/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 41 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsTAProof [FINISHED, 105 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(a)) -> c(n__f(g(f(a)))) f(X) -> n__f(X) activate(n__f(X)) -> f(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(a) -> a encArg(c(x_1)) -> c(encArg(x_1)) encArg(n__f(x_1)) -> n__f(encArg(x_1)) encArg(g(x_1)) -> g(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_a -> a encode_c(x_1) -> c(encArg(x_1)) encode_n__f(x_1) -> n__f(encArg(x_1)) 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(f(a)) -> c(n__f(g(f(a)))) f(X) -> n__f(X) activate(n__f(X)) -> f(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(c(x_1)) -> c(encArg(x_1)) encArg(n__f(x_1)) -> n__f(encArg(x_1)) encArg(g(x_1)) -> g(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_a -> a encode_c(x_1) -> c(encArg(x_1)) encode_n__f(x_1) -> n__f(encArg(x_1)) 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(f(a)) -> c(n__f(g(f(a)))) f(X) -> n__f(X) activate(n__f(X)) -> f(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(c(x_1)) -> c(encArg(x_1)) encArg(n__f(x_1)) -> n__f(encArg(x_1)) encArg(g(x_1)) -> g(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_a -> a encode_c(x_1) -> c(encArg(x_1)) encode_n__f(x_1) -> n__f(encArg(x_1)) 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(f(a)) -> c(n__f(g(f(a)))) f(X) -> n__f(X) activate(n__f(X)) -> f(X) activate(X) -> X encArg(a) -> a encArg(c(x_1)) -> c(encArg(x_1)) encArg(n__f(x_1)) -> n__f(encArg(x_1)) encArg(g(x_1)) -> g(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_a -> a encode_c(x_1) -> c(encArg(x_1)) encode_n__f(x_1) -> n__f(encArg(x_1)) 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 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: a0() -> 0 c0(0) -> 0 n__f0(0) -> 0 g0(0) -> 0 cons_f0(0) -> 0 cons_activate0(0) -> 0 f0(0) -> 1 activate0(0) -> 2 encArg0(0) -> 3 encode_f0(0) -> 4 encode_a0() -> 5 encode_c0(0) -> 6 encode_n__f0(0) -> 7 encode_g0(0) -> 8 encode_activate0(0) -> 9 n__f1(0) -> 1 f1(0) -> 2 a1() -> 3 encArg1(0) -> 10 c1(10) -> 3 encArg1(0) -> 11 n__f1(11) -> 3 encArg1(0) -> 12 g1(12) -> 3 encArg1(0) -> 13 f1(13) -> 3 encArg1(0) -> 14 activate1(14) -> 3 f1(13) -> 4 a1() -> 5 c1(10) -> 6 n__f1(11) -> 7 g1(12) -> 8 activate1(14) -> 9 n__f2(0) -> 2 n__f2(13) -> 3 n__f2(13) -> 4 a1() -> 10 a1() -> 11 a1() -> 12 a1() -> 13 a1() -> 14 c1(10) -> 10 c1(10) -> 11 c1(10) -> 12 c1(10) -> 13 c1(10) -> 14 n__f1(11) -> 10 n__f1(11) -> 11 n__f1(11) -> 12 n__f1(11) -> 13 n__f1(11) -> 14 g1(12) -> 10 g1(12) -> 11 g1(12) -> 12 g1(12) -> 13 g1(12) -> 14 f1(13) -> 10 f1(13) -> 11 f1(13) -> 12 f1(13) -> 13 f1(13) -> 14 activate1(14) -> 10 activate1(14) -> 11 activate1(14) -> 12 activate1(14) -> 13 activate1(14) -> 14 n__f2(13) -> 9 n__f2(13) -> 10 n__f2(13) -> 11 n__f2(13) -> 12 n__f2(13) -> 13 n__f2(13) -> 14 f2(11) -> 3 f2(11) -> 9 f2(11) -> 10 f2(11) -> 11 f2(11) -> 12 f2(11) -> 13 f2(11) -> 14 a2() -> 18 f2(18) -> 17 g2(17) -> 16 n__f2(16) -> 15 c2(15) -> 3 c2(15) -> 4 c2(15) -> 9 c2(15) -> 10 c2(15) -> 11 c2(15) -> 12 c2(15) -> 13 c2(15) -> 14 f2(13) -> 3 f2(13) -> 9 f2(13) -> 10 f2(13) -> 11 f2(13) -> 12 f2(13) -> 13 f2(13) -> 14 n__f3(11) -> 3 n__f3(11) -> 9 n__f3(11) -> 10 n__f3(11) -> 11 n__f3(11) -> 12 n__f3(11) -> 13 n__f3(11) -> 14 n__f3(18) -> 17 n__f3(13) -> 3 n__f3(13) -> 9 n__f3(13) -> 10 n__f3(13) -> 11 n__f3(13) -> 12 n__f3(13) -> 13 n__f3(13) -> 14 0 -> 2 14 -> 3 14 -> 9 14 -> 10 14 -> 11 14 -> 12 14 -> 13 ---------------------------------------- (8) BOUNDS(1, n^1)