/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsMatchBoundsProof [FINISHED, 2 ms] (6) BOUNDS(1, n^1) (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (8) TRS for Loop Detection (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (10) BEST (11) proven lower bound (12) LowerBoundPropagationProof [FINISHED, 0 ms] (13) BOUNDS(n^1, INF) (14) 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: active(g(X)) -> mark(h(X)) active(c) -> mark(d) active(h(d)) -> mark(g(c)) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) proper(c) -> ok(c) proper(d) -> ok(d) g(ok(X)) -> ok(g(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) The following defined symbols can occur below the 0th argument of top: proper, active The following defined symbols can occur below the 0th argument of proper: proper, active The following defined symbols can occur below the 0th argument of active: proper, active Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: active(g(X)) -> mark(h(X)) active(h(d)) -> mark(g(c)) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(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: active(c) -> mark(d) proper(c) -> ok(c) proper(d) -> ok(d) g(ok(X)) -> ok(g(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (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: active(c) -> mark(d) proper(c) -> ok(c) proper(d) -> ok(d) g(ok(X)) -> ok(g(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (5) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 4. The certificate found is represented by the following graph. "[21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] {(21,22,[active_1|0, proper_1|0, g_1|0, h_1|0, top_1|0]), (21,23,[mark_1|1]), (21,24,[ok_1|1]), (21,25,[ok_1|1]), (21,26,[ok_1|1]), (21,27,[ok_1|1]), (21,28,[top_1|1]), (21,29,[top_1|1]), (21,30,[top_1|2]), (21,31,[top_1|2]), (21,34,[top_1|3]), (21,35,[top_1|3]), (21,37,[top_1|4]), (22,22,[c|0, mark_1|0, d|0, ok_1|0]), (23,22,[d|1]), (24,22,[c|1]), (25,22,[d|1]), (26,22,[g_1|1]), (26,26,[ok_1|1]), (27,22,[h_1|1]), (27,27,[ok_1|1]), (28,22,[proper_1|1]), (28,24,[ok_1|1]), (28,25,[ok_1|1]), (29,22,[active_1|1]), (29,23,[mark_1|1]), (30,24,[active_1|2]), (30,25,[active_1|2]), (30,32,[mark_1|2]), (31,23,[proper_1|2]), (31,33,[ok_1|2]), (32,22,[d|2]), (33,22,[d|2]), (34,32,[proper_1|3]), (34,36,[ok_1|3]), (35,33,[active_1|3]), (36,22,[d|3]), (37,36,[active_1|4])}" ---------------------------------------- (6) BOUNDS(1, n^1) ---------------------------------------- (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (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, n^1). The TRS R consists of the following rules: active(g(X)) -> mark(h(X)) active(c) -> mark(d) active(h(d)) -> mark(g(c)) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) proper(c) -> ok(c) proper(d) -> ok(d) g(ok(X)) -> ok(g(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence g(ok(X)) ->^+ ok(g(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / ok(X)]. The result substitution is [ ]. ---------------------------------------- (10) Complex Obligation (BEST) ---------------------------------------- (11) 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: active(g(X)) -> mark(h(X)) active(c) -> mark(d) active(h(d)) -> mark(g(c)) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) proper(c) -> ok(c) proper(d) -> ok(d) g(ok(X)) -> ok(g(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (12) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (13) BOUNDS(n^1, INF) ---------------------------------------- (14) 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: active(g(X)) -> mark(h(X)) active(c) -> mark(d) active(h(d)) -> mark(g(c)) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) proper(c) -> ok(c) proper(d) -> ok(d) g(ok(X)) -> ok(g(X)) h(ok(X)) -> ok(h(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL