/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).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, 51 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: walk#1(Leaf(x2)) -> cons_x(x2) walk#1(Node(x5, x3)) -> comp_f_g(walk#1(x5), walk#1(x3)) comp_f_g#1(comp_f_g(x4, x5), comp_f_g(x2, x3), x1) -> comp_f_g#1(x4, x5, comp_f_g#1(x2, x3, x1)) comp_f_g#1(comp_f_g(x7, x9), cons_x(x2), x4) -> comp_f_g#1(x7, x9, Cons(x2, x4)) comp_f_g#1(cons_x(x2), comp_f_g(x5, x7), x3) -> Cons(x2, comp_f_g#1(x5, x7, x3)) comp_f_g#1(cons_x(x5), cons_x(x2), x4) -> Cons(x5, Cons(x2, x4)) main(Leaf(x4)) -> Cons(x4, Nil) main(Node(x9, x5)) -> comp_f_g#1(walk#1(x9), walk#1(x5), 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: walk#1(Leaf(x2)) -> cons_x(x2) walk#1(Node(x5, x3)) -> comp_f_g(walk#1(x5), walk#1(x3)) comp_f_g#1(comp_f_g(x4, x5), comp_f_g(x2, x3), x1) -> comp_f_g#1(x4, x5, comp_f_g#1(x2, x3, x1)) comp_f_g#1(comp_f_g(x7, x9), cons_x(x2), x4) -> comp_f_g#1(x7, x9, Cons(x2, x4)) comp_f_g#1(cons_x(x2), comp_f_g(x5, x7), x3) -> Cons(x2, comp_f_g#1(x5, x7, x3)) comp_f_g#1(cons_x(x5), cons_x(x2), x4) -> Cons(x5, Cons(x2, x4)) main(Leaf(x4)) -> Cons(x4, Nil) main(Node(x9, x5)) -> comp_f_g#1(walk#1(x9), walk#1(x5), 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: Leaf0(0) -> 0 cons_x0(0) -> 0 Node0(0, 0) -> 0 comp_f_g0(0, 0) -> 0 Cons0(0, 0) -> 0 Nil0() -> 0 walk#10(0) -> 1 comp_f_g#10(0, 0, 0) -> 2 main0(0) -> 3 cons_x1(0) -> 1 walk#11(0) -> 4 walk#11(0) -> 5 comp_f_g1(4, 5) -> 1 comp_f_g#11(0, 0, 0) -> 6 comp_f_g#11(0, 0, 6) -> 2 Cons1(0, 0) -> 7 comp_f_g#11(0, 0, 7) -> 2 comp_f_g#11(0, 0, 0) -> 8 Cons1(0, 8) -> 2 Cons1(0, 0) -> 9 Cons1(0, 9) -> 2 Nil1() -> 10 Cons1(0, 10) -> 3 walk#11(0) -> 11 walk#11(0) -> 12 Nil1() -> 13 comp_f_g#11(11, 12, 13) -> 3 cons_x1(0) -> 4 cons_x1(0) -> 5 cons_x1(0) -> 11 cons_x1(0) -> 12 comp_f_g1(4, 5) -> 4 comp_f_g1(4, 5) -> 5 comp_f_g1(4, 5) -> 11 comp_f_g1(4, 5) -> 12 comp_f_g#11(0, 0, 6) -> 6 comp_f_g#11(0, 0, 7) -> 6 comp_f_g#11(0, 0, 6) -> 8 Cons1(0, 6) -> 7 Cons1(0, 7) -> 7 comp_f_g#11(0, 0, 7) -> 8 Cons1(0, 8) -> 6 Cons1(0, 8) -> 8 Cons1(0, 6) -> 9 Cons1(0, 7) -> 9 Cons1(0, 9) -> 6 Cons1(0, 9) -> 8 comp_f_g#12(4, 5, 13) -> 14 comp_f_g#12(4, 5, 14) -> 3 Cons2(0, 13) -> 15 comp_f_g#12(4, 5, 15) -> 3 comp_f_g#12(4, 5, 13) -> 16 Cons2(0, 16) -> 3 Cons2(0, 13) -> 17 Cons2(0, 17) -> 3 comp_f_g#12(4, 5, 14) -> 14 comp_f_g#12(4, 5, 15) -> 14 comp_f_g#12(4, 5, 14) -> 16 Cons2(0, 14) -> 15 Cons2(0, 15) -> 15 comp_f_g#12(4, 5, 15) -> 16 Cons2(0, 16) -> 14 Cons2(0, 16) -> 16 Cons2(0, 14) -> 17 Cons2(0, 15) -> 17 Cons2(0, 17) -> 14 Cons2(0, 17) -> 16 ---------------------------------------- (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: walk#1(Leaf(x2)) -> cons_x(x2) walk#1(Node(x5, x3)) -> comp_f_g(walk#1(x5), walk#1(x3)) comp_f_g#1(comp_f_g(x4, x5), comp_f_g(x2, x3), x1) -> comp_f_g#1(x4, x5, comp_f_g#1(x2, x3, x1)) comp_f_g#1(comp_f_g(x7, x9), cons_x(x2), x4) -> comp_f_g#1(x7, x9, Cons(x2, x4)) comp_f_g#1(cons_x(x2), comp_f_g(x5, x7), x3) -> Cons(x2, comp_f_g#1(x5, x7, x3)) comp_f_g#1(cons_x(x5), cons_x(x2), x4) -> Cons(x5, Cons(x2, x4)) main(Leaf(x4)) -> Cons(x4, Nil) main(Node(x9, x5)) -> comp_f_g#1(walk#1(x9), walk#1(x5), 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 walk#1(Node(x5, x3)) ->^+ comp_f_g(walk#1(x5), walk#1(x3)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [x5 / Node(x5, x3)]. The result substitution is [ ]. ---------------------------------------- (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: walk#1(Leaf(x2)) -> cons_x(x2) walk#1(Node(x5, x3)) -> comp_f_g(walk#1(x5), walk#1(x3)) comp_f_g#1(comp_f_g(x4, x5), comp_f_g(x2, x3), x1) -> comp_f_g#1(x4, x5, comp_f_g#1(x2, x3, x1)) comp_f_g#1(comp_f_g(x7, x9), cons_x(x2), x4) -> comp_f_g#1(x7, x9, Cons(x2, x4)) comp_f_g#1(cons_x(x2), comp_f_g(x5, x7), x3) -> Cons(x2, comp_f_g#1(x5, x7, x3)) comp_f_g#1(cons_x(x5), cons_x(x2), x4) -> Cons(x5, Cons(x2, x4)) main(Leaf(x4)) -> Cons(x4, Nil) main(Node(x9, x5)) -> comp_f_g#1(walk#1(x9), walk#1(x5), 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: walk#1(Leaf(x2)) -> cons_x(x2) walk#1(Node(x5, x3)) -> comp_f_g(walk#1(x5), walk#1(x3)) comp_f_g#1(comp_f_g(x4, x5), comp_f_g(x2, x3), x1) -> comp_f_g#1(x4, x5, comp_f_g#1(x2, x3, x1)) comp_f_g#1(comp_f_g(x7, x9), cons_x(x2), x4) -> comp_f_g#1(x7, x9, Cons(x2, x4)) comp_f_g#1(cons_x(x2), comp_f_g(x5, x7), x3) -> Cons(x2, comp_f_g#1(x5, x7, x3)) comp_f_g#1(cons_x(x5), cons_x(x2), x4) -> Cons(x5, Cons(x2, x4)) main(Leaf(x4)) -> Cons(x4, Nil) main(Node(x9, x5)) -> comp_f_g#1(walk#1(x9), walk#1(x5), Nil) S is empty. Rewrite Strategy: INNERMOST