/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 193 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), 277 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), 80 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 53 ms] (20) 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(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) plus(minus(x, s(0)), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0))) 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(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) ---------------------------------------- (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(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) plus(minus(x, s(0)), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(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^1, INF). The TRS R consists of the following rules: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) plus(minus(x, s(0)), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(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^1, INF). The TRS R consists of the following rules: minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) quot(0', s(y)) -> 0' quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) plus(minus(x, s(0')), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0'))) The (relative) TRS S consists of the following rules: encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) quot(0', s(y)) -> 0' quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) plus(minus(x, s(0')), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0'))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) Types: minus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus 0' :: 0':s:cons_minus:cons_quot:cons_plus s :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus quot :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus plus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encArg :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus cons_minus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus cons_quot :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus cons_plus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_minus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_0 :: 0':s:cons_minus:cons_quot:cons_plus encode_s :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_quot :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_plus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus hole_0':s:cons_minus:cons_quot:cons_plus1_3 :: 0':s:cons_minus:cons_quot:cons_plus gen_0':s:cons_minus:cons_quot:cons_plus2_3 :: Nat -> 0':s:cons_minus:cons_quot:cons_plus ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: minus, quot, plus, encArg They will be analysed ascendingly in the following order: minus < quot minus < plus minus < encArg quot < encArg plus < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) quot(0', s(y)) -> 0' quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) plus(minus(x, s(0')), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0'))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) Types: minus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus 0' :: 0':s:cons_minus:cons_quot:cons_plus s :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus quot :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus plus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encArg :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus cons_minus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus cons_quot :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus cons_plus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_minus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_0 :: 0':s:cons_minus:cons_quot:cons_plus encode_s :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_quot :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_plus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus hole_0':s:cons_minus:cons_quot:cons_plus1_3 :: 0':s:cons_minus:cons_quot:cons_plus gen_0':s:cons_minus:cons_quot:cons_plus2_3 :: Nat -> 0':s:cons_minus:cons_quot:cons_plus Generator Equations: gen_0':s:cons_minus:cons_quot:cons_plus2_3(0) <=> 0' gen_0':s:cons_minus:cons_quot:cons_plus2_3(+(x, 1)) <=> s(gen_0':s:cons_minus:cons_quot:cons_plus2_3(x)) The following defined symbols remain to be analysed: minus, quot, plus, encArg They will be analysed ascendingly in the following order: minus < quot minus < plus minus < encArg quot < encArg plus < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: minus(gen_0':s:cons_minus:cons_quot:cons_plus2_3(n4_3), gen_0':s:cons_minus:cons_quot:cons_plus2_3(n4_3)) -> gen_0':s:cons_minus:cons_quot:cons_plus2_3(0), rt in Omega(1 + n4_3) Induction Base: minus(gen_0':s:cons_minus:cons_quot:cons_plus2_3(0), gen_0':s:cons_minus:cons_quot:cons_plus2_3(0)) ->_R^Omega(1) gen_0':s:cons_minus:cons_quot:cons_plus2_3(0) Induction Step: minus(gen_0':s:cons_minus:cons_quot:cons_plus2_3(+(n4_3, 1)), gen_0':s:cons_minus:cons_quot:cons_plus2_3(+(n4_3, 1))) ->_R^Omega(1) minus(gen_0':s:cons_minus:cons_quot:cons_plus2_3(n4_3), gen_0':s:cons_minus:cons_quot:cons_plus2_3(n4_3)) ->_IH gen_0':s:cons_minus:cons_quot:cons_plus2_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(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) quot(0', s(y)) -> 0' quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) plus(minus(x, s(0')), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0'))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) Types: minus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus 0' :: 0':s:cons_minus:cons_quot:cons_plus s :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus quot :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus plus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encArg :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus cons_minus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus cons_quot :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus cons_plus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_minus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_0 :: 0':s:cons_minus:cons_quot:cons_plus encode_s :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_quot :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_plus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus hole_0':s:cons_minus:cons_quot:cons_plus1_3 :: 0':s:cons_minus:cons_quot:cons_plus gen_0':s:cons_minus:cons_quot:cons_plus2_3 :: Nat -> 0':s:cons_minus:cons_quot:cons_plus Generator Equations: gen_0':s:cons_minus:cons_quot:cons_plus2_3(0) <=> 0' gen_0':s:cons_minus:cons_quot:cons_plus2_3(+(x, 1)) <=> s(gen_0':s:cons_minus:cons_quot:cons_plus2_3(x)) The following defined symbols remain to be analysed: minus, quot, plus, encArg They will be analysed ascendingly in the following order: minus < quot minus < plus minus < encArg quot < encArg plus < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) quot(0', s(y)) -> 0' quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) plus(minus(x, s(0')), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0'))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) Types: minus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus 0' :: 0':s:cons_minus:cons_quot:cons_plus s :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus quot :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus plus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encArg :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus cons_minus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus cons_quot :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus cons_plus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_minus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_0 :: 0':s:cons_minus:cons_quot:cons_plus encode_s :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_quot :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_plus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus hole_0':s:cons_minus:cons_quot:cons_plus1_3 :: 0':s:cons_minus:cons_quot:cons_plus gen_0':s:cons_minus:cons_quot:cons_plus2_3 :: Nat -> 0':s:cons_minus:cons_quot:cons_plus Lemmas: minus(gen_0':s:cons_minus:cons_quot:cons_plus2_3(n4_3), gen_0':s:cons_minus:cons_quot:cons_plus2_3(n4_3)) -> gen_0':s:cons_minus:cons_quot:cons_plus2_3(0), rt in Omega(1 + n4_3) Generator Equations: gen_0':s:cons_minus:cons_quot:cons_plus2_3(0) <=> 0' gen_0':s:cons_minus:cons_quot:cons_plus2_3(+(x, 1)) <=> s(gen_0':s:cons_minus:cons_quot:cons_plus2_3(x)) The following defined symbols remain to be analysed: quot, plus, encArg They will be analysed ascendingly in the following order: quot < encArg plus < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: plus(gen_0':s:cons_minus:cons_quot:cons_plus2_3(n512_3), gen_0':s:cons_minus:cons_quot:cons_plus2_3(b)) -> gen_0':s:cons_minus:cons_quot:cons_plus2_3(+(n512_3, b)), rt in Omega(1 + n512_3) Induction Base: plus(gen_0':s:cons_minus:cons_quot:cons_plus2_3(0), gen_0':s:cons_minus:cons_quot:cons_plus2_3(b)) ->_R^Omega(1) gen_0':s:cons_minus:cons_quot:cons_plus2_3(b) Induction Step: plus(gen_0':s:cons_minus:cons_quot:cons_plus2_3(+(n512_3, 1)), gen_0':s:cons_minus:cons_quot:cons_plus2_3(b)) ->_R^Omega(1) s(plus(gen_0':s:cons_minus:cons_quot:cons_plus2_3(n512_3), gen_0':s:cons_minus:cons_quot:cons_plus2_3(b))) ->_IH s(gen_0':s:cons_minus:cons_quot:cons_plus2_3(+(b, c513_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: minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) quot(0', s(y)) -> 0' quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) plus(minus(x, s(0')), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0'))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) Types: minus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus 0' :: 0':s:cons_minus:cons_quot:cons_plus s :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus quot :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus plus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encArg :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus cons_minus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus cons_quot :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus cons_plus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_minus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_0 :: 0':s:cons_minus:cons_quot:cons_plus encode_s :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_quot :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus encode_plus :: 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus -> 0':s:cons_minus:cons_quot:cons_plus hole_0':s:cons_minus:cons_quot:cons_plus1_3 :: 0':s:cons_minus:cons_quot:cons_plus gen_0':s:cons_minus:cons_quot:cons_plus2_3 :: Nat -> 0':s:cons_minus:cons_quot:cons_plus Lemmas: minus(gen_0':s:cons_minus:cons_quot:cons_plus2_3(n4_3), gen_0':s:cons_minus:cons_quot:cons_plus2_3(n4_3)) -> gen_0':s:cons_minus:cons_quot:cons_plus2_3(0), rt in Omega(1 + n4_3) plus(gen_0':s:cons_minus:cons_quot:cons_plus2_3(n512_3), gen_0':s:cons_minus:cons_quot:cons_plus2_3(b)) -> gen_0':s:cons_minus:cons_quot:cons_plus2_3(+(n512_3, b)), rt in Omega(1 + n512_3) Generator Equations: gen_0':s:cons_minus:cons_quot:cons_plus2_3(0) <=> 0' gen_0':s:cons_minus:cons_quot:cons_plus2_3(+(x, 1)) <=> s(gen_0':s:cons_minus:cons_quot:cons_plus2_3(x)) The following defined symbols remain to be analysed: encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':s:cons_minus:cons_quot:cons_plus2_3(n1446_3)) -> gen_0':s:cons_minus:cons_quot:cons_plus2_3(n1446_3), rt in Omega(0) Induction Base: encArg(gen_0':s:cons_minus:cons_quot:cons_plus2_3(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':s:cons_minus:cons_quot:cons_plus2_3(+(n1446_3, 1))) ->_R^Omega(0) s(encArg(gen_0':s:cons_minus:cons_quot:cons_plus2_3(n1446_3))) ->_IH s(gen_0':s:cons_minus:cons_quot:cons_plus2_3(c1447_3)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (20) BOUNDS(1, INF)