/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) CpxTrsMatchBoundsTAProof [FINISHED, 29 ms] (4) BOUNDS(1, n^1) (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: revApp#2(Nil, x16) -> x16 revApp#2(Cons(x6, x4), x2) -> revApp#2(x4, Cons(x6, x2)) dfsAcc#3(Leaf(x8), x16) -> Cons(x8, x16) dfsAcc#3(Node(x6, x4), x2) -> dfsAcc#3(x4, dfsAcc#3(x6, x2)) main(x1) -> revApp#2(dfsAcc#3(x1, Nil), Nil) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: revApp#2(Nil, x16) -> x16 revApp#2(Cons(x6, x4), x2) -> revApp#2(x4, Cons(x6, x2)) dfsAcc#3(Leaf(x8), x16) -> Cons(x8, x16) dfsAcc#3(Node(x6, x4), x2) -> dfsAcc#3(x4, dfsAcc#3(x6, x2)) main(x1) -> revApp#2(dfsAcc#3(x1, Nil), Nil) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (3) 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 2. 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 Cons0(0, 0) -> 0 Leaf0(0) -> 0 Node0(0, 0) -> 0 revApp#20(0, 0) -> 1 dfsAcc#30(0, 0) -> 2 main0(0) -> 3 Cons1(0, 0) -> 4 revApp#21(0, 4) -> 1 Cons1(0, 0) -> 2 dfsAcc#31(0, 0) -> 5 dfsAcc#31(0, 5) -> 2 Nil1() -> 7 dfsAcc#31(0, 7) -> 6 Nil1() -> 8 revApp#21(6, 8) -> 3 Cons1(0, 4) -> 4 Cons1(0, 5) -> 2 Cons1(0, 0) -> 5 Cons1(0, 7) -> 6 dfsAcc#31(0, 5) -> 5 dfsAcc#31(0, 7) -> 5 dfsAcc#31(0, 5) -> 6 Cons2(0, 8) -> 9 revApp#22(7, 9) -> 3 Cons1(0, 5) -> 5 Cons1(0, 7) -> 5 Cons1(0, 5) -> 6 revApp#22(5, 9) -> 3 Cons2(0, 9) -> 9 revApp#22(0, 9) -> 3 Cons1(0, 9) -> 4 revApp#21(0, 4) -> 3 0 -> 1 4 -> 1 4 -> 3 9 -> 3 ---------------------------------------- (4) BOUNDS(1, n^1) ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: revApp#2(Nil, x16) -> x16 revApp#2(Cons(x6, x4), x2) -> revApp#2(x4, Cons(x6, x2)) dfsAcc#3(Leaf(x8), x16) -> Cons(x8, x16) dfsAcc#3(Node(x6, x4), x2) -> dfsAcc#3(x4, dfsAcc#3(x6, x2)) main(x1) -> revApp#2(dfsAcc#3(x1, Nil), Nil) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence dfsAcc#3(Node(x6, x4), x2) ->^+ dfsAcc#3(x4, dfsAcc#3(x6, x2)) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [x4 / Node(x6, x4)]. The result substitution is [x2 / dfsAcc#3(x6, x2)]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) 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, n^1). The TRS R consists of the following rules: revApp#2(Nil, x16) -> x16 revApp#2(Cons(x6, x4), x2) -> revApp#2(x4, Cons(x6, x2)) dfsAcc#3(Leaf(x8), x16) -> Cons(x8, x16) dfsAcc#3(Node(x6, x4), x2) -> dfsAcc#3(x4, dfsAcc#3(x6, x2)) main(x1) -> revApp#2(dfsAcc#3(x1, Nil), Nil) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: revApp#2(Nil, x16) -> x16 revApp#2(Cons(x6, x4), x2) -> revApp#2(x4, Cons(x6, x2)) dfsAcc#3(Leaf(x8), x16) -> Cons(x8, x16) dfsAcc#3(Node(x6, x4), x2) -> dfsAcc#3(x4, dfsAcc#3(x6, x2)) main(x1) -> revApp#2(dfsAcc#3(x1, Nil), Nil) S is empty. Rewrite Strategy: INNERMOST