/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^2), ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^2, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 196 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), 269 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), 52 ms] (18) BEST (19) proven lower bound (20) LowerBoundPropagationProof [FINISHED, 0 ms] (21) BOUNDS(n^2, INF) (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 670 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 0 ms] (26) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^2, INF). The TRS R consists of the following rules: p(s(x)) -> x fact(0) -> s(0) fact(s(x)) -> *(s(x), fact(p(s(x)))) *(0, y) -> 0 *(s(x), y) -> +(*(x, y), y) +(x, 0) -> x +(x, s(y)) -> s(+(x, 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(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fact(x_1)) -> fact(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fact(x_1) -> fact(encArg(x_1)) encode_0 -> 0 encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) 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^2, INF). The TRS R consists of the following rules: p(s(x)) -> x fact(0) -> s(0) fact(s(x)) -> *(s(x), fact(p(s(x)))) *(0, y) -> 0 *(s(x), y) -> +(*(x, y), y) +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fact(x_1)) -> fact(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fact(x_1) -> fact(encArg(x_1)) encode_0 -> 0 encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) 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^2, INF). The TRS R consists of the following rules: p(s(x)) -> x fact(0) -> s(0) fact(s(x)) -> *(s(x), fact(p(s(x)))) *(0, y) -> 0 *(s(x), y) -> +(*(x, y), y) +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fact(x_1)) -> fact(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fact(x_1) -> fact(encArg(x_1)) encode_0 -> 0 encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) 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^2, INF). The TRS R consists of the following rules: p(s(x)) -> x fact(0') -> s(0') fact(s(x)) -> *'(s(x), fact(p(s(x)))) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) +'(x, 0') -> x +'(x, s(y)) -> s(+'(x, y)) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fact(x_1)) -> fact(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fact(x_1) -> fact(encArg(x_1)) encode_0 -> 0' encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) 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: p(s(x)) -> x fact(0') -> s(0') fact(s(x)) -> *'(s(x), fact(p(s(x)))) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) +'(x, 0') -> x +'(x, s(y)) -> s(+'(x, y)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fact(x_1)) -> fact(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fact(x_1) -> fact(encArg(x_1)) encode_0 -> 0' encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) Types: p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ s :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ 0' :: s:0':cons_p:cons_fact:cons_*:cons_+ *' :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ +' :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encArg :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_* :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_+ :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_s :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_0 :: s:0':cons_p:cons_fact:cons_*:cons_+ encode_* :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_+ :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ hole_s:0':cons_p:cons_fact:cons_*:cons_+1_3 :: s:0':cons_p:cons_fact:cons_*:cons_+ gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3 :: Nat -> s:0':cons_p:cons_fact:cons_*:cons_+ ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: fact, *', +', encArg They will be analysed ascendingly in the following order: *' < fact fact < encArg +' < *' *' < encArg +' < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: p(s(x)) -> x fact(0') -> s(0') fact(s(x)) -> *'(s(x), fact(p(s(x)))) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) +'(x, 0') -> x +'(x, s(y)) -> s(+'(x, y)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fact(x_1)) -> fact(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fact(x_1) -> fact(encArg(x_1)) encode_0 -> 0' encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) Types: p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ s :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ 0' :: s:0':cons_p:cons_fact:cons_*:cons_+ *' :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ +' :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encArg :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_* :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_+ :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_s :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_0 :: s:0':cons_p:cons_fact:cons_*:cons_+ encode_* :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_+ :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ hole_s:0':cons_p:cons_fact:cons_*:cons_+1_3 :: s:0':cons_p:cons_fact:cons_*:cons_+ gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3 :: Nat -> s:0':cons_p:cons_fact:cons_*:cons_+ Generator Equations: gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(0) <=> 0' gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(x, 1)) <=> s(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(x)) The following defined symbols remain to be analysed: +', fact, *', encArg They will be analysed ascendingly in the following order: *' < fact fact < encArg +' < *' *' < encArg +' < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: +'(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(a), gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n4_3)) -> gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(n4_3, a)), rt in Omega(1 + n4_3) Induction Base: +'(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(a), gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(0)) ->_R^Omega(1) gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(a) Induction Step: +'(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(a), gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(n4_3, 1))) ->_R^Omega(1) s(+'(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(a), gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n4_3))) ->_IH s(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(a, 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: p(s(x)) -> x fact(0') -> s(0') fact(s(x)) -> *'(s(x), fact(p(s(x)))) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) +'(x, 0') -> x +'(x, s(y)) -> s(+'(x, y)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fact(x_1)) -> fact(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fact(x_1) -> fact(encArg(x_1)) encode_0 -> 0' encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) Types: p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ s :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ 0' :: s:0':cons_p:cons_fact:cons_*:cons_+ *' :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ +' :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encArg :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_* :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_+ :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_s :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_0 :: s:0':cons_p:cons_fact:cons_*:cons_+ encode_* :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_+ :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ hole_s:0':cons_p:cons_fact:cons_*:cons_+1_3 :: s:0':cons_p:cons_fact:cons_*:cons_+ gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3 :: Nat -> s:0':cons_p:cons_fact:cons_*:cons_+ Generator Equations: gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(0) <=> 0' gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(x, 1)) <=> s(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(x)) The following defined symbols remain to be analysed: +', fact, *', encArg They will be analysed ascendingly in the following order: *' < fact fact < encArg +' < *' *' < encArg +' < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: p(s(x)) -> x fact(0') -> s(0') fact(s(x)) -> *'(s(x), fact(p(s(x)))) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) +'(x, 0') -> x +'(x, s(y)) -> s(+'(x, y)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fact(x_1)) -> fact(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fact(x_1) -> fact(encArg(x_1)) encode_0 -> 0' encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) Types: p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ s :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ 0' :: s:0':cons_p:cons_fact:cons_*:cons_+ *' :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ +' :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encArg :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_* :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_+ :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_s :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_0 :: s:0':cons_p:cons_fact:cons_*:cons_+ encode_* :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_+ :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ hole_s:0':cons_p:cons_fact:cons_*:cons_+1_3 :: s:0':cons_p:cons_fact:cons_*:cons_+ gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3 :: Nat -> s:0':cons_p:cons_fact:cons_*:cons_+ Lemmas: +'(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(a), gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n4_3)) -> gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(n4_3, a)), rt in Omega(1 + n4_3) Generator Equations: gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(0) <=> 0' gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(x, 1)) <=> s(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(x)) The following defined symbols remain to be analysed: *', fact, encArg They will be analysed ascendingly in the following order: *' < fact fact < encArg *' < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: *'(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n825_3), gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(b)) -> gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(*(n825_3, b)), rt in Omega(1 + b*n825_3 + n825_3) Induction Base: *'(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(0), gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(b)) ->_R^Omega(1) 0' Induction Step: *'(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(n825_3, 1)), gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(b)) ->_R^Omega(1) +'(*'(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n825_3), gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(b)), gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(b)) ->_IH +'(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(*(c826_3, b)), gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(b)) ->_L^Omega(1 + b) gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(b, *(n825_3, b))) We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). ---------------------------------------- (18) Complex Obligation (BEST) ---------------------------------------- (19) Obligation: Proved the lower bound n^2 for the following obligation: Innermost TRS: Rules: p(s(x)) -> x fact(0') -> s(0') fact(s(x)) -> *'(s(x), fact(p(s(x)))) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) +'(x, 0') -> x +'(x, s(y)) -> s(+'(x, y)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fact(x_1)) -> fact(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fact(x_1) -> fact(encArg(x_1)) encode_0 -> 0' encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) Types: p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ s :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ 0' :: s:0':cons_p:cons_fact:cons_*:cons_+ *' :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ +' :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encArg :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_* :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_+ :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_s :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_0 :: s:0':cons_p:cons_fact:cons_*:cons_+ encode_* :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_+ :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ hole_s:0':cons_p:cons_fact:cons_*:cons_+1_3 :: s:0':cons_p:cons_fact:cons_*:cons_+ gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3 :: Nat -> s:0':cons_p:cons_fact:cons_*:cons_+ Lemmas: +'(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(a), gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n4_3)) -> gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(n4_3, a)), rt in Omega(1 + n4_3) Generator Equations: gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(0) <=> 0' gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(x, 1)) <=> s(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(x)) The following defined symbols remain to be analysed: *', fact, encArg They will be analysed ascendingly in the following order: *' < fact fact < encArg *' < encArg ---------------------------------------- (20) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (21) BOUNDS(n^2, INF) ---------------------------------------- (22) Obligation: Innermost TRS: Rules: p(s(x)) -> x fact(0') -> s(0') fact(s(x)) -> *'(s(x), fact(p(s(x)))) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) +'(x, 0') -> x +'(x, s(y)) -> s(+'(x, y)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fact(x_1)) -> fact(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fact(x_1) -> fact(encArg(x_1)) encode_0 -> 0' encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) Types: p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ s :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ 0' :: s:0':cons_p:cons_fact:cons_*:cons_+ *' :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ +' :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encArg :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_* :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_+ :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_s :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_0 :: s:0':cons_p:cons_fact:cons_*:cons_+ encode_* :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_+ :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ hole_s:0':cons_p:cons_fact:cons_*:cons_+1_3 :: s:0':cons_p:cons_fact:cons_*:cons_+ gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3 :: Nat -> s:0':cons_p:cons_fact:cons_*:cons_+ Lemmas: +'(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(a), gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n4_3)) -> gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(n4_3, a)), rt in Omega(1 + n4_3) *'(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n825_3), gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(b)) -> gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(*(n825_3, b)), rt in Omega(1 + b*n825_3 + n825_3) Generator Equations: gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(0) <=> 0' gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(x, 1)) <=> s(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(x)) The following defined symbols remain to be analysed: fact, encArg They will be analysed ascendingly in the following order: fact < encArg ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: fact(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n1810_3)) -> *3_3, rt in Omega(n1810_3) Induction Base: fact(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(0)) Induction Step: fact(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(n1810_3, 1))) ->_R^Omega(1) *'(s(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n1810_3)), fact(p(s(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n1810_3))))) ->_R^Omega(1) *'(s(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n1810_3)), fact(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n1810_3))) ->_IH *'(s(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n1810_3)), *3_3) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (24) Obligation: Innermost TRS: Rules: p(s(x)) -> x fact(0') -> s(0') fact(s(x)) -> *'(s(x), fact(p(s(x)))) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) +'(x, 0') -> x +'(x, s(y)) -> s(+'(x, y)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fact(x_1)) -> fact(encArg(x_1)) encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fact(x_1) -> fact(encArg(x_1)) encode_0 -> 0' encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) Types: p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ s :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ 0' :: s:0':cons_p:cons_fact:cons_*:cons_+ *' :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ +' :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encArg :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_* :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ cons_+ :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_p :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_s :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_fact :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_0 :: s:0':cons_p:cons_fact:cons_*:cons_+ encode_* :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ encode_+ :: s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ -> s:0':cons_p:cons_fact:cons_*:cons_+ hole_s:0':cons_p:cons_fact:cons_*:cons_+1_3 :: s:0':cons_p:cons_fact:cons_*:cons_+ gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3 :: Nat -> s:0':cons_p:cons_fact:cons_*:cons_+ Lemmas: +'(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(a), gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n4_3)) -> gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(n4_3, a)), rt in Omega(1 + n4_3) *'(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n825_3), gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(b)) -> gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(*(n825_3, b)), rt in Omega(1 + b*n825_3 + n825_3) fact(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n1810_3)) -> *3_3, rt in Omega(n1810_3) Generator Equations: gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(0) <=> 0' gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(x, 1)) <=> s(gen_s:0':cons_p:cons_fact:cons_*: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_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n3205_3)) -> gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n3205_3), rt in Omega(0) Induction Base: encArg(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(+(n3205_3, 1))) ->_R^Omega(0) s(encArg(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(n3205_3))) ->_IH s(gen_s:0':cons_p:cons_fact:cons_*:cons_+2_3(c3206_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)