(encArg(x_1)) encode_isNePal(x_1) -> isNePal(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isPalListKind(x_1) -> isPalListKind(encArg(x_1)) encode_isPal(x_1) -> isPal(encArg(x_1)) encode_a -> a encode_e -> e encode_i -> i encode_o -> o encode_u -> u 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(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) active(__(X, nil)) -> mark(X) active(__(nil, X)) -> mark(X) active(U11(tt, V)) -> mark(U12(isNeList(V))) active(U12(tt)) -> mark(tt) active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2)) active(U22(tt, V2)) -> mark(U23(isList(V2))) active(U23(tt)) -> mark(tt) active(U31(tt, V)) -> mark(U32(isQid(V))) active(U32(tt)) -> mark(tt) active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2)) active(U42(tt, V2)) -> mark(U43(isNeList(V2))) active(U43(tt)) -> mark(tt) active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2)) active(U52(tt, V2)) -> mark(U53(isList(V2))) active(U53(tt)) -> mark(tt) active(U61(tt, V)) -> mark(U62(isQid(V))) active(U62(tt)) -> mark(tt) active(U71(tt, V)) -> mark(U72(isNePal(V))) active(U72(tt)) -> mark(tt) active(and(tt, X)) -> mark(X) active(isList(V)) -> mark(U11(isPalListKind(V), V)) active(isList(nil)) -> mark(tt) active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNeList(V)) -> mark(U31(isPalListKind(V), V)) active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNePal(V)) -> mark(U61(isPalListKind(V), V)) active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))) active(isPal(V)) -> mark(U71(isPalListKind(V), V)) active(isPal(nil)) -> mark(tt) active(isPalListKind(a)) -> mark(tt) active(isPalListKind(e)) -> mark(tt) active(isPalListKind(i)) -> mark(tt) active(isPalListKind(nil)) -> mark(tt) active(isPalListKind(o)) -> mark(tt) active(isPalListKind(u)) -> mark(tt) active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2))) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) active(__(X1, X2)) -> __(active(X1), X2) active(__(X1, X2)) -> __(X1, active(X2)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X)) -> U32(active(X)) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2)) -> U42(active(X1), X2) active(U43(X)) -> U43(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(U52(X1, X2)) -> U52(active(X1), X2) active(U53(X)) -> U53(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X)) -> U72(active(X)) active(and(X1, X2)) -> and(active(X1), X2) __(mark(X1), X2) -> mark(__(X1, X2)) __(X1, mark(X2)) -> mark(__(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2) -> mark(U42(X1, X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X)) -> mark(U72(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(__(X1, X2)) -> __(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X)) -> U12(proper(X)) proper(isNeList(X)) -> isNeList(proper(X)) proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(isList(X)) -> isList(proper(X)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(isQid(X)) -> isQid(proper(X)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X)) -> U72(proper(X)) proper(isNePal(X)) -> isNePal(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isPalListKind(X)) -> isPalListKind(proper(X)) proper(isPal(X)) -> isPal(proper(X)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(o) -> ok(o) proper(u) -> ok(u) __(ok(X1), ok(X2)) -> ok(__(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNeList(ok(X)) -> ok(isNeList(X)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) isList(ok(X)) -> ok(isList(X)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X)) -> ok(U32(X)) isQid(ok(X)) -> ok(isQid(X)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) U43(ok(X)) -> ok(U43(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X)) -> ok(U72(X)) isNePal(ok(X)) -> ok(isNePal(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isPalListKind(ok(X)) -> ok(isPalListKind(X)) isPal(ok(X)) -> ok(isPal(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(tt) -> tt encArg(a) -> a encArg(e) -> e encArg(i) -> i encArg(o) -> o encArg(u) -> u encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1)) -> U12(encArg(x_1)) encArg(cons_U21(x_1, x_2, x_3)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U22(x_1, x_2)) -> U22(encArg(x_1), encArg(x_2)) encArg(cons_U23(x_1)) -> U23(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1)) -> U32(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2)) -> U42(encArg(x_1), encArg(x_2)) encArg(cons_U43(x_1)) -> U43(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U52(x_1, x_2)) -> U52(encArg(x_1), encArg(x_2)) encArg(cons_U53(x_1)) -> U53(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U62(x_1)) -> U62(encArg(x_1)) encArg(cons_U71(x_1, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_U72(x_1)) -> U72(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNeList(x_1)) -> isNeList(encArg(x_1)) encArg(cons_isList(x_1)) -> isList(encArg(x_1)) encArg(cons_isQid(x_1)) -> isQid(encArg(x_1)) encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1)) encArg(cons_isPalListKind(x_1)) -> isPalListKind(encArg(x_1)) encArg(cons_isPal(x_1)) -> isPal(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_nil -> nil encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_U12(x_1) -> U12(encArg(x_1)) encode_isNeList(x_1) -> isNeList(encArg(x_1)) encode_U21(x_1, x_2, x_3) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U22(x_1, x_2) -> U22(encArg(x_1), encArg(x_2)) encode_isList(x_1) -> isList(encArg(x_1)) encode_U23(x_1) -> U23(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1) -> U32(encArg(x_1)) encode_isQid(x_1) -> isQid(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2) -> U42(encArg(x_1), encArg(x_2)) encode_U43(x_1) -> U43(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U52(x_1, x_2) -> U52(encArg(x_1), encArg(x_2)) encode_U53(x_1) -> U53(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U62(x_1) -> U62(encArg(x_1)) encode_U71(x_1, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_U72(x_1) -> U72(encArg(x_1)) encode_isNePal(x_1) -> isNePal(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isPalListKind(x_1) -> isPalListKind(encArg(x_1)) encode_isPal(x_1) -> isPal(encArg(x_1)) encode_a -> a encode_e -> e encode_i -> i encode_o -> o encode_u -> u 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(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) active(__(X, nil)) -> mark(X) active(__(nil, X)) -> mark(X) active(U11(tt, V)) -> mark(U12(isNeList(V))) active(U12(tt)) -> mark(tt) active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2)) active(U22(tt, V2)) -> mark(U23(isList(V2))) active(U23(tt)) -> mark(tt) active(U31(tt, V)) -> mark(U32(isQid(V))) active(U32(tt)) -> mark(tt) active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2)) active(U42(tt, V2)) -> mark(U43(isNeList(V2))) active(U43(tt)) -> mark(tt) active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2)) active(U52(tt, V2)) -> mark(U53(isList(V2))) active(U53(tt)) -> mark(tt) active(U61(tt, V)) -> mark(U62(isQid(V))) active(U62(tt)) -> mark(tt) active(U71(tt, V)) -> mark(U72(isNePal(V))) active(U72(tt)) -> mark(tt) active(and(tt, X)) -> mark(X) active(isList(V)) -> mark(U11(isPalListKind(V), V)) active(isList(nil)) -> mark(tt) active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNeList(V)) -> mark(U31(isPalListKind(V), V)) active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNePal(V)) -> mark(U61(isPalListKind(V), V)) active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))) active(isPal(V)) -> mark(U71(isPalListKind(V), V)) active(isPal(nil)) -> mark(tt) active(isPalListKind(a)) -> mark(tt) active(isPalListKind(e)) -> mark(tt) active(isPalListKind(i)) -> mark(tt) active(isPalListKind(nil)) -> mark(tt) active(isPalListKind(o)) -> mark(tt) active(isPalListKind(u)) -> mark(tt) active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2))) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) active(__(X1, X2)) -> __(active(X1), X2) active(__(X1, X2)) -> __(X1, active(X2)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X)) -> U32(active(X)) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2)) -> U42(active(X1), X2) active(U43(X)) -> U43(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(U52(X1, X2)) -> U52(active(X1), X2) active(U53(X)) -> U53(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X)) -> U72(active(X)) active(and(X1, X2)) -> and(active(X1), X2) __(mark(X1), X2) -> mark(__(X1, X2)) __(X1, mark(X2)) -> mark(__(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2) -> mark(U42(X1, X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X)) -> mark(U72(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(__(X1, X2)) -> __(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X)) -> U12(proper(X)) proper(isNeList(X)) -> isNeList(proper(X)) proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(isList(X)) -> isList(proper(X)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(isQid(X)) -> isQid(proper(X)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X)) -> U72(proper(X)) proper(isNePal(X)) -> isNePal(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isPalListKind(X)) -> isPalListKind(proper(X)) proper(isPal(X)) -> isPal(proper(X)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(o) -> ok(o) proper(u) -> ok(u) __(ok(X1), ok(X2)) -> ok(__(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNeList(ok(X)) -> ok(isNeList(X)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) isList(ok(X)) -> ok(isList(X)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X)) -> ok(U32(X)) isQid(ok(X)) -> ok(isQid(X)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) U43(ok(X)) -> ok(U43(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X)) -> ok(U72(X)) isNePal(ok(X)) -> ok(isNePal(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isPalListKind(ok(X)) -> ok(isPalListKind(X)) isPal(ok(X)) -> ok(isPal(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(tt) -> tt encArg(a) -> a encArg(e) -> e encArg(i) -> i encArg(o) -> o encArg(u) -> u encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1)) -> U12(encArg(x_1)) encArg(cons_U21(x_1, x_2, x_3)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U22(x_1, x_2)) -> U22(encArg(x_1), encArg(x_2)) encArg(cons_U23(x_1)) -> U23(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1)) -> U32(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2)) -> U42(encArg(x_1), encArg(x_2)) encArg(cons_U43(x_1)) -> U43(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U52(x_1, x_2)) -> U52(encArg(x_1), encArg(x_2)) encArg(cons_U53(x_1)) -> U53(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U62(x_1)) -> U62(encArg(x_1)) encArg(cons_U71(x_1, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_U72(x_1)) -> U72(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNeList(x_1)) -> isNeList(encArg(x_1)) encArg(cons_isList(x_1)) -> isList(encArg(x_1)) encArg(cons_isQid(x_1)) -> isQid(encArg(x_1)) encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1)) encArg(cons_isPalListKind(x_1)) -> isPalListKind(encArg(x_1)) encArg(cons_isPal(x_1)) -> isPal(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_nil -> nil encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_U12(x_1) -> U12(encArg(x_1)) encode_isNeList(x_1) -> isNeList(encArg(x_1)) encode_U21(x_1, x_2, x_3) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U22(x_1, x_2) -> U22(encArg(x_1), encArg(x_2)) encode_isList(x_1) -> isList(encArg(x_1)) encode_U23(x_1) -> U23(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1) -> U32(encArg(x_1)) encode_isQid(x_1) -> isQid(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2) -> U42(encArg(x_1), encArg(x_2)) encode_U43(x_1) -> U43(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U52(x_1, x_2) -> U52(encArg(x_1), encArg(x_2)) encode_U53(x_1) -> U53(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U62(x_1) -> U62(encArg(x_1)) encode_U71(x_1, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_U72(x_1) -> U72(encArg(x_1)) encode_isNePal(x_1) -> isNePal(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isPalListKind(x_1) -> isPalListKind(encArg(x_1)) encode_isPal(x_1) -> isPal(encArg(x_1)) encode_a -> a encode_e -> e encode_i -> i encode_o -> o encode_u -> u 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(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) active(__(X, nil)) -> mark(X) active(__(nil, X)) -> mark(X) active(U11(tt, V)) -> mark(U12(isNeList(V))) active(U12(tt)) -> mark(tt) active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2)) active(U22(tt, V2)) -> mark(U23(isList(V2))) active(U23(tt)) -> mark(tt) active(U31(tt, V)) -> mark(U32(isQid(V))) active(U32(tt)) -> mark(tt) active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2)) active(U42(tt, V2)) -> mark(U43(isNeList(V2))) active(U43(tt)) -> mark(tt) active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2)) active(U52(tt, V2)) -> mark(U53(isList(V2))) active(U53(tt)) -> mark(tt) active(U61(tt, V)) -> mark(U62(isQid(V))) active(U62(tt)) -> mark(tt) active(U71(tt, V)) -> mark(U72(isNePal(V))) active(U72(tt)) -> mark(tt) active(and(tt, X)) -> mark(X) active(isList(V)) -> mark(U11(isPalListKind(V), V)) active(isList(nil)) -> mark(tt) active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNeList(V)) -> mark(U31(isPalListKind(V), V)) active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNePal(V)) -> mark(U61(isPalListKind(V), V)) active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))) active(isPal(V)) -> mark(U71(isPalListKind(V), V)) active(isPal(nil)) -> mark(tt) active(isPalListKind(a)) -> mark(tt) active(isPalListKind(e)) -> mark(tt) active(isPalListKind(i)) -> mark(tt) active(isPalListKind(nil)) -> mark(tt) active(isPalListKind(o)) -> mark(tt) active(isPalListKind(u)) -> mark(tt) active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2))) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) active(__(X1, X2)) -> __(active(X1), X2) active(__(X1, X2)) -> __(X1, active(X2)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X)) -> U32(active(X)) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2)) -> U42(active(X1), X2) active(U43(X)) -> U43(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(U52(X1, X2)) -> U52(active(X1), X2) active(U53(X)) -> U53(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X)) -> U72(active(X)) active(and(X1, X2)) -> and(active(X1), X2) __(mark(X1), X2) -> mark(__(X1, X2)) __(X1, mark(X2)) -> mark(__(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2) -> mark(U42(X1, X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X)) -> mark(U72(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(__(X1, X2)) -> __(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X)) -> U12(proper(X)) proper(isNeList(X)) -> isNeList(proper(X)) proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(isList(X)) -> isList(proper(X)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(isQid(X)) -> isQid(proper(X)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X)) -> U72(proper(X)) proper(isNePal(X)) -> isNePal(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isPalListKind(X)) -> isPalListKind(proper(X)) proper(isPal(X)) -> isPal(proper(X)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(o) -> ok(o) proper(u) -> ok(u) __(ok(X1), ok(X2)) -> ok(__(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNeList(ok(X)) -> ok(isNeList(X)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) isList(ok(X)) -> ok(isList(X)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X)) -> ok(U32(X)) isQid(ok(X)) -> ok(isQid(X)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) U43(ok(X)) -> ok(U43(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X)) -> ok(U72(X)) isNePal(ok(X)) -> ok(isNePal(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isPalListKind(ok(X)) -> ok(isPalListKind(X)) isPal(ok(X)) -> ok(isPal(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(tt) -> tt encArg(a) -> a encArg(e) -> e encArg(i) -> i encArg(o) -> o encArg(u) -> u encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1)) -> U12(encArg(x_1)) encArg(cons_U21(x_1, x_2, x_3)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U22(x_1, x_2)) -> U22(encArg(x_1), encArg(x_2)) encArg(cons_U23(x_1)) -> U23(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1)) -> U32(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2)) -> U42(encArg(x_1), encArg(x_2)) encArg(cons_U43(x_1)) -> U43(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U52(x_1, x_2)) -> U52(encArg(x_1), encArg(x_2)) encArg(cons_U53(x_1)) -> U53(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U62(x_1)) -> U62(encArg(x_1)) encArg(cons_U71(x_1, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_U72(x_1)) -> U72(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNeList(x_1)) -> isNeList(encArg(x_1)) encArg(cons_isList(x_1)) -> isList(encArg(x_1)) encArg(cons_isQid(x_1)) -> isQid(encArg(x_1)) encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1)) encArg(cons_isPalListKind(x_1)) -> isPalListKind(encArg(x_1)) encArg(cons_isPal(x_1)) -> isPal(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_nil -> nil encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_U12(x_1) -> U12(encArg(x_1)) encode_isNeList(x_1) -> isNeList(encArg(x_1)) encode_U21(x_1, x_2, x_3) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U22(x_1, x_2) -> U22(encArg(x_1), encArg(x_2)) encode_isList(x_1) -> isList(encArg(x_1)) encode_U23(x_1) -> U23(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1) -> U32(encArg(x_1)) encode_isQid(x_1) -> isQid(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2) -> U42(encArg(x_1), encArg(x_2)) encode_U43(x_1) -> U43(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U52(x_1, x_2) -> U52(encArg(x_1), encArg(x_2)) encode_U53(x_1) -> U53(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U62(x_1) -> U62(encArg(x_1)) encode_U71(x_1, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_U72(x_1) -> U72(encArg(x_1)) encode_isNePal(x_1) -> isNePal(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isPalListKind(x_1) -> isPalListKind(encArg(x_1)) encode_isPal(x_1) -> isPal(encArg(x_1)) encode_a -> a encode_e -> e encode_i -> i encode_o -> o encode_u -> u 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 U52(ok(X1), ok(X2)) ->^+ ok(U52(X1, X2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / ok(X1), X2 / ok(X2)]. 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(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) active(__(X, nil)) -> mark(X) active(__(nil, X)) -> mark(X) active(U11(tt, V)) -> mark(U12(isNeList(V))) active(U12(tt)) -> mark(tt) active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2)) active(U22(tt, V2)) -> mark(U23(isList(V2))) active(U23(tt)) -> mark(tt) active(U31(tt, V)) -> mark(U32(isQid(V))) active(U32(tt)) -> mark(tt) active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2)) active(U42(tt, V2)) -> mark(U43(isNeList(V2))) active(U43(tt)) -> mark(tt) active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2)) active(U52(tt, V2)) -> mark(U53(isList(V2))) active(U53(tt)) -> mark(tt) active(U61(tt, V)) -> mark(U62(isQid(V))) active(U62(tt)) -> mark(tt) active(U71(tt, V)) -> mark(U72(isNePal(V))) active(U72(tt)) -> mark(tt) active(and(tt, X)) -> mark(X) active(isList(V)) -> mark(U11(isPalListKind(V), V)) active(isList(nil)) -> mark(tt) active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNeList(V)) -> mark(U31(isPalListKind(V), V)) active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNePal(V)) -> mark(U61(isPalListKind(V), V)) active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))) active(isPal(V)) -> mark(U71(isPalListKind(V), V)) active(isPal(nil)) -> mark(tt) active(isPalListKind(a)) -> mark(tt) active(isPalListKind(e)) -> mark(tt) active(isPalListKind(i)) -> mark(tt) active(isPalListKind(nil)) -> mark(tt) active(isPalListKind(o)) -> mark(tt) active(isPalListKind(u)) -> mark(tt) active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2))) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) active(__(X1, X2)) -> __(active(X1), X2) active(__(X1, X2)) -> __(X1, active(X2)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X)) -> U32(active(X)) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2)) -> U42(active(X1), X2) active(U43(X)) -> U43(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(U52(X1, X2)) -> U52(active(X1), X2) active(U53(X)) -> U53(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X)) -> U72(active(X)) active(and(X1, X2)) -> and(active(X1), X2) __(mark(X1), X2) -> mark(__(X1, X2)) __(X1, mark(X2)) -> mark(__(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2) -> mark(U42(X1, X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X)) -> mark(U72(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(__(X1, X2)) -> __(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X)) -> U12(proper(X)) proper(isNeList(X)) -> isNeList(proper(X)) proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(isList(X)) -> isList(proper(X)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(isQid(X)) -> isQid(proper(X)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X)) -> U72(proper(X)) proper(isNePal(X)) -> isNePal(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isPalListKind(X)) -> isPalListKind(proper(X)) proper(isPal(X)) -> isPal(proper(X)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(o) -> ok(o) proper(u) -> ok(u) __(ok(X1), ok(X2)) -> ok(__(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNeList(ok(X)) -> ok(isNeList(X)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) isList(ok(X)) -> ok(isList(X)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X)) -> ok(U32(X)) isQid(ok(X)) -> ok(isQid(X)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) U43(ok(X)) -> ok(U43(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X)) -> ok(U72(X)) isNePal(ok(X)) -> ok(isNePal(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isPalListKind(ok(X)) -> ok(isPalListKind(X)) isPal(ok(X)) -> ok(isPal(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(tt) -> tt encArg(a) -> a encArg(e) -> e encArg(i) -> i encArg(o) -> o encArg(u) -> u encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1)) -> U12(encArg(x_1)) encArg(cons_U21(x_1, x_2, x_3)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U22(x_1, x_2)) -> U22(encArg(x_1), encArg(x_2)) encArg(cons_U23(x_1)) -> U23(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1)) -> U32(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2)) -> U42(encArg(x_1), encArg(x_2)) encArg(cons_U43(x_1)) -> U43(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U52(x_1, x_2)) -> U52(encArg(x_1), encArg(x_2)) encArg(cons_U53(x_1)) -> U53(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U62(x_1)) -> U62(encArg(x_1)) encArg(cons_U71(x_1, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_U72(x_1)) -> U72(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNeList(x_1)) -> isNeList(encArg(x_1)) encArg(cons_isList(x_1)) -> isList(encArg(x_1)) encArg(cons_isQid(x_1)) -> isQid(encArg(x_1)) encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1)) encArg(cons_isPalListKind(x_1)) -> isPalListKind(encArg(x_1)) encArg(cons_isPal(x_1)) -> isPal(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_nil -> nil encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_U12(x_1) -> U12(encArg(x_1)) encode_isNeList(x_1) -> isNeList(encArg(x_1)) encode_U21(x_1, x_2, x_3) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U22(x_1, x_2) -> U22(encArg(x_1), encArg(x_2)) encode_isList(x_1) -> isList(encArg(x_1)) encode_U23(x_1) -> U23(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1) -> U32(encArg(x_1)) encode_isQid(x_1) -> isQid(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2) -> U42(encArg(x_1), encArg(x_2)) encode_U43(x_1) -> U43(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U52(x_1, x_2) -> U52(encArg(x_1), encArg(x_2)) encode_U53(x_1) -> U53(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U62(x_1) -> U62(encArg(x_1)) encode_U71(x_1, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_U72(x_1) -> U72(encArg(x_1)) encode_isNePal(x_1) -> isNePal(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isPalListKind(x_1) -> isPalListKind(encArg(x_1)) encode_isPal(x_1) -> isPal(encArg(x_1)) encode_a -> a encode_e -> e encode_i -> i encode_o -> o encode_u -> u 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(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) active(__(X, nil)) -> mark(X) active(__(nil, X)) -> mark(X) active(U11(tt, V)) -> mark(U12(isNeList(V))) active(U12(tt)) -> mark(tt) active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2)) active(U22(tt, V2)) -> mark(U23(isList(V2))) active(U23(tt)) -> mark(tt) active(U31(tt, V)) -> mark(U32(isQid(V))) active(U32(tt)) -> mark(tt) active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2)) active(U42(tt, V2)) -> mark(U43(isNeList(V2))) active(U43(tt)) -> mark(tt) active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2)) active(U52(tt, V2)) -> mark(U53(isList(V2))) active(U53(tt)) -> mark(tt) active(U61(tt, V)) -> mark(U62(isQid(V))) active(U62(tt)) -> mark(tt) active(U71(tt, V)) -> mark(U72(isNePal(V))) active(U72(tt)) -> mark(tt) active(and(tt, X)) -> mark(X) active(isList(V)) -> mark(U11(isPalListKind(V), V)) active(isList(nil)) -> mark(tt) active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNeList(V)) -> mark(U31(isPalListKind(V), V)) active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNePal(V)) -> mark(U61(isPalListKind(V), V)) active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))) active(isPal(V)) -> mark(U71(isPalListKind(V), V)) active(isPal(nil)) -> mark(tt) active(isPalListKind(a)) -> mark(tt) active(isPalListKind(e)) -> mark(tt) active(isPalListKind(i)) -> mark(tt) active(isPalListKind(nil)) -> mark(tt) active(isPalListKind(o)) -> mark(tt) active(isPalListKind(u)) -> mark(tt) active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2))) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) active(__(X1, X2)) -> __(active(X1), X2) active(__(X1, X2)) -> __(X1, active(X2)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X)) -> U12(active(X)) active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U32(X)) -> U32(active(X)) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(U42(X1, X2)) -> U42(active(X1), X2) active(U43(X)) -> U43(active(X)) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(U52(X1, X2)) -> U52(active(X1), X2) active(U53(X)) -> U53(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X)) -> U72(active(X)) active(and(X1, X2)) -> and(active(X1), X2) __(mark(X1), X2) -> mark(__(X1, X2)) __(X1, mark(X2)) -> mark(__(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) U42(mark(X1), X2) -> mark(U42(X1, X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) U52(mark(X1), X2) -> mark(U52(X1, X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X)) -> mark(U72(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(__(X1, X2)) -> __(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X)) -> U12(proper(X)) proper(isNeList(X)) -> isNeList(proper(X)) proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(isList(X)) -> isList(proper(X)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(isQid(X)) -> isQid(proper(X)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X)) -> U72(proper(X)) proper(isNePal(X)) -> isNePal(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isPalListKind(X)) -> isPalListKind(proper(X)) proper(isPal(X)) -> isPal(proper(X)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(o) -> ok(o) proper(u) -> ok(u) __(ok(X1), ok(X2)) -> ok(__(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X)) -> ok(U12(X)) isNeList(ok(X)) -> ok(isNeList(X)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) isList(ok(X)) -> ok(isList(X)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U32(ok(X)) -> ok(U32(X)) isQid(ok(X)) -> ok(isQid(X)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) U43(ok(X)) -> ok(U43(X)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X)) -> ok(U72(X)) isNePal(ok(X)) -> ok(isNePal(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isPalListKind(ok(X)) -> ok(isPalListKind(X)) isPal(ok(X)) -> ok(isPal(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(tt) -> tt encArg(a) -> a encArg(e) -> e encArg(i) -> i encArg(o) -> o encArg(u) -> u encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(cons_U12(x_1)) -> U12(encArg(x_1)) encArg(cons_U21(x_1, x_2, x_3)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U22(x_1, x_2)) -> U22(encArg(x_1), encArg(x_2)) encArg(cons_U23(x_1)) -> U23(encArg(x_1)) encArg(cons_U31(x_1, x_2)) -> U31(encArg(x_1), encArg(x_2)) encArg(cons_U32(x_1)) -> U32(encArg(x_1)) encArg(cons_U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U42(x_1, x_2)) -> U42(encArg(x_1), encArg(x_2)) encArg(cons_U43(x_1)) -> U43(encArg(x_1)) encArg(cons_U51(x_1, x_2, x_3)) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_U52(x_1, x_2)) -> U52(encArg(x_1), encArg(x_2)) encArg(cons_U53(x_1)) -> U53(encArg(x_1)) encArg(cons_U61(x_1, x_2)) -> U61(encArg(x_1), encArg(x_2)) encArg(cons_U62(x_1)) -> U62(encArg(x_1)) encArg(cons_U71(x_1, x_2)) -> U71(encArg(x_1), encArg(x_2)) encArg(cons_U72(x_1)) -> U72(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_isNeList(x_1)) -> isNeList(encArg(x_1)) encArg(cons_isList(x_1)) -> isList(encArg(x_1)) encArg(cons_isQid(x_1)) -> isQid(encArg(x_1)) encArg(cons_isNePal(x_1)) -> isNePal(encArg(x_1)) encArg(cons_isPalListKind(x_1)) -> isPalListKind(encArg(x_1)) encArg(cons_isPal(x_1)) -> isPal(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_nil -> nil encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_U12(x_1) -> U12(encArg(x_1)) encode_isNeList(x_1) -> isNeList(encArg(x_1)) encode_U21(x_1, x_2, x_3) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U22(x_1, x_2) -> U22(encArg(x_1), encArg(x_2)) encode_isList(x_1) -> isList(encArg(x_1)) encode_U23(x_1) -> U23(encArg(x_1)) encode_U31(x_1, x_2) -> U31(encArg(x_1), encArg(x_2)) encode_U32(x_1) -> U32(encArg(x_1)) encode_isQid(x_1) -> isQid(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U42(x_1, x_2) -> U42(encArg(x_1), encArg(x_2)) encode_U43(x_1) -> U43(encArg(x_1)) encode_U51(x_1, x_2, x_3) -> U51(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U52(x_1, x_2) -> U52(encArg(x_1), encArg(x_2)) encode_U53(x_1) -> U53(encArg(x_1)) encode_U61(x_1, x_2) -> U61(encArg(x_1), encArg(x_2)) encode_U62(x_1) -> U62(encArg(x_1)) encode_U71(x_1, x_2) -> U71(encArg(x_1), encArg(x_2)) encode_U72(x_1) -> U72(encArg(x_1)) encode_isNePal(x_1) -> isNePal(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_isPalListKind(x_1) -> isPalListKind(encArg(x_1)) encode_isPal(x_1) -> isPal(encArg(x_1)) encode_a -> a encode_e -> e encode_i -> i encode_o -> o encode_u -> u 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