/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).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, 238 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: comp_f_g#1(plus_x(x3), comp_f_g(x1, x2), 0) -> plus_x#1(x3, comp_f_g#1(x1, x2, 0)) comp_f_g#1(plus_x(x3), id, 0) -> plus_x#1(x3, 0) map#2(Nil) -> Nil map#2(Cons(x16, x6)) -> Cons(plus_x(x16), map#2(x6)) plus_x#1(0, x6) -> x6 plus_x#1(S(x8), x10) -> S(plus_x#1(x8, x10)) foldr_f#3(Nil, 0) -> 0 foldr_f#3(Cons(x16, x5), x24) -> comp_f_g#1(x16, foldr#3(x5), x24) foldr#3(Nil) -> id foldr#3(Cons(x32, x6)) -> comp_f_g(x32, foldr#3(x6)) main(x3) -> foldr_f#3(map#2(x3), 0) 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: comp_f_g#1(plus_x(x3), comp_f_g(x1, x2), 0) -> plus_x#1(x3, comp_f_g#1(x1, x2, 0)) comp_f_g#1(plus_x(x3), id, 0) -> plus_x#1(x3, 0) map#2(Nil) -> Nil map#2(Cons(x16, x6)) -> Cons(plus_x(x16), map#2(x6)) plus_x#1(0, x6) -> x6 plus_x#1(S(x8), x10) -> S(plus_x#1(x8, x10)) foldr_f#3(Nil, 0) -> 0 foldr_f#3(Cons(x16, x5), x24) -> comp_f_g#1(x16, foldr#3(x5), x24) foldr#3(Nil) -> id foldr#3(Cons(x32, x6)) -> comp_f_g(x32, foldr#3(x6)) main(x3) -> foldr_f#3(map#2(x3), 0) 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, 4, 5, 6] transitions: plus_x0(0) -> 0 comp_f_g0(0, 0) -> 0 00() -> 0 id0() -> 0 Nil0() -> 0 Cons0(0, 0) -> 0 S0(0) -> 0 comp_f_g#10(0, 0, 0) -> 1 map#20(0) -> 2 plus_x#10(0, 0) -> 3 foldr_f#30(0, 0) -> 4 foldr#30(0) -> 5 main0(0) -> 6 01() -> 8 comp_f_g#11(0, 0, 8) -> 7 plus_x#11(0, 7) -> 1 01() -> 9 plus_x#11(0, 9) -> 1 Nil1() -> 2 plus_x1(0) -> 10 map#21(0) -> 11 Cons1(10, 11) -> 2 plus_x#11(0, 0) -> 12 S1(12) -> 3 01() -> 4 foldr#31(0) -> 13 comp_f_g#11(0, 13, 0) -> 4 id1() -> 5 foldr#31(0) -> 14 comp_f_g1(0, 14) -> 5 map#21(0) -> 15 01() -> 16 foldr_f#31(15, 16) -> 6 plus_x#11(0, 7) -> 7 plus_x#11(0, 9) -> 7 Nil1() -> 11 Nil1() -> 15 Cons1(10, 11) -> 11 Cons1(10, 11) -> 15 plus_x#11(0, 7) -> 12 S1(12) -> 1 plus_x#11(0, 9) -> 12 S1(12) -> 12 id1() -> 13 id1() -> 14 comp_f_g1(0, 14) -> 13 comp_f_g1(0, 14) -> 14 comp_f_g#11(0, 14, 8) -> 7 plus_x#11(0, 7) -> 4 plus_x#11(0, 9) -> 4 S1(12) -> 7 02() -> 6 foldr#32(11) -> 17 comp_f_g#12(10, 17, 16) -> 6 id2() -> 17 foldr#32(11) -> 18 comp_f_g2(10, 18) -> 17 id2() -> 18 comp_f_g2(10, 18) -> 18 02() -> 20 comp_f_g#12(10, 18, 20) -> 19 plus_x#12(0, 19) -> 6 02() -> 21 plus_x#12(0, 21) -> 6 plus_x#12(0, 19) -> 19 plus_x#12(0, 21) -> 19 plus_x#11(0, 19) -> 12 S1(12) -> 6 plus_x#11(0, 21) -> 12 S1(12) -> 19 0 -> 3 0 -> 12 7 -> 1 7 -> 12 7 -> 4 9 -> 1 9 -> 7 19 -> 6 19 -> 12 21 -> 6 21 -> 12 21 -> 19 ---------------------------------------- (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: comp_f_g#1(plus_x(x3), comp_f_g(x1, x2), 0) -> plus_x#1(x3, comp_f_g#1(x1, x2, 0)) comp_f_g#1(plus_x(x3), id, 0) -> plus_x#1(x3, 0) map#2(Nil) -> Nil map#2(Cons(x16, x6)) -> Cons(plus_x(x16), map#2(x6)) plus_x#1(0, x6) -> x6 plus_x#1(S(x8), x10) -> S(plus_x#1(x8, x10)) foldr_f#3(Nil, 0) -> 0 foldr_f#3(Cons(x16, x5), x24) -> comp_f_g#1(x16, foldr#3(x5), x24) foldr#3(Nil) -> id foldr#3(Cons(x32, x6)) -> comp_f_g(x32, foldr#3(x6)) main(x3) -> foldr_f#3(map#2(x3), 0) 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 map#2(Cons(x16, x6)) ->^+ Cons(plus_x(x16), map#2(x6)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [x6 / Cons(x16, x6)]. 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: comp_f_g#1(plus_x(x3), comp_f_g(x1, x2), 0) -> plus_x#1(x3, comp_f_g#1(x1, x2, 0)) comp_f_g#1(plus_x(x3), id, 0) -> plus_x#1(x3, 0) map#2(Nil) -> Nil map#2(Cons(x16, x6)) -> Cons(plus_x(x16), map#2(x6)) plus_x#1(0, x6) -> x6 plus_x#1(S(x8), x10) -> S(plus_x#1(x8, x10)) foldr_f#3(Nil, 0) -> 0 foldr_f#3(Cons(x16, x5), x24) -> comp_f_g#1(x16, foldr#3(x5), x24) foldr#3(Nil) -> id foldr#3(Cons(x32, x6)) -> comp_f_g(x32, foldr#3(x6)) main(x3) -> foldr_f#3(map#2(x3), 0) 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: comp_f_g#1(plus_x(x3), comp_f_g(x1, x2), 0) -> plus_x#1(x3, comp_f_g#1(x1, x2, 0)) comp_f_g#1(plus_x(x3), id, 0) -> plus_x#1(x3, 0) map#2(Nil) -> Nil map#2(Cons(x16, x6)) -> Cons(plus_x(x16), map#2(x6)) plus_x#1(0, x6) -> x6 plus_x#1(S(x8), x10) -> S(plus_x#1(x8, x10)) foldr_f#3(Nil, 0) -> 0 foldr_f#3(Cons(x16, x5), x24) -> comp_f_g#1(x16, foldr#3(x5), x24) foldr#3(Nil) -> id foldr#3(Cons(x32, x6)) -> comp_f_g(x32, foldr#3(x6)) main(x3) -> foldr_f#3(map#2(x3), 0) S is empty. Rewrite Strategy: INNERMOST