/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 50 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsTAProof [FINISHED, 500 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: f(n__f(n__a)) -> f(n__g(f(n__a))) f(X) -> n__f(X) a -> n__a g(X) -> n__g(X) activate(n__f(X)) -> f(X) activate(n__a) -> a activate(n__g(X)) -> g(X) activate(X) -> X S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(n__f(x_1)) -> n__f(encArg(x_1)) encArg(n__a) -> n__a encArg(n__g(x_1)) -> n__g(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_a) -> a encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_n__f(x_1) -> n__f(encArg(x_1)) encode_n__a -> n__a encode_n__g(x_1) -> n__g(encArg(x_1)) encode_a -> a encode_g(x_1) -> g(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: f(n__f(n__a)) -> f(n__g(f(n__a))) f(X) -> n__f(X) a -> n__a g(X) -> n__g(X) activate(n__f(X)) -> f(X) activate(n__a) -> a activate(n__g(X)) -> g(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(n__f(x_1)) -> n__f(encArg(x_1)) encArg(n__a) -> n__a encArg(n__g(x_1)) -> n__g(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_a) -> a encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_n__f(x_1) -> n__f(encArg(x_1)) encode_n__a -> n__a encode_n__g(x_1) -> n__g(encArg(x_1)) encode_a -> a encode_g(x_1) -> g(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: f(n__f(n__a)) -> f(n__g(f(n__a))) f(X) -> n__f(X) a -> n__a g(X) -> n__g(X) activate(n__f(X)) -> f(X) activate(n__a) -> a activate(n__g(X)) -> g(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(n__f(x_1)) -> n__f(encArg(x_1)) encArg(n__a) -> n__a encArg(n__g(x_1)) -> n__g(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_a) -> a encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_n__f(x_1) -> n__f(encArg(x_1)) encode_n__a -> n__a encode_n__g(x_1) -> n__g(encArg(x_1)) encode_a -> a encode_g(x_1) -> g(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) 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: f(n__f(n__a)) -> f(n__g(f(n__a))) f(X) -> n__f(X) a -> n__a g(X) -> n__g(X) activate(n__f(X)) -> f(X) activate(n__a) -> a activate(n__g(X)) -> g(X) activate(X) -> X encArg(n__f(x_1)) -> n__f(encArg(x_1)) encArg(n__a) -> n__a encArg(n__g(x_1)) -> n__g(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_a) -> a encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_n__f(x_1) -> n__f(encArg(x_1)) encode_n__a -> n__a encode_n__g(x_1) -> n__g(encArg(x_1)) encode_a -> a encode_g(x_1) -> g(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (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 4. 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, 7, 8, 9, 10, 11, 12] transitions: n__f0(0) -> 0 n__a0() -> 0 n__g0(0) -> 0 cons_f0(0) -> 0 cons_a0() -> 0 cons_g0(0) -> 0 cons_activate0(0) -> 0 f0(0) -> 1 a0() -> 2 g0(0) -> 3 activate0(0) -> 4 encArg0(0) -> 5 encode_f0(0) -> 6 encode_n__f0(0) -> 7 encode_n__a0() -> 8 encode_n__g0(0) -> 9 encode_a0() -> 10 encode_g0(0) -> 11 encode_activate0(0) -> 12 n__a1() -> 15 f1(15) -> 14 n__g1(14) -> 13 f1(13) -> 1 n__f1(0) -> 1 n__a1() -> 2 n__g1(0) -> 3 f1(0) -> 4 a1() -> 4 g1(0) -> 4 encArg1(0) -> 16 n__f1(16) -> 5 n__a1() -> 5 encArg1(0) -> 17 n__g1(17) -> 5 encArg1(0) -> 18 f1(18) -> 5 a1() -> 5 encArg1(0) -> 19 g1(19) -> 5 encArg1(0) -> 20 activate1(20) -> 5 f1(18) -> 6 n__f1(16) -> 7 n__a1() -> 8 n__g1(17) -> 9 a1() -> 10 g1(19) -> 11 activate1(20) -> 12 f1(13) -> 4 n__f2(13) -> 1 n__f2(0) -> 4 n__f2(18) -> 5 n__f2(18) -> 6 n__f2(15) -> 14 n__a2() -> 4 n__a2() -> 5 n__a2() -> 10 n__g2(0) -> 4 n__g2(19) -> 5 n__g2(19) -> 11 n__f1(16) -> 16 n__f1(16) -> 17 n__f1(16) -> 18 n__f1(16) -> 19 n__f1(16) -> 20 n__a1() -> 16 n__a1() -> 17 n__a1() -> 18 n__a1() -> 19 n__a1() -> 20 n__g1(17) -> 16 n__g1(17) -> 17 n__g1(17) -> 18 n__g1(17) -> 19 n__g1(17) -> 20 f1(18) -> 16 f1(18) -> 17 f1(18) -> 18 f1(18) -> 19 f1(18) -> 20 a1() -> 16 a1() -> 17 a1() -> 18 a1() -> 19 a1() -> 20 g1(19) -> 16 g1(19) -> 17 g1(19) -> 18 g1(19) -> 19 g1(19) -> 20 activate1(20) -> 16 activate1(20) -> 17 activate1(20) -> 18 activate1(20) -> 19 activate1(20) -> 20 n__a2() -> 23 f2(23) -> 22 n__g2(22) -> 21 f2(21) -> 5 f2(21) -> 6 f2(21) -> 12 f2(21) -> 16 f2(21) -> 17 f2(21) -> 18 f2(21) -> 19 f2(21) -> 20 n__f2(13) -> 4 n__f2(18) -> 12 n__f2(18) -> 16 n__f2(18) -> 17 n__f2(18) -> 18 n__f2(18) -> 19 n__f2(18) -> 20 n__a2() -> 12 n__a2() -> 16 n__a2() -> 17 n__a2() -> 18 n__a2() -> 19 n__a2() -> 20 n__g2(19) -> 12 n__g2(19) -> 16 n__g2(19) -> 17 n__g2(19) -> 18 n__g2(19) -> 19 n__g2(19) -> 20 f2(16) -> 5 f2(16) -> 12 f2(16) -> 16 f2(16) -> 17 f2(16) -> 18 f2(16) -> 19 f2(16) -> 20 a2() -> 5 a2() -> 12 a2() -> 16 a2() -> 17 a2() -> 18 a2() -> 19 a2() -> 20 g2(17) -> 5 g2(17) -> 12 g2(17) -> 16 g2(17) -> 17 g2(17) -> 18 g2(17) -> 19 g2(17) -> 20 f2(18) -> 5 f2(18) -> 12 f2(18) -> 16 f2(18) -> 17 f2(18) -> 18 f2(18) -> 19 f2(18) -> 20 g2(19) -> 5 g2(19) -> 12 g2(19) -> 16 g2(19) -> 17 g2(19) -> 18 g2(19) -> 19 g2(19) -> 20 n__a3() -> 26 f3(26) -> 25 n__g3(25) -> 24 f3(24) -> 5 f3(24) -> 12 f3(24) -> 16 f3(24) -> 17 f3(24) -> 18 f3(24) -> 19 f3(24) -> 20 n__f3(21) -> 5 n__f3(16) -> 5 n__f3(21) -> 6 n__f3(21) -> 12 n__f3(16) -> 12 n__f3(21) -> 16 n__f3(16) -> 16 n__f3(21) -> 17 n__f3(16) -> 17 n__f3(21) -> 18 n__f3(16) -> 18 n__f3(21) -> 19 n__f3(16) -> 19 n__f3(21) -> 20 n__f3(16) -> 20 n__f3(23) -> 22 n__a3() -> 5 n__a3() -> 12 n__a3() -> 16 n__a3() -> 17 n__a3() -> 18 n__a3() -> 19 n__a3() -> 20 n__g3(17) -> 5 n__g3(17) -> 12 n__g3(17) -> 16 n__g3(17) -> 17 n__g3(17) -> 18 n__g3(17) -> 19 n__g3(17) -> 20 n__f3(18) -> 5 n__f3(18) -> 12 n__f3(18) -> 16 n__f3(18) -> 17 n__f3(18) -> 18 n__f3(18) -> 19 n__f3(18) -> 20 n__g3(19) -> 5 n__g3(19) -> 12 n__g3(19) -> 16 n__g3(19) -> 17 n__g3(19) -> 18 n__g3(19) -> 19 n__g3(19) -> 20 n__f4(24) -> 5 n__f4(24) -> 12 n__f4(24) -> 16 n__f4(24) -> 17 n__f4(24) -> 18 n__f4(24) -> 19 n__f4(24) -> 20 n__f4(26) -> 25 f2(24) -> 5 f2(24) -> 12 f2(24) -> 16 f2(24) -> 17 f2(24) -> 18 f2(24) -> 19 f2(24) -> 20 n__f3(24) -> 5 n__f3(24) -> 12 n__f3(24) -> 16 n__f3(24) -> 17 n__f3(24) -> 18 n__f3(24) -> 19 n__f3(24) -> 20 0 -> 4 20 -> 5 20 -> 12 20 -> 16 20 -> 17 20 -> 18 20 -> 19 ---------------------------------------- (8) BOUNDS(1, n^1)