/export/starexec/sandbox2/solver/bin/starexec_run_complexity /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).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 ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt, L)) -> mark(U12(tt, L)) active(U12(tt, L)) -> mark(s(length(L))) active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N)) active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N)) active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U11(tt, L)) active(take(0, IL)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X1, X2)) -> U12(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4) active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4) active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X1), X2) -> mark(U12(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4)) U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4)) U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4)) U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4)) U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(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(n^1, INF). The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt, L)) -> mark(U12(tt, L)) active(U12(tt, L)) -> mark(s(length(L))) active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N)) active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N)) active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U11(tt, L)) active(take(0, IL)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X1, X2)) -> U12(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4) active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4) active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X1), X2) -> mark(U12(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4)) U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4)) U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4)) U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4)) U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(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 take(ok(X1), ok(X2)) ->^+ ok(take(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 [ ]. ---------------------------------------- (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(n^1, INF). The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt, L)) -> mark(U12(tt, L)) active(U12(tt, L)) -> mark(s(length(L))) active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N)) active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N)) active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U11(tt, L)) active(take(0, IL)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X1, X2)) -> U12(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4) active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4) active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X1), X2) -> mark(U12(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4)) U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4)) U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4)) U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4)) U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(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(n^1, INF). The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt, L)) -> mark(U12(tt, L)) active(U12(tt, L)) -> mark(s(length(L))) active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N)) active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N)) active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U11(tt, L)) active(take(0, IL)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(U12(X1, X2)) -> U12(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4) active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4) active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) U12(mark(X1), X2) -> mark(U12(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4)) U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4)) U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4)) proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4)) U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4)) U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL