/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 184 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsTAProof [FINISHED, 160 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(f(x)) -> f(c(f(x))) f(f(x)) -> f(d(f(x))) g(c(x)) -> x g(d(x)) -> x g(c(0)) -> g(d(1)) g(c(1)) -> g(d(0)) 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(c(x_1)) -> c(encArg(x_1)) encArg(d(x_1)) -> d(encArg(x_1)) encArg(0) -> 0 encArg(1) -> 1 encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_0 -> 0 encode_1 -> 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(f(x)) -> f(c(f(x))) f(f(x)) -> f(d(f(x))) g(c(x)) -> x g(d(x)) -> x g(c(0)) -> g(d(1)) g(c(1)) -> g(d(0)) The (relative) TRS S consists of the following rules: encArg(c(x_1)) -> c(encArg(x_1)) encArg(d(x_1)) -> d(encArg(x_1)) encArg(0) -> 0 encArg(1) -> 1 encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_0 -> 0 encode_1 -> 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(f(x)) -> f(c(f(x))) f(f(x)) -> f(d(f(x))) g(c(x)) -> x g(d(x)) -> x g(c(0)) -> g(d(1)) g(c(1)) -> g(d(0)) The (relative) TRS S consists of the following rules: encArg(c(x_1)) -> c(encArg(x_1)) encArg(d(x_1)) -> d(encArg(x_1)) encArg(0) -> 0 encArg(1) -> 1 encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_0 -> 0 encode_1 -> 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(f(x)) -> f(c(f(x))) f(f(x)) -> f(d(f(x))) g(c(x)) -> x g(d(x)) -> x g(c(0)) -> g(d(1)) g(c(1)) -> g(d(0)) encArg(c(x_1)) -> c(encArg(x_1)) encArg(d(x_1)) -> d(encArg(x_1)) encArg(0) -> 0 encArg(1) -> 1 encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_0 -> 0 encode_1 -> 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 3. 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] transitions: c0(0) -> 0 d0(0) -> 0 00() -> 0 10() -> 0 cons_f0(0) -> 0 cons_g0(0) -> 0 f0(0) -> 1 g0(0) -> 2 encArg0(0) -> 3 encode_f0(0) -> 4 encode_c0(0) -> 5 encode_d0(0) -> 6 encode_g0(0) -> 7 encode_00() -> 8 encode_10() -> 9 11() -> 11 d1(11) -> 10 g1(10) -> 2 01() -> 13 d1(13) -> 12 g1(12) -> 2 encArg1(0) -> 14 c1(14) -> 3 encArg1(0) -> 15 d1(15) -> 3 01() -> 3 11() -> 3 encArg1(0) -> 16 f1(16) -> 3 encArg1(0) -> 17 g1(17) -> 3 f1(16) -> 4 c1(14) -> 5 d1(15) -> 6 g1(17) -> 7 01() -> 8 11() -> 9 c1(14) -> 14 c1(14) -> 15 c1(14) -> 16 c1(14) -> 17 d1(15) -> 14 d1(15) -> 15 d1(15) -> 16 d1(15) -> 17 01() -> 14 01() -> 15 01() -> 16 01() -> 17 11() -> 14 11() -> 15 11() -> 16 11() -> 17 f1(16) -> 14 f1(16) -> 15 f1(16) -> 16 f1(16) -> 17 g1(17) -> 14 g1(17) -> 15 g1(17) -> 16 g1(17) -> 17 12() -> 19 d2(19) -> 18 g2(18) -> 3 g2(18) -> 7 g2(18) -> 14 02() -> 21 d2(21) -> 20 g2(20) -> 3 g2(20) -> 7 g2(20) -> 14 f2(16) -> 23 c2(23) -> 22 f2(22) -> 3 f2(22) -> 4 f2(22) -> 14 f2(16) -> 25 d2(25) -> 24 f2(24) -> 3 f2(24) -> 4 f2(24) -> 14 f2(22) -> 23 f2(24) -> 23 f2(22) -> 25 f3(22) -> 27 c3(27) -> 26 f3(26) -> 23 f3(24) -> 27 f3(26) -> 25 f2(24) -> 25 f3(22) -> 29 d3(29) -> 28 f3(28) -> 23 f3(24) -> 29 f3(28) -> 25 0 -> 2 11 -> 2 13 -> 2 14 -> 3 14 -> 7 14 -> 15 14 -> 16 14 -> 17 15 -> 3 15 -> 7 15 -> 14 19 -> 3 19 -> 7 19 -> 14 21 -> 3 21 -> 7 21 -> 14 ---------------------------------------- (8) BOUNDS(1, n^1)