/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^2), ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^2, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 194 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), 291 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), 93 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), 58 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 20 ms] (26) typed CpxTrs (27) RewriteLemmaProof [LOWER BOUND(ID), 95 ms] (28) 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: +(0, y) -> y +(s(x), y) -> s(+(x, y)) *(x, 0) -> 0 *(x, s(y)) -> +(x, *(x, y)) twice(0) -> 0 twice(s(x)) -> s(s(twice(x))) -(x, 0) -> x -(s(x), s(y)) -> -(x, y) f(s(x)) -> f(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(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_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) 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^2, INF). The TRS R consists of the following rules: +(0, y) -> y +(s(x), y) -> s(+(x, y)) *(x, 0) -> 0 *(x, s(y)) -> +(x, *(x, y)) twice(0) -> 0 twice(s(x)) -> s(s(twice(x))) -(x, 0) -> x -(s(x), s(y)) -> -(x, y) f(s(x)) -> f(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0))))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(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)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) 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^2, INF). The TRS R consists of the following rules: +(0, y) -> y +(s(x), y) -> s(+(x, y)) *(x, 0) -> 0 *(x, s(y)) -> +(x, *(x, y)) twice(0) -> 0 twice(s(x)) -> s(s(twice(x))) -(x, 0) -> x -(s(x), s(y)) -> -(x, y) f(s(x)) -> f(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0))))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(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)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) 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^2, INF). The TRS R consists of the following rules: +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) *'(x, 0') -> 0' *'(x, s(y)) -> +'(x, *'(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) -(x, 0') -> x -(s(x), s(y)) -> -(x, y) f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0'))))) The (relative) TRS S consists of the following rules: encArg(0') -> 0' encArg(s(x_1)) -> s(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)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) 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: +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) *'(x, 0') -> 0' *'(x, s(y)) -> +'(x, *'(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) -(x, 0') -> x -(s(x), s(y)) -> -(x, y) f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0'))))) encArg(0') -> 0' encArg(s(x_1)) -> s(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)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) Types: +' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f 0' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f s :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f *' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f - :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encArg :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_+ :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_* :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_- :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_+ :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_0 :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_s :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_* :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_- :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f hole_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f1_3 :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3 :: Nat -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: +', *', twice, -, f, encArg They will be analysed ascendingly in the following order: +' < *' +' < f +' < encArg *' < f *' < encArg twice < encArg - < f - < encArg f < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) *'(x, 0') -> 0' *'(x, s(y)) -> +'(x, *'(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) -(x, 0') -> x -(s(x), s(y)) -> -(x, y) f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0'))))) encArg(0') -> 0' encArg(s(x_1)) -> s(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)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) Types: +' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f 0' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f s :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f *' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f - :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encArg :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_+ :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_* :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_- :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_+ :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_0 :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_s :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_* :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_- :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f hole_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f1_3 :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3 :: Nat -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f Generator Equations: gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0) <=> 0' gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(x)) The following defined symbols remain to be analysed: +', *', twice, -, f, encArg They will be analysed ascendingly in the following order: +' < *' +' < f +' < encArg *' < f *' < encArg twice < encArg - < f - < encArg f < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: +'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n4_3), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(b)) -> gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(n4_3, b)), rt in Omega(1 + n4_3) Induction Base: +'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(b)) ->_R^Omega(1) gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(b) Induction Step: +'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(n4_3, 1)), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(b)) ->_R^Omega(1) s(+'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n4_3), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(b))) ->_IH s(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_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)) *'(x, 0') -> 0' *'(x, s(y)) -> +'(x, *'(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) -(x, 0') -> x -(s(x), s(y)) -> -(x, y) f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0'))))) encArg(0') -> 0' encArg(s(x_1)) -> s(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)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) Types: +' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f 0' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f s :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f *' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f - :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encArg :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_+ :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_* :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_- :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_+ :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_0 :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_s :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_* :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_- :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f hole_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f1_3 :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3 :: Nat -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f Generator Equations: gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0) <=> 0' gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(x)) The following defined symbols remain to be analysed: +', *', twice, -, f, encArg They will be analysed ascendingly in the following order: +' < *' +' < f +' < encArg *' < f *' < encArg twice < encArg - < f - < encArg f < 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)) *'(x, 0') -> 0' *'(x, s(y)) -> +'(x, *'(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) -(x, 0') -> x -(s(x), s(y)) -> -(x, y) f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0'))))) encArg(0') -> 0' encArg(s(x_1)) -> s(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)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) Types: +' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f 0' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f s :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f *' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f - :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encArg :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_+ :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_* :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_- :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_+ :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_0 :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_s :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_* :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_- :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f hole_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f1_3 :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3 :: Nat -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f Lemmas: +'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n4_3), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(b)) -> gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(n4_3, b)), rt in Omega(1 + n4_3) Generator Equations: gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0) <=> 0' gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(x)) The following defined symbols remain to be analysed: *', twice, -, f, encArg They will be analysed ascendingly in the following order: *' < f *' < encArg twice < encArg - < f - < encArg f < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: *'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(a), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n929_3)) -> gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(*(n929_3, a)), rt in Omega(1 + a*n929_3 + n929_3) Induction Base: *'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(a), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0)) ->_R^Omega(1) 0' Induction Step: *'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(a), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(n929_3, 1))) ->_R^Omega(1) +'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(a), *'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(a), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n929_3))) ->_IH +'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(a), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(*(c930_3, a))) ->_L^Omega(1 + a) gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(a, *(n929_3, a))) 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: +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) *'(x, 0') -> 0' *'(x, s(y)) -> +'(x, *'(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) -(x, 0') -> x -(s(x), s(y)) -> -(x, y) f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0'))))) encArg(0') -> 0' encArg(s(x_1)) -> s(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)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) Types: +' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f 0' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f s :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f *' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f - :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encArg :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_+ :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_* :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_- :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_+ :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_0 :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_s :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_* :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_- :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f hole_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f1_3 :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3 :: Nat -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f Lemmas: +'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n4_3), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(b)) -> gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(n4_3, b)), rt in Omega(1 + n4_3) Generator Equations: gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0) <=> 0' gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(x)) The following defined symbols remain to be analysed: *', twice, -, f, encArg They will be analysed ascendingly in the following order: *' < f *' < encArg twice < encArg - < f - < encArg f < encArg ---------------------------------------- (20) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (21) BOUNDS(n^2, INF) ---------------------------------------- (22) Obligation: Innermost TRS: Rules: +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) *'(x, 0') -> 0' *'(x, s(y)) -> +'(x, *'(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) -(x, 0') -> x -(s(x), s(y)) -> -(x, y) f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0'))))) encArg(0') -> 0' encArg(s(x_1)) -> s(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)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) Types: +' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f 0' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f s :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f *' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f - :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encArg :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_+ :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_* :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_- :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_+ :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_0 :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_s :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_* :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_- :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f hole_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f1_3 :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3 :: Nat -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f Lemmas: +'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n4_3), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(b)) -> gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(n4_3, b)), rt in Omega(1 + n4_3) *'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(a), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n929_3)) -> gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(*(n929_3, a)), rt in Omega(1 + a*n929_3 + n929_3) Generator Equations: gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0) <=> 0' gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(x)) The following defined symbols remain to be analysed: twice, -, f, encArg They will be analysed ascendingly in the following order: twice < encArg - < f - < encArg f < encArg ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: twice(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n2111_3)) -> gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(*(2, n2111_3)), rt in Omega(1 + n2111_3) Induction Base: twice(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0)) ->_R^Omega(1) 0' Induction Step: twice(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(n2111_3, 1))) ->_R^Omega(1) s(s(twice(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n2111_3)))) ->_IH s(s(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(*(2, c2112_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: +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) *'(x, 0') -> 0' *'(x, s(y)) -> +'(x, *'(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) -(x, 0') -> x -(s(x), s(y)) -> -(x, y) f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0'))))) encArg(0') -> 0' encArg(s(x_1)) -> s(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)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) Types: +' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f 0' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f s :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f *' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f - :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encArg :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_+ :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_* :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_- :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_+ :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_0 :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_s :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_* :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_- :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f hole_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f1_3 :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3 :: Nat -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f Lemmas: +'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n4_3), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(b)) -> gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(n4_3, b)), rt in Omega(1 + n4_3) *'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(a), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n929_3)) -> gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(*(n929_3, a)), rt in Omega(1 + a*n929_3 + n929_3) twice(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n2111_3)) -> gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(*(2, n2111_3)), rt in Omega(1 + n2111_3) Generator Equations: gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0) <=> 0' gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(x)) The following defined symbols remain to be analysed: -, f, encArg They will be analysed ascendingly in the following order: - < f - < encArg f < encArg ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: -(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n2507_3), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n2507_3)) -> gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0), rt in Omega(1 + n2507_3) Induction Base: -(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0)) ->_R^Omega(1) gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0) Induction Step: -(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(n2507_3, 1)), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(n2507_3, 1))) ->_R^Omega(1) -(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n2507_3), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n2507_3)) ->_IH gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (26) Obligation: Innermost TRS: Rules: +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) *'(x, 0') -> 0' *'(x, s(y)) -> +'(x, *'(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) -(x, 0') -> x -(s(x), s(y)) -> -(x, y) f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0'))))) encArg(0') -> 0' encArg(s(x_1)) -> s(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)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) Types: +' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f 0' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f s :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f *' :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f - :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encArg :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_+ :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_* :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_- :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f cons_f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_+ :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_0 :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_s :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_* :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_twice :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_- :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f encode_f :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f hole_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f1_3 :: 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3 :: Nat -> 0':s:cons_+:cons_*:cons_twice:cons_-:cons_f Lemmas: +'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n4_3), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(b)) -> gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(n4_3, b)), rt in Omega(1 + n4_3) *'(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(a), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n929_3)) -> gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(*(n929_3, a)), rt in Omega(1 + a*n929_3 + n929_3) twice(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n2111_3)) -> gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(*(2, n2111_3)), rt in Omega(1 + n2111_3) -(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n2507_3), gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n2507_3)) -> gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0), rt in Omega(1 + n2507_3) Generator Equations: gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0) <=> 0' gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(x)) The following defined symbols remain to be analysed: f, encArg They will be analysed ascendingly in the following order: f < encArg ---------------------------------------- (27) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n3245_3)) -> gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n3245_3), rt in Omega(0) Induction Base: encArg(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(+(n3245_3, 1))) ->_R^Omega(0) s(encArg(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(n3245_3))) ->_IH s(gen_0':s:cons_+:cons_*:cons_twice:cons_-:cons_f2_3(c3246_3)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (28) BOUNDS(1, INF)