/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^3), ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^3, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 172 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), 270 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), 1914 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 45 ms] (20) BEST (21) proven lower bound (22) LowerBoundPropagationProof [FINISHED, 0 ms] (23) BOUNDS(n^3, INF) (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 37 ms] (26) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^3, INF). The TRS R consists of the following rules: +(0, y) -> y +(s(x), y) -> s(+(x, y)) +(p(x), y) -> p(+(x, y)) minus(0) -> 0 minus(s(x)) -> p(minus(x)) minus(p(x)) -> s(minus(x)) *(0, y) -> 0 *(s(x), y) -> +(*(x, y), y) *(p(x), y) -> +(*(x, y), minus(y)) 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(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, INF). The TRS R consists of the following rules: +(0, y) -> y +(s(x), y) -> s(+(x, y)) +(p(x), y) -> p(+(x, y)) minus(0) -> 0 minus(s(x)) -> p(minus(x)) minus(p(x)) -> s(minus(x)) *(0, y) -> 0 *(s(x), y) -> +(*(x, y), y) *(p(x), y) -> +(*(x, y), minus(y)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) 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^3, INF). The TRS R consists of the following rules: +(0, y) -> y +(s(x), y) -> s(+(x, y)) +(p(x), y) -> p(+(x, y)) minus(0) -> 0 minus(s(x)) -> p(minus(x)) minus(p(x)) -> s(minus(x)) *(0, y) -> 0 *(s(x), y) -> +(*(x, y), y) *(p(x), y) -> +(*(x, y), minus(y)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) 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^3, INF). The TRS R consists of the following rules: +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) +'(p(x), y) -> p(+'(x, y)) minus(0') -> 0' minus(s(x)) -> p(minus(x)) minus(p(x)) -> s(minus(x)) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) *'(p(x), y) -> +'(*'(x, y), minus(y)) The (relative) TRS S consists of the following rules: encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) +'(p(x), y) -> p(+'(x, y)) minus(0') -> 0' minus(s(x)) -> p(minus(x)) minus(p(x)) -> s(minus(x)) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) *'(p(x), y) -> +'(*'(x, y), minus(y)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) Types: +' :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* 0' :: 0':s:p:cons_+:cons_minus:cons_* s :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* p :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* *' :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encArg :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_+ :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_* :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_+ :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_0 :: 0':s:p:cons_+:cons_minus:cons_* encode_s :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_p :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_* :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* hole_0':s:p:cons_+:cons_minus:cons_*1_3 :: 0':s:p:cons_+:cons_minus:cons_* gen_0':s:p:cons_+:cons_minus:cons_*2_3 :: Nat -> 0':s:p:cons_+:cons_minus:cons_* ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: +', minus, *', encArg They will be analysed ascendingly in the following order: +' < *' +' < encArg minus < *' minus < encArg *' < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) +'(p(x), y) -> p(+'(x, y)) minus(0') -> 0' minus(s(x)) -> p(minus(x)) minus(p(x)) -> s(minus(x)) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) *'(p(x), y) -> +'(*'(x, y), minus(y)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) Types: +' :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* 0' :: 0':s:p:cons_+:cons_minus:cons_* s :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* p :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* *' :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encArg :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_+ :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_* :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_+ :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_0 :: 0':s:p:cons_+:cons_minus:cons_* encode_s :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_p :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_* :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* hole_0':s:p:cons_+:cons_minus:cons_*1_3 :: 0':s:p:cons_+:cons_minus:cons_* gen_0':s:p:cons_+:cons_minus:cons_*2_3 :: Nat -> 0':s:p:cons_+:cons_minus:cons_* Generator Equations: gen_0':s:p:cons_+:cons_minus:cons_*2_3(0) <=> 0' gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(x, 1)) <=> s(gen_0':s:p:cons_+:cons_minus:cons_*2_3(x)) The following defined symbols remain to be analysed: +', minus, *', encArg They will be analysed ascendingly in the following order: +' < *' +' < encArg minus < *' minus < encArg *' < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: +'(gen_0':s:p:cons_+:cons_minus:cons_*2_3(n4_3), gen_0':s:p:cons_+:cons_minus:cons_*2_3(b)) -> gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(n4_3, b)), rt in Omega(1 + n4_3) Induction Base: +'(gen_0':s:p:cons_+:cons_minus:cons_*2_3(0), gen_0':s:p:cons_+:cons_minus:cons_*2_3(b)) ->_R^Omega(1) gen_0':s:p:cons_+:cons_minus:cons_*2_3(b) Induction Step: +'(gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(n4_3, 1)), gen_0':s:p:cons_+:cons_minus:cons_*2_3(b)) ->_R^Omega(1) s(+'(gen_0':s:p:cons_+:cons_minus:cons_*2_3(n4_3), gen_0':s:p:cons_+:cons_minus:cons_*2_3(b))) ->_IH s(gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(b, c5_3))) 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: +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) +'(p(x), y) -> p(+'(x, y)) minus(0') -> 0' minus(s(x)) -> p(minus(x)) minus(p(x)) -> s(minus(x)) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) *'(p(x), y) -> +'(*'(x, y), minus(y)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) Types: +' :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* 0' :: 0':s:p:cons_+:cons_minus:cons_* s :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* p :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* *' :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encArg :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_+ :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_* :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_+ :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_0 :: 0':s:p:cons_+:cons_minus:cons_* encode_s :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_p :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_* :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* hole_0':s:p:cons_+:cons_minus:cons_*1_3 :: 0':s:p:cons_+:cons_minus:cons_* gen_0':s:p:cons_+:cons_minus:cons_*2_3 :: Nat -> 0':s:p:cons_+:cons_minus:cons_* Generator Equations: gen_0':s:p:cons_+:cons_minus:cons_*2_3(0) <=> 0' gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(x, 1)) <=> s(gen_0':s:p:cons_+:cons_minus:cons_*2_3(x)) The following defined symbols remain to be analysed: +', minus, *', encArg They will be analysed ascendingly in the following order: +' < *' +' < encArg minus < *' minus < encArg *' < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) +'(p(x), y) -> p(+'(x, y)) minus(0') -> 0' minus(s(x)) -> p(minus(x)) minus(p(x)) -> s(minus(x)) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) *'(p(x), y) -> +'(*'(x, y), minus(y)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) Types: +' :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* 0' :: 0':s:p:cons_+:cons_minus:cons_* s :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* p :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* *' :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encArg :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_+ :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_* :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_+ :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_0 :: 0':s:p:cons_+:cons_minus:cons_* encode_s :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_p :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_* :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* hole_0':s:p:cons_+:cons_minus:cons_*1_3 :: 0':s:p:cons_+:cons_minus:cons_* gen_0':s:p:cons_+:cons_minus:cons_*2_3 :: Nat -> 0':s:p:cons_+:cons_minus:cons_* Lemmas: +'(gen_0':s:p:cons_+:cons_minus:cons_*2_3(n4_3), gen_0':s:p:cons_+:cons_minus:cons_*2_3(b)) -> gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(n4_3, b)), rt in Omega(1 + n4_3) Generator Equations: gen_0':s:p:cons_+:cons_minus:cons_*2_3(0) <=> 0' gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(x, 1)) <=> s(gen_0':s:p:cons_+:cons_minus:cons_*2_3(x)) The following defined symbols remain to be analysed: minus, *', encArg They will be analysed ascendingly in the following order: minus < *' minus < encArg *' < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: minus(gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(1, n979_3))) -> *3_3, rt in Omega(n979_3) Induction Base: minus(gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(1, 0))) Induction Step: minus(gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(1, +(n979_3, 1)))) ->_R^Omega(1) p(minus(gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(1, n979_3)))) ->_IH p(*3_3) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: Innermost TRS: Rules: +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) +'(p(x), y) -> p(+'(x, y)) minus(0') -> 0' minus(s(x)) -> p(minus(x)) minus(p(x)) -> s(minus(x)) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) *'(p(x), y) -> +'(*'(x, y), minus(y)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) Types: +' :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* 0' :: 0':s:p:cons_+:cons_minus:cons_* s :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* p :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* *' :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encArg :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_+ :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_* :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_+ :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_0 :: 0':s:p:cons_+:cons_minus:cons_* encode_s :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_p :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_* :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* hole_0':s:p:cons_+:cons_minus:cons_*1_3 :: 0':s:p:cons_+:cons_minus:cons_* gen_0':s:p:cons_+:cons_minus:cons_*2_3 :: Nat -> 0':s:p:cons_+:cons_minus:cons_* Lemmas: +'(gen_0':s:p:cons_+:cons_minus:cons_*2_3(n4_3), gen_0':s:p:cons_+:cons_minus:cons_*2_3(b)) -> gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(n4_3, b)), rt in Omega(1 + n4_3) minus(gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(1, n979_3))) -> *3_3, rt in Omega(n979_3) Generator Equations: gen_0':s:p:cons_+:cons_minus:cons_*2_3(0) <=> 0' gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(x, 1)) <=> s(gen_0':s:p:cons_+:cons_minus:cons_*2_3(x)) The following defined symbols remain to be analysed: *', encArg They will be analysed ascendingly in the following order: *' < encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: *'(gen_0':s:p:cons_+:cons_minus:cons_*2_3(n2211_3), gen_0':s:p:cons_+:cons_minus:cons_*2_3(b)) -> gen_0':s:p:cons_+:cons_minus:cons_*2_3(*(n2211_3, b)), rt in Omega(1 + b*n2211_3^2 + n2211_3) Induction Base: *'(gen_0':s:p:cons_+:cons_minus:cons_*2_3(0), gen_0':s:p:cons_+:cons_minus:cons_*2_3(b)) ->_R^Omega(1) 0' Induction Step: *'(gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(n2211_3, 1)), gen_0':s:p:cons_+:cons_minus:cons_*2_3(b)) ->_R^Omega(1) +'(*'(gen_0':s:p:cons_+:cons_minus:cons_*2_3(n2211_3), gen_0':s:p:cons_+:cons_minus:cons_*2_3(b)), gen_0':s:p:cons_+:cons_minus:cons_*2_3(b)) ->_IH +'(gen_0':s:p:cons_+:cons_minus:cons_*2_3(*(c2212_3, b)), gen_0':s:p:cons_+:cons_minus:cons_*2_3(b)) ->_L^Omega(1 + b*n2211_3) gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(*(n2211_3, b), b)) We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3). ---------------------------------------- (20) Complex Obligation (BEST) ---------------------------------------- (21) Obligation: Proved the lower bound n^3 for the following obligation: Innermost TRS: Rules: +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) +'(p(x), y) -> p(+'(x, y)) minus(0') -> 0' minus(s(x)) -> p(minus(x)) minus(p(x)) -> s(minus(x)) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) *'(p(x), y) -> +'(*'(x, y), minus(y)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) Types: +' :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* 0' :: 0':s:p:cons_+:cons_minus:cons_* s :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* p :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* *' :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encArg :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_+ :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_* :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_+ :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_0 :: 0':s:p:cons_+:cons_minus:cons_* encode_s :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_p :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_* :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* hole_0':s:p:cons_+:cons_minus:cons_*1_3 :: 0':s:p:cons_+:cons_minus:cons_* gen_0':s:p:cons_+:cons_minus:cons_*2_3 :: Nat -> 0':s:p:cons_+:cons_minus:cons_* Lemmas: +'(gen_0':s:p:cons_+:cons_minus:cons_*2_3(n4_3), gen_0':s:p:cons_+:cons_minus:cons_*2_3(b)) -> gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(n4_3, b)), rt in Omega(1 + n4_3) minus(gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(1, n979_3))) -> *3_3, rt in Omega(n979_3) Generator Equations: gen_0':s:p:cons_+:cons_minus:cons_*2_3(0) <=> 0' gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(x, 1)) <=> s(gen_0':s:p:cons_+:cons_minus:cons_*2_3(x)) The following defined symbols remain to be analysed: *', encArg They will be analysed ascendingly in the following order: *' < encArg ---------------------------------------- (22) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (23) BOUNDS(n^3, INF) ---------------------------------------- (24) Obligation: Innermost TRS: Rules: +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) +'(p(x), y) -> p(+'(x, y)) minus(0') -> 0' minus(s(x)) -> p(minus(x)) minus(p(x)) -> s(minus(x)) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) *'(p(x), y) -> +'(*'(x, y), minus(y)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1)) -> minus(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_minus(x_1) -> minus(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) Types: +' :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* 0' :: 0':s:p:cons_+:cons_minus:cons_* s :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* p :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* *' :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encArg :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_+ :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* cons_* :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_+ :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_0 :: 0':s:p:cons_+:cons_minus:cons_* encode_s :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_p :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_minus :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* encode_* :: 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* -> 0':s:p:cons_+:cons_minus:cons_* hole_0':s:p:cons_+:cons_minus:cons_*1_3 :: 0':s:p:cons_+:cons_minus:cons_* gen_0':s:p:cons_+:cons_minus:cons_*2_3 :: Nat -> 0':s:p:cons_+:cons_minus:cons_* Lemmas: +'(gen_0':s:p:cons_+:cons_minus:cons_*2_3(n4_3), gen_0':s:p:cons_+:cons_minus:cons_*2_3(b)) -> gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(n4_3, b)), rt in Omega(1 + n4_3) minus(gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(1, n979_3))) -> *3_3, rt in Omega(n979_3) *'(gen_0':s:p:cons_+:cons_minus:cons_*2_3(n2211_3), gen_0':s:p:cons_+:cons_minus:cons_*2_3(b)) -> gen_0':s:p:cons_+:cons_minus:cons_*2_3(*(n2211_3, b)), rt in Omega(1 + b*n2211_3^2 + n2211_3) Generator Equations: gen_0':s:p:cons_+:cons_minus:cons_*2_3(0) <=> 0' gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(x, 1)) <=> s(gen_0':s:p:cons_+:cons_minus:cons_*2_3(x)) The following defined symbols remain to be analysed: encArg ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':s:p:cons_+:cons_minus:cons_*2_3(n3709_3)) -> gen_0':s:p:cons_+:cons_minus:cons_*2_3(n3709_3), rt in Omega(0) Induction Base: encArg(gen_0':s:p:cons_+:cons_minus:cons_*2_3(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':s:p:cons_+:cons_minus:cons_*2_3(+(n3709_3, 1))) ->_R^Omega(0) s(encArg(gen_0':s:p:cons_+:cons_minus:cons_*2_3(n3709_3))) ->_IH s(gen_0':s:p:cons_+:cons_minus:cons_*2_3(c3710_3)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (26) BOUNDS(1, INF)