/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 48 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsTAProof [FINISHED, 141 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: d(x) -> e(u(x)) d(u(x)) -> c(x) c(u(x)) -> b(x) v(e(x)) -> x b(u(x)) -> a(e(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(e(x_1)) -> e(encArg(x_1)) encArg(u(x_1)) -> u(encArg(x_1)) encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_v(x_1)) -> v(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) encode_e(x_1) -> e(encArg(x_1)) encode_u(x_1) -> u(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_v(x_1) -> v(encArg(x_1)) encode_a(x_1) -> a(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: d(x) -> e(u(x)) d(u(x)) -> c(x) c(u(x)) -> b(x) v(e(x)) -> x b(u(x)) -> a(e(x)) The (relative) TRS S consists of the following rules: encArg(e(x_1)) -> e(encArg(x_1)) encArg(u(x_1)) -> u(encArg(x_1)) encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_v(x_1)) -> v(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) encode_e(x_1) -> e(encArg(x_1)) encode_u(x_1) -> u(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_v(x_1) -> v(encArg(x_1)) encode_a(x_1) -> a(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: d(x) -> e(u(x)) d(u(x)) -> c(x) c(u(x)) -> b(x) v(e(x)) -> x b(u(x)) -> a(e(x)) The (relative) TRS S consists of the following rules: encArg(e(x_1)) -> e(encArg(x_1)) encArg(u(x_1)) -> u(encArg(x_1)) encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_v(x_1)) -> v(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) encode_e(x_1) -> e(encArg(x_1)) encode_u(x_1) -> u(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_v(x_1) -> v(encArg(x_1)) encode_a(x_1) -> a(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: d(x) -> e(u(x)) d(u(x)) -> c(x) c(u(x)) -> b(x) v(e(x)) -> x b(u(x)) -> a(e(x)) encArg(e(x_1)) -> e(encArg(x_1)) encArg(u(x_1)) -> u(encArg(x_1)) encArg(a(x_1)) -> a(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_v(x_1)) -> v(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) encode_e(x_1) -> e(encArg(x_1)) encode_u(x_1) -> u(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_v(x_1) -> v(encArg(x_1)) encode_a(x_1) -> a(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 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, 10, 11, 12] transitions: e0(0) -> 0 u0(0) -> 0 a0(0) -> 0 cons_d0(0) -> 0 cons_c0(0) -> 0 cons_v0(0) -> 0 cons_b0(0) -> 0 d0(0) -> 1 c0(0) -> 2 v0(0) -> 3 b0(0) -> 4 encArg0(0) -> 5 encode_d0(0) -> 6 encode_e0(0) -> 7 encode_u0(0) -> 8 encode_c0(0) -> 9 encode_b0(0) -> 10 encode_v0(0) -> 11 encode_a0(0) -> 12 u1(0) -> 13 e1(13) -> 1 c1(0) -> 1 b1(0) -> 2 e1(0) -> 14 a1(14) -> 4 encArg1(0) -> 15 e1(15) -> 5 encArg1(0) -> 16 u1(16) -> 5 encArg1(0) -> 17 a1(17) -> 5 encArg1(0) -> 18 d1(18) -> 5 encArg1(0) -> 19 c1(19) -> 5 encArg1(0) -> 20 v1(20) -> 5 encArg1(0) -> 21 b1(21) -> 5 d1(18) -> 6 e1(15) -> 7 u1(16) -> 8 c1(19) -> 9 b1(21) -> 10 v1(20) -> 11 a1(17) -> 12 u2(18) -> 22 e2(22) -> 5 e2(22) -> 6 b1(0) -> 1 a1(14) -> 2 e1(15) -> 15 e1(15) -> 16 e1(15) -> 17 e1(15) -> 18 e1(15) -> 19 e1(15) -> 20 e1(15) -> 21 u1(16) -> 15 u1(16) -> 16 u1(16) -> 17 u1(16) -> 18 u1(16) -> 19 u1(16) -> 20 u1(16) -> 21 a1(17) -> 15 a1(17) -> 16 a1(17) -> 17 a1(17) -> 18 a1(17) -> 19 a1(17) -> 20 a1(17) -> 21 d1(18) -> 15 d1(18) -> 16 d1(18) -> 17 d1(18) -> 18 d1(18) -> 19 d1(18) -> 20 d1(18) -> 21 c1(19) -> 15 c1(19) -> 16 c1(19) -> 17 c1(19) -> 18 c1(19) -> 19 c1(19) -> 20 c1(19) -> 21 v1(20) -> 15 v1(20) -> 16 v1(20) -> 17 v1(20) -> 18 v1(20) -> 19 v1(20) -> 20 v1(20) -> 21 b1(21) -> 15 b1(21) -> 16 b1(21) -> 17 b1(21) -> 18 b1(21) -> 19 b1(21) -> 20 b1(21) -> 21 e2(22) -> 15 e2(22) -> 16 e2(22) -> 17 e2(22) -> 18 e2(22) -> 19 e2(22) -> 20 e2(22) -> 21 c2(16) -> 5 c2(16) -> 6 c2(16) -> 15 c2(16) -> 16 c2(16) -> 17 c2(16) -> 18 c2(16) -> 19 c2(16) -> 20 c2(16) -> 21 b2(16) -> 5 b2(16) -> 9 b2(16) -> 15 b2(16) -> 16 b2(16) -> 17 b2(16) -> 18 b2(16) -> 19 b2(16) -> 20 b2(16) -> 21 a1(14) -> 1 e2(16) -> 23 a2(23) -> 5 a2(23) -> 10 a2(23) -> 15 b2(16) -> 6 a2(23) -> 9 c2(18) -> 5 c2(18) -> 6 c2(18) -> 11 c2(18) -> 15 b2(18) -> 5 b2(18) -> 9 b2(18) -> 11 b2(18) -> 15 e2(18) -> 23 b3(18) -> 5 b3(18) -> 6 b3(18) -> 11 b3(18) -> 15 a2(23) -> 6 e3(18) -> 24 a3(24) -> 5 a3(24) -> 6 a3(24) -> 9 a3(24) -> 11 a3(24) -> 15 0 -> 3 15 -> 5 15 -> 11 15 -> 16 15 -> 17 15 -> 18 15 -> 19 15 -> 20 15 -> 21 22 -> 5 22 -> 11 22 -> 15 ---------------------------------------- (8) BOUNDS(1, n^1)