/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 231 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), 445 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), 581 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: minus(minus(x)) -> x minus(+(x, y)) -> *(minus(minus(minus(x))), minus(minus(minus(y)))) minus(*(x, y)) -> +(minus(minus(minus(x))), minus(minus(minus(y)))) f(minus(x)) -> minus(minus(minus(f(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(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(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: minus(minus(x)) -> x minus(+(x, y)) -> *(minus(minus(minus(x))), minus(minus(minus(y)))) minus(*(x, y)) -> +(minus(minus(minus(x))), minus(minus(minus(y)))) f(minus(x)) -> minus(minus(minus(f(x)))) The (relative) TRS S consists of the following rules: encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(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: minus(minus(x)) -> x minus(+(x, y)) -> *(minus(minus(minus(x))), minus(minus(minus(y)))) minus(*(x, y)) -> +(minus(minus(minus(x))), minus(minus(minus(y)))) f(minus(x)) -> minus(minus(minus(f(x)))) The (relative) TRS S consists of the following rules: encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(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: minus(minus(x)) -> x minus(+'(x, y)) -> *'(minus(minus(minus(x))), minus(minus(minus(y)))) minus(*'(x, y)) -> +'(minus(minus(minus(x))), minus(minus(minus(y)))) f(minus(x)) -> minus(minus(minus(f(x)))) The (relative) TRS S consists of the following rules: encArg(+'(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: minus(minus(x)) -> x minus(+'(x, y)) -> *'(minus(minus(minus(x))), minus(minus(minus(y)))) minus(*'(x, y)) -> +'(minus(minus(minus(x))), minus(minus(minus(y)))) f(minus(x)) -> minus(minus(minus(f(x)))) encArg(+'(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) Types: minus :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f +' :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f *' :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f f :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encArg :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f cons_minus :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f cons_f :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encode_minus :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encode_+ :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encode_* :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encode_f :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f hole_+':*':cons_minus:cons_f1_0 :: +':*':cons_minus:cons_f gen_+':*':cons_minus:cons_f2_0 :: Nat -> +':*':cons_minus:cons_f ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: minus, f, encArg They will be analysed ascendingly in the following order: minus < f minus < encArg f < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: minus(minus(x)) -> x minus(+'(x, y)) -> *'(minus(minus(minus(x))), minus(minus(minus(y)))) minus(*'(x, y)) -> +'(minus(minus(minus(x))), minus(minus(minus(y)))) f(minus(x)) -> minus(minus(minus(f(x)))) encArg(+'(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) Types: minus :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f +' :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f *' :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f f :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encArg :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f cons_minus :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f cons_f :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encode_minus :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encode_+ :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encode_* :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encode_f :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f hole_+':*':cons_minus:cons_f1_0 :: +':*':cons_minus:cons_f gen_+':*':cons_minus:cons_f2_0 :: Nat -> +':*':cons_minus:cons_f Generator Equations: gen_+':*':cons_minus:cons_f2_0(0) <=> hole_+':*':cons_minus:cons_f1_0 gen_+':*':cons_minus:cons_f2_0(+(x, 1)) <=> +'(hole_+':*':cons_minus:cons_f1_0, gen_+':*':cons_minus:cons_f2_0(x)) The following defined symbols remain to be analysed: minus, f, encArg They will be analysed ascendingly in the following order: minus < f minus < encArg f < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: minus(gen_+':*':cons_minus:cons_f2_0(n4_0)) -> *3_0, rt in Omega(n4_0) Induction Base: minus(gen_+':*':cons_minus:cons_f2_0(0)) Induction Step: minus(gen_+':*':cons_minus:cons_f2_0(+(n4_0, 1))) ->_R^Omega(1) *'(minus(minus(minus(hole_+':*':cons_minus:cons_f1_0))), minus(minus(minus(gen_+':*':cons_minus:cons_f2_0(n4_0))))) ->_R^Omega(1) *'(minus(hole_+':*':cons_minus:cons_f1_0), minus(minus(minus(gen_+':*':cons_minus:cons_f2_0(n4_0))))) ->_IH *'(minus(hole_+':*':cons_minus:cons_f1_0), minus(minus(*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: minus(minus(x)) -> x minus(+'(x, y)) -> *'(minus(minus(minus(x))), minus(minus(minus(y)))) minus(*'(x, y)) -> +'(minus(minus(minus(x))), minus(minus(minus(y)))) f(minus(x)) -> minus(minus(minus(f(x)))) encArg(+'(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) Types: minus :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f +' :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f *' :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f f :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encArg :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f cons_minus :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f cons_f :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encode_minus :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encode_+ :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encode_* :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encode_f :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f hole_+':*':cons_minus:cons_f1_0 :: +':*':cons_minus:cons_f gen_+':*':cons_minus:cons_f2_0 :: Nat -> +':*':cons_minus:cons_f Generator Equations: gen_+':*':cons_minus:cons_f2_0(0) <=> hole_+':*':cons_minus:cons_f1_0 gen_+':*':cons_minus:cons_f2_0(+(x, 1)) <=> +'(hole_+':*':cons_minus:cons_f1_0, gen_+':*':cons_minus:cons_f2_0(x)) The following defined symbols remain to be analysed: minus, f, encArg They will be analysed ascendingly in the following order: minus < f minus < encArg f < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: minus(minus(x)) -> x minus(+'(x, y)) -> *'(minus(minus(minus(x))), minus(minus(minus(y)))) minus(*'(x, y)) -> +'(minus(minus(minus(x))), minus(minus(minus(y)))) f(minus(x)) -> minus(minus(minus(f(x)))) encArg(+'(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) Types: minus :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f +' :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f *' :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f f :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encArg :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f cons_minus :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f cons_f :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encode_minus :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encode_+ :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encode_* :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f encode_f :: +':*':cons_minus:cons_f -> +':*':cons_minus:cons_f hole_+':*':cons_minus:cons_f1_0 :: +':*':cons_minus:cons_f gen_+':*':cons_minus:cons_f2_0 :: Nat -> +':*':cons_minus:cons_f Lemmas: minus(gen_+':*':cons_minus:cons_f2_0(n4_0)) -> *3_0, rt in Omega(n4_0) Generator Equations: gen_+':*':cons_minus:cons_f2_0(0) <=> hole_+':*':cons_minus:cons_f1_0 gen_+':*':cons_minus:cons_f2_0(+(x, 1)) <=> +'(hole_+':*':cons_minus:cons_f1_0, gen_+':*':cons_minus:cons_f2_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_+':*':cons_minus:cons_f2_0(+(1, n9943_0))) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_+':*':cons_minus:cons_f2_0(+(1, 0))) Induction Step: encArg(gen_+':*':cons_minus:cons_f2_0(+(1, +(n9943_0, 1)))) ->_R^Omega(0) +'(encArg(hole_+':*':cons_minus:cons_f1_0), encArg(gen_+':*':cons_minus:cons_f2_0(+(1, n9943_0)))) ->_IH +'(encArg(hole_+':*':cons_minus:cons_f1_0), *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)