/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 215 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 456 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 103 ms] (18) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: f(g(x)) -> g(f(f(x))) f(h(x)) -> h(g(x)) f'(s(x), y, y) -> f'(y, x, s(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(g(x_1)) -> g(encArg(x_1)) encArg(h(x_1)) -> h(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_f'(x_1, x_2, x_3)) -> f'(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f(x_1) -> f(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_f'(x_1, x_2, x_3) -> f'(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: f(g(x)) -> g(f(f(x))) f(h(x)) -> h(g(x)) f'(s(x), y, y) -> f'(y, x, s(x)) The (relative) TRS S consists of the following rules: encArg(g(x_1)) -> g(encArg(x_1)) encArg(h(x_1)) -> h(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_f'(x_1, x_2, x_3)) -> f'(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f(x_1) -> f(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_f'(x_1, x_2, x_3) -> f'(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(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(n^1, INF). The TRS R consists of the following rules: f(g(x)) -> g(f(f(x))) f(h(x)) -> h(g(x)) f'(s(x), y, y) -> f'(y, x, s(x)) The (relative) TRS S consists of the following rules: encArg(g(x_1)) -> g(encArg(x_1)) encArg(h(x_1)) -> h(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_f'(x_1, x_2, x_3)) -> f'(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f(x_1) -> f(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_f'(x_1, x_2, x_3) -> f'(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: f(g(x)) -> g(f(f(x))) f(h(x)) -> h(g(x)) f'(s(x), y, y) -> f'(y, x, s(x)) The (relative) TRS S consists of the following rules: encArg(g(x_1)) -> g(encArg(x_1)) encArg(h(x_1)) -> h(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_f'(x_1, x_2, x_3)) -> f'(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f(x_1) -> f(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_f'(x_1, x_2, x_3) -> f'(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: f(g(x)) -> g(f(f(x))) f(h(x)) -> h(g(x)) f'(s(x), y, y) -> f'(y, x, s(x)) encArg(g(x_1)) -> g(encArg(x_1)) encArg(h(x_1)) -> h(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_f'(x_1, x_2, x_3)) -> f'(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f(x_1) -> f(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_f'(x_1, x_2, x_3) -> f'(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) Types: f :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' g :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' h :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' f' :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' s :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encArg :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' cons_f :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' cons_f' :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_f :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_g :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_h :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_f' :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_s :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' hole_g:h:s:cons_f:cons_f'1_0 :: g:h:s:cons_f:cons_f' gen_g:h:s:cons_f:cons_f'2_0 :: Nat -> g:h:s:cons_f:cons_f' ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: f, f', encArg They will be analysed ascendingly in the following order: f < encArg f' < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: f(g(x)) -> g(f(f(x))) f(h(x)) -> h(g(x)) f'(s(x), y, y) -> f'(y, x, s(x)) encArg(g(x_1)) -> g(encArg(x_1)) encArg(h(x_1)) -> h(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_f'(x_1, x_2, x_3)) -> f'(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f(x_1) -> f(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_f'(x_1, x_2, x_3) -> f'(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) Types: f :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' g :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' h :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' f' :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' s :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encArg :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' cons_f :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' cons_f' :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_f :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_g :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_h :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_f' :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_s :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' hole_g:h:s:cons_f:cons_f'1_0 :: g:h:s:cons_f:cons_f' gen_g:h:s:cons_f:cons_f'2_0 :: Nat -> g:h:s:cons_f:cons_f' Generator Equations: gen_g:h:s:cons_f:cons_f'2_0(0) <=> hole_g:h:s:cons_f:cons_f'1_0 gen_g:h:s:cons_f:cons_f'2_0(+(x, 1)) <=> g(gen_g:h:s:cons_f:cons_f'2_0(x)) The following defined symbols remain to be analysed: f, f', encArg They will be analysed ascendingly in the following order: f < encArg f' < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: f(gen_g:h:s:cons_f:cons_f'2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) Induction Base: f(gen_g:h:s:cons_f:cons_f'2_0(+(1, 0))) Induction Step: f(gen_g:h:s:cons_f:cons_f'2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1) g(f(f(gen_g:h:s:cons_f:cons_f'2_0(+(1, n4_0))))) ->_IH g(f(*3_0)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: f(g(x)) -> g(f(f(x))) f(h(x)) -> h(g(x)) f'(s(x), y, y) -> f'(y, x, s(x)) encArg(g(x_1)) -> g(encArg(x_1)) encArg(h(x_1)) -> h(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_f'(x_1, x_2, x_3)) -> f'(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f(x_1) -> f(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_f'(x_1, x_2, x_3) -> f'(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) Types: f :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' g :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' h :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' f' :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' s :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encArg :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' cons_f :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' cons_f' :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_f :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_g :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_h :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_f' :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_s :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' hole_g:h:s:cons_f:cons_f'1_0 :: g:h:s:cons_f:cons_f' gen_g:h:s:cons_f:cons_f'2_0 :: Nat -> g:h:s:cons_f:cons_f' Generator Equations: gen_g:h:s:cons_f:cons_f'2_0(0) <=> hole_g:h:s:cons_f:cons_f'1_0 gen_g:h:s:cons_f:cons_f'2_0(+(x, 1)) <=> g(gen_g:h:s:cons_f:cons_f'2_0(x)) The following defined symbols remain to be analysed: f, f', encArg They will be analysed ascendingly in the following order: f < encArg f' < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: f(g(x)) -> g(f(f(x))) f(h(x)) -> h(g(x)) f'(s(x), y, y) -> f'(y, x, s(x)) encArg(g(x_1)) -> g(encArg(x_1)) encArg(h(x_1)) -> h(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_f'(x_1, x_2, x_3)) -> f'(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f(x_1) -> f(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_f'(x_1, x_2, x_3) -> f'(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) Types: f :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' g :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' h :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' f' :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' s :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encArg :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' cons_f :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' cons_f' :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_f :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_g :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_h :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_f' :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' encode_s :: g:h:s:cons_f:cons_f' -> g:h:s:cons_f:cons_f' hole_g:h:s:cons_f:cons_f'1_0 :: g:h:s:cons_f:cons_f' gen_g:h:s:cons_f:cons_f'2_0 :: Nat -> g:h:s:cons_f:cons_f' Lemmas: f(gen_g:h:s:cons_f:cons_f'2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) Generator Equations: gen_g:h:s:cons_f:cons_f'2_0(0) <=> hole_g:h:s:cons_f:cons_f'1_0 gen_g:h:s:cons_f:cons_f'2_0(+(x, 1)) <=> g(gen_g:h:s:cons_f:cons_f'2_0(x)) The following defined symbols remain to be analysed: f', encArg They will be analysed ascendingly in the following order: f' < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_g:h:s:cons_f:cons_f'2_0(+(1, n402_0))) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_g:h:s:cons_f:cons_f'2_0(+(1, 0))) Induction Step: encArg(gen_g:h:s:cons_f:cons_f'2_0(+(1, +(n402_0, 1)))) ->_R^Omega(0) g(encArg(gen_g:h:s:cons_f:cons_f'2_0(+(1, n402_0)))) ->_IH g(*3_0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (18) BOUNDS(1, INF)