/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(EXP, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 638 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 9 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection (13) DecreasingLoopProof [FINISHED, 567 ms] (14) BOUNDS(EXP, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U11(tt, N) -> mark(N) a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U31(tt) -> 0 a__U41(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) a__plus(N, 0) -> a__U11(a__isNat(N), N) a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N) a__x(N, 0) -> a__U31(a__isNat(N)) a__x(N, s(M)) -> a__U41(a__and(a__isNat(M), isNat(N)), M, N) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1, X2) -> U11(X1, X2) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U31(X) -> U31(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(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(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(isNat(x_1)) -> isNat(encArg(x_1)) encArg(x(x_1, x_2)) -> x(encArg(x_1), encArg(x_2)) encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U21(x_1, x_2, x_3)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(U31(x_1)) -> U31(encArg(x_1)) encArg(U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_a__U11(x_1, x_2)) -> a__U11(encArg(x_1), encArg(x_2)) encArg(cons_a__U21(x_1, x_2, x_3)) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__U31(x_1)) -> a__U31(encArg(x_1)) encArg(cons_a__U41(x_1, x_2, x_3)) -> a__U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__and(x_1, x_2)) -> a__and(encArg(x_1), encArg(x_2)) encArg(cons_a__isNat(x_1)) -> a__isNat(encArg(x_1)) encArg(cons_a__plus(x_1, x_2)) -> a__plus(encArg(x_1), encArg(x_2)) encArg(cons_a__x(x_1, x_2)) -> a__x(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__U11(x_1, x_2) -> a__U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_a__U21(x_1, x_2, x_3) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_a__plus(x_1, x_2) -> a__plus(encArg(x_1), encArg(x_2)) encode_a__U31(x_1) -> a__U31(encArg(x_1)) encode_0 -> 0 encode_a__U41(x_1, x_2, x_3) -> a__U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a__x(x_1, x_2) -> a__x(encArg(x_1), encArg(x_2)) encode_a__and(x_1, x_2) -> a__and(encArg(x_1), encArg(x_2)) encode_a__isNat(x_1) -> a__isNat(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_x(x_1, x_2) -> x(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U21(x_1, x_2, x_3) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U31(x_1) -> U31(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U11(tt, N) -> mark(N) a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U31(tt) -> 0 a__U41(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) a__plus(N, 0) -> a__U11(a__isNat(N), N) a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N) a__x(N, 0) -> a__U31(a__isNat(N)) a__x(N, s(M)) -> a__U41(a__and(a__isNat(M), isNat(N)), M, N) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1, X2) -> U11(X1, X2) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U31(X) -> U31(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(X) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(isNat(x_1)) -> isNat(encArg(x_1)) encArg(x(x_1, x_2)) -> x(encArg(x_1), encArg(x_2)) encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U21(x_1, x_2, x_3)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(U31(x_1)) -> U31(encArg(x_1)) encArg(U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_a__U11(x_1, x_2)) -> a__U11(encArg(x_1), encArg(x_2)) encArg(cons_a__U21(x_1, x_2, x_3)) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__U31(x_1)) -> a__U31(encArg(x_1)) encArg(cons_a__U41(x_1, x_2, x_3)) -> a__U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__and(x_1, x_2)) -> a__and(encArg(x_1), encArg(x_2)) encArg(cons_a__isNat(x_1)) -> a__isNat(encArg(x_1)) encArg(cons_a__plus(x_1, x_2)) -> a__plus(encArg(x_1), encArg(x_2)) encArg(cons_a__x(x_1, x_2)) -> a__x(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__U11(x_1, x_2) -> a__U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_a__U21(x_1, x_2, x_3) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_a__plus(x_1, x_2) -> a__plus(encArg(x_1), encArg(x_2)) encode_a__U31(x_1) -> a__U31(encArg(x_1)) encode_0 -> 0 encode_a__U41(x_1, x_2, x_3) -> a__U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a__x(x_1, x_2) -> a__x(encArg(x_1), encArg(x_2)) encode_a__and(x_1, x_2) -> a__and(encArg(x_1), encArg(x_2)) encode_a__isNat(x_1) -> a__isNat(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_x(x_1, x_2) -> x(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U21(x_1, x_2, x_3) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U31(x_1) -> U31(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) 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(EXP, INF). The TRS R consists of the following rules: a__U11(tt, N) -> mark(N) a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U31(tt) -> 0 a__U41(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) a__plus(N, 0) -> a__U11(a__isNat(N), N) a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N) a__x(N, 0) -> a__U31(a__isNat(N)) a__x(N, s(M)) -> a__U41(a__and(a__isNat(M), isNat(N)), M, N) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1, X2) -> U11(X1, X2) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U31(X) -> U31(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(X) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(isNat(x_1)) -> isNat(encArg(x_1)) encArg(x(x_1, x_2)) -> x(encArg(x_1), encArg(x_2)) encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U21(x_1, x_2, x_3)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(U31(x_1)) -> U31(encArg(x_1)) encArg(U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_a__U11(x_1, x_2)) -> a__U11(encArg(x_1), encArg(x_2)) encArg(cons_a__U21(x_1, x_2, x_3)) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__U31(x_1)) -> a__U31(encArg(x_1)) encArg(cons_a__U41(x_1, x_2, x_3)) -> a__U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__and(x_1, x_2)) -> a__and(encArg(x_1), encArg(x_2)) encArg(cons_a__isNat(x_1)) -> a__isNat(encArg(x_1)) encArg(cons_a__plus(x_1, x_2)) -> a__plus(encArg(x_1), encArg(x_2)) encArg(cons_a__x(x_1, x_2)) -> a__x(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__U11(x_1, x_2) -> a__U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_a__U21(x_1, x_2, x_3) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_a__plus(x_1, x_2) -> a__plus(encArg(x_1), encArg(x_2)) encode_a__U31(x_1) -> a__U31(encArg(x_1)) encode_0 -> 0 encode_a__U41(x_1, x_2, x_3) -> a__U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a__x(x_1, x_2) -> a__x(encArg(x_1), encArg(x_2)) encode_a__and(x_1, x_2) -> a__and(encArg(x_1), encArg(x_2)) encode_a__isNat(x_1) -> a__isNat(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_x(x_1, x_2) -> x(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U21(x_1, x_2, x_3) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U31(x_1) -> U31(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) 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(EXP, INF). The TRS R consists of the following rules: a__U11(tt, N) -> mark(N) a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U31(tt) -> 0 a__U41(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) a__plus(N, 0) -> a__U11(a__isNat(N), N) a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N) a__x(N, 0) -> a__U31(a__isNat(N)) a__x(N, s(M)) -> a__U41(a__and(a__isNat(M), isNat(N)), M, N) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1, X2) -> U11(X1, X2) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U31(X) -> U31(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(X) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(isNat(x_1)) -> isNat(encArg(x_1)) encArg(x(x_1, x_2)) -> x(encArg(x_1), encArg(x_2)) encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U21(x_1, x_2, x_3)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(U31(x_1)) -> U31(encArg(x_1)) encArg(U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_a__U11(x_1, x_2)) -> a__U11(encArg(x_1), encArg(x_2)) encArg(cons_a__U21(x_1, x_2, x_3)) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__U31(x_1)) -> a__U31(encArg(x_1)) encArg(cons_a__U41(x_1, x_2, x_3)) -> a__U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__and(x_1, x_2)) -> a__and(encArg(x_1), encArg(x_2)) encArg(cons_a__isNat(x_1)) -> a__isNat(encArg(x_1)) encArg(cons_a__plus(x_1, x_2)) -> a__plus(encArg(x_1), encArg(x_2)) encArg(cons_a__x(x_1, x_2)) -> a__x(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__U11(x_1, x_2) -> a__U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_a__U21(x_1, x_2, x_3) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_a__plus(x_1, x_2) -> a__plus(encArg(x_1), encArg(x_2)) encode_a__U31(x_1) -> a__U31(encArg(x_1)) encode_0 -> 0 encode_a__U41(x_1, x_2, x_3) -> a__U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a__x(x_1, x_2) -> a__x(encArg(x_1), encArg(x_2)) encode_a__and(x_1, x_2) -> a__and(encArg(x_1), encArg(x_2)) encode_a__isNat(x_1) -> a__isNat(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_x(x_1, x_2) -> x(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U21(x_1, x_2, x_3) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U31(x_1) -> U31(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) 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 mark(U41(X1, X2, X3)) ->^+ a__U41(mark(X1), X2, X3) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / U41(X1, X2, X3)]. 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(EXP, INF). The TRS R consists of the following rules: a__U11(tt, N) -> mark(N) a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U31(tt) -> 0 a__U41(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) a__plus(N, 0) -> a__U11(a__isNat(N), N) a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N) a__x(N, 0) -> a__U31(a__isNat(N)) a__x(N, s(M)) -> a__U41(a__and(a__isNat(M), isNat(N)), M, N) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1, X2) -> U11(X1, X2) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U31(X) -> U31(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(X) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(isNat(x_1)) -> isNat(encArg(x_1)) encArg(x(x_1, x_2)) -> x(encArg(x_1), encArg(x_2)) encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U21(x_1, x_2, x_3)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(U31(x_1)) -> U31(encArg(x_1)) encArg(U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_a__U11(x_1, x_2)) -> a__U11(encArg(x_1), encArg(x_2)) encArg(cons_a__U21(x_1, x_2, x_3)) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__U31(x_1)) -> a__U31(encArg(x_1)) encArg(cons_a__U41(x_1, x_2, x_3)) -> a__U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__and(x_1, x_2)) -> a__and(encArg(x_1), encArg(x_2)) encArg(cons_a__isNat(x_1)) -> a__isNat(encArg(x_1)) encArg(cons_a__plus(x_1, x_2)) -> a__plus(encArg(x_1), encArg(x_2)) encArg(cons_a__x(x_1, x_2)) -> a__x(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__U11(x_1, x_2) -> a__U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_a__U21(x_1, x_2, x_3) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_a__plus(x_1, x_2) -> a__plus(encArg(x_1), encArg(x_2)) encode_a__U31(x_1) -> a__U31(encArg(x_1)) encode_0 -> 0 encode_a__U41(x_1, x_2, x_3) -> a__U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a__x(x_1, x_2) -> a__x(encArg(x_1), encArg(x_2)) encode_a__and(x_1, x_2) -> a__and(encArg(x_1), encArg(x_2)) encode_a__isNat(x_1) -> a__isNat(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_x(x_1, x_2) -> x(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U21(x_1, x_2, x_3) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U31(x_1) -> U31(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) 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(EXP, INF). The TRS R consists of the following rules: a__U11(tt, N) -> mark(N) a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U31(tt) -> 0 a__U41(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) a__plus(N, 0) -> a__U11(a__isNat(N), N) a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N) a__x(N, 0) -> a__U31(a__isNat(N)) a__x(N, s(M)) -> a__U41(a__and(a__isNat(M), isNat(N)), M, N) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1, X2) -> U11(X1, X2) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U31(X) -> U31(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(X) The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(isNat(x_1)) -> isNat(encArg(x_1)) encArg(x(x_1, x_2)) -> x(encArg(x_1), encArg(x_2)) encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U21(x_1, x_2, x_3)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(U31(x_1)) -> U31(encArg(x_1)) encArg(U41(x_1, x_2, x_3)) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_a__U11(x_1, x_2)) -> a__U11(encArg(x_1), encArg(x_2)) encArg(cons_a__U21(x_1, x_2, x_3)) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__U31(x_1)) -> a__U31(encArg(x_1)) encArg(cons_a__U41(x_1, x_2, x_3)) -> a__U41(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__and(x_1, x_2)) -> a__and(encArg(x_1), encArg(x_2)) encArg(cons_a__isNat(x_1)) -> a__isNat(encArg(x_1)) encArg(cons_a__plus(x_1, x_2)) -> a__plus(encArg(x_1), encArg(x_2)) encArg(cons_a__x(x_1, x_2)) -> a__x(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__U11(x_1, x_2) -> a__U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_mark(x_1) -> mark(encArg(x_1)) encode_a__U21(x_1, x_2, x_3) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_a__plus(x_1, x_2) -> a__plus(encArg(x_1), encArg(x_2)) encode_a__U31(x_1) -> a__U31(encArg(x_1)) encode_0 -> 0 encode_a__U41(x_1, x_2, x_3) -> a__U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a__x(x_1, x_2) -> a__x(encArg(x_1), encArg(x_2)) encode_a__and(x_1, x_2) -> a__and(encArg(x_1), encArg(x_2)) encode_a__isNat(x_1) -> a__isNat(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_x(x_1, x_2) -> x(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U21(x_1, x_2, x_3) -> U21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_U31(x_1) -> U31(encArg(x_1)) encode_U41(x_1, x_2, x_3) -> U41(encArg(x_1), encArg(x_2), encArg(x_3)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (13) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence mark(U41(tt, X2, X3)) ->^+ a__plus(a__x(mark(X3), mark(X2)), mark(X3)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. The pumping substitution is [X3 / U41(tt, X2, X3)]. The result substitution is [ ]. The rewrite sequence mark(U41(tt, X2, X3)) ->^+ a__plus(a__x(mark(X3), mark(X2)), mark(X3)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [X3 / U41(tt, X2, X3)]. The result substitution is [ ]. ---------------------------------------- (14) BOUNDS(EXP, INF)