/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), O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) DependencyGraphProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsTAProof [FINISHED, 9 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (10) TRS for Loop Detection (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: rev(nil) -> nil rev(rev(x)) -> x rev(++(x, y)) -> ++(rev(y), rev(x)) ++(nil, y) -> y ++(x, nil) -> x ++(.(x, y), z) -> .(x, ++(y, z)) ++(x, ++(y, z)) -> ++(++(x, y), z) make(x) -> .(x, nil) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) DependencyGraphProof (UPPER BOUND(ID)) The following rules are not reachable from basic terms in the dependency graph and can be removed: rev(rev(x)) -> x rev(++(x, y)) -> ++(rev(y), rev(x)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: rev(nil) -> nil ++(nil, y) -> y ++(x, nil) -> x ++(.(x, y), z) -> .(x, ++(y, z)) ++(x, ++(y, z)) -> ++(++(x, y), z) make(x) -> .(x, nil) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) NestedDefinedSymbolProof (UPPER BOUND(ID)) The TRS does not nest defined symbols. Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: ++(x, ++(y, z)) -> ++(++(x, y), z) ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: rev(nil) -> nil ++(nil, y) -> y ++(x, nil) -> x ++(.(x, y), z) -> .(x, ++(y, z)) make(x) -> .(x, nil) S is empty. Rewrite Strategy: FULL ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: rev(nil) -> nil ++(nil, y) -> y ++(x, nil) -> x ++(.(x, y), z) -> .(x, ++(y, z)) make(x) -> .(x, nil) S is empty. Rewrite Strategy: FULL ---------------------------------------- (7) CpxTrsMatchBoundsTAProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1, 2, 3] transitions: nil0() -> 0 .0(0, 0) -> 0 rev0(0) -> 1 ++0(0, 0) -> 2 make0(0) -> 3 nil1() -> 1 ++1(0, 0) -> 4 .1(0, 4) -> 2 nil1() -> 5 .1(0, 5) -> 3 .1(0, 4) -> 4 0 -> 2 0 -> 4 ---------------------------------------- (8) BOUNDS(1, n^1) ---------------------------------------- (9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (10) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: rev(nil) -> nil rev(rev(x)) -> x rev(++(x, y)) -> ++(rev(y), rev(x)) ++(nil, y) -> y ++(x, nil) -> x ++(.(x, y), z) -> .(x, ++(y, z)) ++(x, ++(y, z)) -> ++(++(x, y), z) make(x) -> .(x, nil) S is empty. Rewrite Strategy: FULL ---------------------------------------- (11) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence ++(.(x, y), z) ->^+ .(x, ++(y, z)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [y / .(x, y)]. The result substitution is [ ]. ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) 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, n^1). The TRS R consists of the following rules: rev(nil) -> nil rev(rev(x)) -> x rev(++(x, y)) -> ++(rev(y), rev(x)) ++(nil, y) -> y ++(x, nil) -> x ++(.(x, y), z) -> .(x, ++(y, z)) ++(x, ++(y, z)) -> ++(++(x, y), z) make(x) -> .(x, nil) S is empty. Rewrite Strategy: FULL ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: rev(nil) -> nil rev(rev(x)) -> x rev(++(x, y)) -> ++(rev(y), rev(x)) ++(nil, y) -> y ++(x, nil) -> x ++(.(x, y), z) -> .(x, ++(y, z)) ++(x, ++(y, z)) -> ++(++(x, y), z) make(x) -> .(x, nil) S is empty. Rewrite Strategy: FULL