/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: divide_ys#1(x2, x1) -> Cons(take#2(x1, x2), Cons(drop#2(x1, x2), Nil)) cond_merge_ys_zs_2(True, Cons(x7, x8), Cons(x5, x6), x4, x3, x2, x1) -> Cons(x4, merge#2(x3, Cons(x5, x6))) cond_merge_ys_zs_2(False, Cons(x7, x8), Cons(x5, x6), x4, x3, x2, x1) -> Cons(x2, merge#2(Cons(x7, x8), x1)) merge#2(Nil, x2) -> x2 merge#2(Cons(x4, x2), Nil) -> Cons(x4, x2) merge#2(Cons(x8, x6), Cons(x4, x2)) -> cond_merge_ys_zs_2(leq#2(x8, x4), Cons(x8, x6), Cons(x4, x2), x8, x6, x4, x2) dc#1(map, divisible, mergesort_zs_3, divide, const_f, Nil) -> Nil dc#1(map, divisible, mergesort_zs_3, divide, const_f, Cons(x229, Nil)) -> Cons(x229, Nil) dc#1(map, divisible, mergesort_zs_3, divide, const_f, Cons(x51, Cons(x25, x33))) -> const_f#2(Cons(x51, Cons(x25, x33)), map#2(dc(map, divisible, mergesort_zs_3, divide, const_f), divide_ys#1(Cons(x51, Cons(x25, x33)), S(halve#1(length#1(x33)))))) drop#2(0, x2) -> x2 drop#2(S(0), Nil) -> bot[1] drop#2(S(x10), Cons(x56, x64)) -> drop#2(x10, x64) take#2(0, x2) -> Nil take#2(S(0), Nil) -> Cons(bot[0], Nil) take#2(S(x22), Cons(x56, x64)) -> Cons(x56, take#2(x22, x64)) halve#1(0) -> 0 halve#1(S(0)) -> S(0) halve#1(S(S(x14))) -> S(halve#1(x14)) const_f#2(x3, Cons(x6, Cons(x4, x2))) -> merge#2(x6, x4) leq#2(0, x16) -> True leq#2(S(x20), 0) -> False leq#2(S(x4), S(x2)) -> leq#2(x4, x2) length#1(Nil) -> 0 length#1(Cons(x6, x8)) -> S(length#1(x8)) map#2(dc(x2, x4, x6, x8, x10), Nil) -> Nil map#2(dc(x6, x8, x10, x12, x14), Cons(x4, x2)) -> Cons(dc#1(x6, x8, x10, x12, x14, x4), map#2(dc(x6, x8, x10, x12, x14), x2)) main(x113) -> dc#1(map, divisible, mergesort_zs_3, divide, const_f, x113) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: divide_ys#1(x2, x1) -> Cons(take#2(x1, x2), Cons(drop#2(x1, x2), Nil)) cond_merge_ys_zs_2(True, Cons(x7, x8), Cons(x5, x6), x4, x3, x2, x1) -> Cons(x4, merge#2(x3, Cons(x5, x6))) cond_merge_ys_zs_2(False, Cons(x7, x8), Cons(x5, x6), x4, x3, x2, x1) -> Cons(x2, merge#2(Cons(x7, x8), x1)) merge#2(Nil, x2) -> x2 merge#2(Cons(x4, x2), Nil) -> Cons(x4, x2) merge#2(Cons(x8, x6), Cons(x4, x2)) -> cond_merge_ys_zs_2(leq#2(x8, x4), Cons(x8, x6), Cons(x4, x2), x8, x6, x4, x2) dc#1(map, divisible, mergesort_zs_3, divide, const_f, Nil) -> Nil dc#1(map, divisible, mergesort_zs_3, divide, const_f, Cons(x229, Nil)) -> Cons(x229, Nil) dc#1(map, divisible, mergesort_zs_3, divide, const_f, Cons(x51, Cons(x25, x33))) -> const_f#2(Cons(x51, Cons(x25, x33)), map#2(dc(map, divisible, mergesort_zs_3, divide, const_f), divide_ys#1(Cons(x51, Cons(x25, x33)), S(halve#1(length#1(x33)))))) drop#2(0, x2) -> x2 drop#2(S(0), Nil) -> bot[1] drop#2(S(x10), Cons(x56, x64)) -> drop#2(x10, x64) take#2(0, x2) -> Nil take#2(S(0), Nil) -> Cons(bot[0], Nil) take#2(S(x22), Cons(x56, x64)) -> Cons(x56, take#2(x22, x64)) halve#1(0) -> 0 halve#1(S(0)) -> S(0) halve#1(S(S(x14))) -> S(halve#1(x14)) const_f#2(x3, Cons(x6, Cons(x4, x2))) -> merge#2(x6, x4) leq#2(0, x16) -> True leq#2(S(x20), 0) -> False leq#2(S(x4), S(x2)) -> leq#2(x4, x2) length#1(Nil) -> 0 length#1(Cons(x6, x8)) -> S(length#1(x8)) map#2(dc(x2, x4, x6, x8, x10), Nil) -> Nil map#2(dc(x6, x8, x10, x12, x14), Cons(x4, x2)) -> Cons(dc#1(x6, x8, x10, x12, x14, x4), map#2(dc(x6, x8, x10, x12, x14), x2)) main(x113) -> dc#1(map, divisible, mergesort_zs_3, divide, const_f, x113) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (3) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence length#1(Cons(x6, x8)) ->^+ S(length#1(x8)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [x8 / Cons(x6, x8)]. The result substitution is [ ]. ---------------------------------------- (4) Complex Obligation (BEST) ---------------------------------------- (5) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: divide_ys#1(x2, x1) -> Cons(take#2(x1, x2), Cons(drop#2(x1, x2), Nil)) cond_merge_ys_zs_2(True, Cons(x7, x8), Cons(x5, x6), x4, x3, x2, x1) -> Cons(x4, merge#2(x3, Cons(x5, x6))) cond_merge_ys_zs_2(False, Cons(x7, x8), Cons(x5, x6), x4, x3, x2, x1) -> Cons(x2, merge#2(Cons(x7, x8), x1)) merge#2(Nil, x2) -> x2 merge#2(Cons(x4, x2), Nil) -> Cons(x4, x2) merge#2(Cons(x8, x6), Cons(x4, x2)) -> cond_merge_ys_zs_2(leq#2(x8, x4), Cons(x8, x6), Cons(x4, x2), x8, x6, x4, x2) dc#1(map, divisible, mergesort_zs_3, divide, const_f, Nil) -> Nil dc#1(map, divisible, mergesort_zs_3, divide, const_f, Cons(x229, Nil)) -> Cons(x229, Nil) dc#1(map, divisible, mergesort_zs_3, divide, const_f, Cons(x51, Cons(x25, x33))) -> const_f#2(Cons(x51, Cons(x25, x33)), map#2(dc(map, divisible, mergesort_zs_3, divide, const_f), divide_ys#1(Cons(x51, Cons(x25, x33)), S(halve#1(length#1(x33)))))) drop#2(0, x2) -> x2 drop#2(S(0), Nil) -> bot[1] drop#2(S(x10), Cons(x56, x64)) -> drop#2(x10, x64) take#2(0, x2) -> Nil take#2(S(0), Nil) -> Cons(bot[0], Nil) take#2(S(x22), Cons(x56, x64)) -> Cons(x56, take#2(x22, x64)) halve#1(0) -> 0 halve#1(S(0)) -> S(0) halve#1(S(S(x14))) -> S(halve#1(x14)) const_f#2(x3, Cons(x6, Cons(x4, x2))) -> merge#2(x6, x4) leq#2(0, x16) -> True leq#2(S(x20), 0) -> False leq#2(S(x4), S(x2)) -> leq#2(x4, x2) length#1(Nil) -> 0 length#1(Cons(x6, x8)) -> S(length#1(x8)) map#2(dc(x2, x4, x6, x8, x10), Nil) -> Nil map#2(dc(x6, x8, x10, x12, x14), Cons(x4, x2)) -> Cons(dc#1(x6, x8, x10, x12, x14, x4), map#2(dc(x6, x8, x10, x12, x14), x2)) main(x113) -> dc#1(map, divisible, mergesort_zs_3, divide, const_f, x113) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (6) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (7) BOUNDS(n^1, INF) ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: divide_ys#1(x2, x1) -> Cons(take#2(x1, x2), Cons(drop#2(x1, x2), Nil)) cond_merge_ys_zs_2(True, Cons(x7, x8), Cons(x5, x6), x4, x3, x2, x1) -> Cons(x4, merge#2(x3, Cons(x5, x6))) cond_merge_ys_zs_2(False, Cons(x7, x8), Cons(x5, x6), x4, x3, x2, x1) -> Cons(x2, merge#2(Cons(x7, x8), x1)) merge#2(Nil, x2) -> x2 merge#2(Cons(x4, x2), Nil) -> Cons(x4, x2) merge#2(Cons(x8, x6), Cons(x4, x2)) -> cond_merge_ys_zs_2(leq#2(x8, x4), Cons(x8, x6), Cons(x4, x2), x8, x6, x4, x2) dc#1(map, divisible, mergesort_zs_3, divide, const_f, Nil) -> Nil dc#1(map, divisible, mergesort_zs_3, divide, const_f, Cons(x229, Nil)) -> Cons(x229, Nil) dc#1(map, divisible, mergesort_zs_3, divide, const_f, Cons(x51, Cons(x25, x33))) -> const_f#2(Cons(x51, Cons(x25, x33)), map#2(dc(map, divisible, mergesort_zs_3, divide, const_f), divide_ys#1(Cons(x51, Cons(x25, x33)), S(halve#1(length#1(x33)))))) drop#2(0, x2) -> x2 drop#2(S(0), Nil) -> bot[1] drop#2(S(x10), Cons(x56, x64)) -> drop#2(x10, x64) take#2(0, x2) -> Nil take#2(S(0), Nil) -> Cons(bot[0], Nil) take#2(S(x22), Cons(x56, x64)) -> Cons(x56, take#2(x22, x64)) halve#1(0) -> 0 halve#1(S(0)) -> S(0) halve#1(S(S(x14))) -> S(halve#1(x14)) const_f#2(x3, Cons(x6, Cons(x4, x2))) -> merge#2(x6, x4) leq#2(0, x16) -> True leq#2(S(x20), 0) -> False leq#2(S(x4), S(x2)) -> leq#2(x4, x2) length#1(Nil) -> 0 length#1(Cons(x6, x8)) -> S(length#1(x8)) map#2(dc(x2, x4, x6, x8, x10), Nil) -> Nil map#2(dc(x6, x8, x10, x12, x14), Cons(x4, x2)) -> Cons(dc#1(x6, x8, x10, x12, x14, x4), map#2(dc(x6, x8, x10, x12, x14), x2)) main(x113) -> dc#1(map, divisible, mergesort_zs_3, divide, const_f, x113) S is empty. Rewrite Strategy: INNERMOST