/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).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, 472 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, M, N) -> a__U12(tt, M, N) a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U21(tt, M, N) -> a__U22(tt, M, N) a__U22(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) a__plus(N, 0) -> mark(N) a__plus(N, s(M)) -> a__U11(tt, M, N) a__x(N, 0) -> 0 a__x(N, s(M)) -> a__U21(tt, M, N) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(U22(X1, X2, X3)) -> a__U22(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 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, X3) -> U12(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__U22(X1, X2, X3) -> U22(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) 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, M, N) -> a__U12(tt, M, N) a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U21(tt, M, N) -> a__U22(tt, M, N) a__U22(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) a__plus(N, 0) -> mark(N) a__plus(N, s(M)) -> a__U11(tt, M, N) a__x(N, 0) -> 0 a__x(N, s(M)) -> a__U21(tt, M, N) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(U22(X1, X2, X3)) -> a__U22(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 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, X3) -> U12(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__U22(X1, X2, X3) -> U22(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) 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(U12(X1, X2, X3)) ->^+ a__U12(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 / U12(X1, X2, X3)]. 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, M, N) -> a__U12(tt, M, N) a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U21(tt, M, N) -> a__U22(tt, M, N) a__U22(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) a__plus(N, 0) -> mark(N) a__plus(N, s(M)) -> a__U11(tt, M, N) a__x(N, 0) -> 0 a__x(N, s(M)) -> a__U21(tt, M, N) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(U22(X1, X2, X3)) -> a__U22(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 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, X3) -> U12(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__U22(X1, X2, X3) -> U22(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) 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, M, N) -> a__U12(tt, M, N) a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U21(tt, M, N) -> a__U22(tt, M, N) a__U22(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) a__plus(N, 0) -> mark(N) a__plus(N, s(M)) -> a__U11(tt, M, N) a__x(N, 0) -> 0 a__x(N, s(M)) -> a__U21(tt, M, N) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(U22(X1, X2, X3)) -> a__U22(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 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, X3) -> U12(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__U22(X1, X2, X3) -> U22(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) 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(U22(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 / U22(tt, X2, X3)]. The result substitution is [ ]. The rewrite sequence mark(U22(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 / U22(tt, X2, X3)]. The result substitution is [ ]. ---------------------------------------- (10) BOUNDS(EXP, INF)