/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). (0) CpxTRS (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (2) TRS for Loop Detection (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (4) BEST (5) proven lower bound (6) LowerBoundPropagationProof [FINISHED, 0 ms] (7) BOUNDS(n^1, INF) (8) TRS for Loop Detection (9) DecreasingLoopProof [FINISHED, 2073 ms] (10) BOUNDS(EXP, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U11(tt, V1, V2) -> a__U12(a__isNat(V1), V2) a__U12(tt, V2) -> a__U13(a__isNat(V2)) a__U13(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V1, V2) -> a__U32(a__isNat(V1), V2) a__U32(tt, V2) -> a__U33(a__isNat(V2)) a__U33(tt) -> tt a__U41(tt, N) -> mark(N) a__U51(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U61(tt) -> 0 a__U71(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__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNat(x(V1, V2)) -> a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) a__isNatKind(0) -> tt a__isNatKind(plus(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(x(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2)) a__plus(N, 0) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N) a__plus(N, s(M)) -> a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) a__x(N, 0) -> a__U61(a__and(a__isNat(N), isNatKind(N))) a__x(N, s(M)) -> a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2)) -> a__U41(mark(X1), X2) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1, X2, X3) -> U11(X1, X2, X3) a__U12(X1, X2) -> U12(X1, X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__U31(X1, X2, X3) -> U31(X1, X2, X3) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2) -> U41(X1, X2) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U61(X) -> U61(X) a__U71(X1, X2, X3) -> U71(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNatKind(X) -> isNatKind(X) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (2) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U11(tt, V1, V2) -> a__U12(a__isNat(V1), V2) a__U12(tt, V2) -> a__U13(a__isNat(V2)) a__U13(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V1, V2) -> a__U32(a__isNat(V1), V2) a__U32(tt, V2) -> a__U33(a__isNat(V2)) a__U33(tt) -> tt a__U41(tt, N) -> mark(N) a__U51(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U61(tt) -> 0 a__U71(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__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNat(x(V1, V2)) -> a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) a__isNatKind(0) -> tt a__isNatKind(plus(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(x(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2)) a__plus(N, 0) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N) a__plus(N, s(M)) -> a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) a__x(N, 0) -> a__U61(a__and(a__isNat(N), isNatKind(N))) a__x(N, s(M)) -> a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2)) -> a__U41(mark(X1), X2) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1, X2, X3) -> U11(X1, X2, X3) a__U12(X1, X2) -> U12(X1, X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__U31(X1, X2, X3) -> U31(X1, X2, X3) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2) -> U41(X1, X2) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U61(X) -> U61(X) a__U71(X1, X2, X3) -> U71(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNatKind(X) -> isNatKind(X) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) 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)) ->^+ a__U41(mark(X1), X2) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / U41(X1, X2)]. The result substitution is [ ]. ---------------------------------------- (4) Complex Obligation (BEST) ---------------------------------------- (5) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U11(tt, V1, V2) -> a__U12(a__isNat(V1), V2) a__U12(tt, V2) -> a__U13(a__isNat(V2)) a__U13(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V1, V2) -> a__U32(a__isNat(V1), V2) a__U32(tt, V2) -> a__U33(a__isNat(V2)) a__U33(tt) -> tt a__U41(tt, N) -> mark(N) a__U51(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U61(tt) -> 0 a__U71(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__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNat(x(V1, V2)) -> a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) a__isNatKind(0) -> tt a__isNatKind(plus(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(x(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2)) a__plus(N, 0) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N) a__plus(N, s(M)) -> a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) a__x(N, 0) -> a__U61(a__and(a__isNat(N), isNatKind(N))) a__x(N, s(M)) -> a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2)) -> a__U41(mark(X1), X2) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1, X2, X3) -> U11(X1, X2, X3) a__U12(X1, X2) -> U12(X1, X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__U31(X1, X2, X3) -> U31(X1, X2, X3) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2) -> U41(X1, X2) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U61(X) -> U61(X) a__U71(X1, X2, X3) -> U71(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNatKind(X) -> isNatKind(X) S is empty. Rewrite Strategy: FULL ---------------------------------------- (6) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (7) BOUNDS(n^1, INF) ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U11(tt, V1, V2) -> a__U12(a__isNat(V1), V2) a__U12(tt, V2) -> a__U13(a__isNat(V2)) a__U13(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V1, V2) -> a__U32(a__isNat(V1), V2) a__U32(tt, V2) -> a__U33(a__isNat(V2)) a__U33(tt) -> tt a__U41(tt, N) -> mark(N) a__U51(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U61(tt) -> 0 a__U71(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__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNat(x(V1, V2)) -> a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) a__isNatKind(0) -> tt a__isNatKind(plus(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2)) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(x(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2)) a__plus(N, 0) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N) a__plus(N, s(M)) -> a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) a__x(N, 0) -> a__U61(a__and(a__isNat(N), isNatKind(N))) a__x(N, s(M)) -> a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2)) -> a__U41(mark(X1), X2) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatKind(X)) -> a__isNatKind(X) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1, X2, X3) -> U11(X1, X2, X3) a__U12(X1, X2) -> U12(X1, X2) a__isNat(X) -> isNat(X) a__U13(X) -> U13(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__U31(X1, X2, X3) -> U31(X1, X2, X3) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2) -> U41(X1, X2) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U61(X) -> U61(X) a__U71(X1, X2, X3) -> U71(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNatKind(X) -> isNatKind(X) S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence mark(plus(X1, s(X1_0))) ->^+ a__U51(a__and(a__and(a__isNat(mark(X1_0)), isNatKind(mark(X1_0))), and(isNat(mark(X1)), isNatKind(mark(X1)))), mark(X1_0), mark(X1)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0]. The pumping substitution is [X1_0 / plus(X1, s(X1_0))]. The result substitution is [ ]. The rewrite sequence mark(plus(X1, s(X1_0))) ->^+ a__U51(a__and(a__and(a__isNat(mark(X1_0)), isNatKind(mark(X1_0))), and(isNat(mark(X1)), isNatKind(mark(X1)))), mark(X1_0), mark(X1)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,1,0]. The pumping substitution is [X1_0 / plus(X1, s(X1_0))]. The result substitution is [ ]. ---------------------------------------- (10) BOUNDS(EXP, INF)