/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 510 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), 71 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 80 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 153 ms] (22) 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: f(tt, x) -> f(eq(x, half(double(x))), s(x)) eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> tt double(s(x)) -> s(s(double(x))) double(0) -> 0 half(s(s(x))) -> s(half(x)) half(0) -> 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(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_half(x_1) -> half(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 ---------------------------------------- (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: f(tt, x) -> f(eq(x, half(double(x))), s(x)) eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> tt double(s(x)) -> s(s(double(x))) double(0) -> 0 half(s(s(x))) -> s(half(x)) half(0) -> 0 The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_half(x_1) -> half(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 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: f(tt, x) -> f(eq(x, half(double(x))), s(x)) eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> tt double(s(x)) -> s(s(double(x))) double(0) -> 0 half(s(s(x))) -> s(half(x)) half(0) -> 0 The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_half(x_1) -> half(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 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: f(tt, x) -> f(eq(x, half(double(x))), s(x)) eq(s(x), s(y)) -> eq(x, y) eq(0', 0') -> tt double(s(x)) -> s(s(double(x))) double(0') -> 0' half(s(s(x))) -> s(half(x)) half(0') -> 0' The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_half(x_1) -> half(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: f(tt, x) -> f(eq(x, half(double(x))), s(x)) eq(s(x), s(y)) -> eq(x, y) eq(0', 0') -> tt double(s(x)) -> s(s(double(x))) double(0') -> 0' half(s(s(x))) -> s(half(x)) half(0') -> 0' encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_half(x_1) -> half(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half tt :: tt:s:0':cons_f:cons_eq:cons_double:cons_half eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half s :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half 0' :: tt:s:0':cons_f:cons_eq:cons_double:cons_half encArg :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_tt :: tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_s :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_0 :: tt:s:0':cons_f:cons_eq:cons_double:cons_half hole_tt:s:0':cons_f:cons_eq:cons_double:cons_half1_3 :: tt:s:0':cons_f:cons_eq:cons_double:cons_half gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3 :: Nat -> tt:s:0':cons_f:cons_eq:cons_double:cons_half ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: f, eq, half, double, encArg They will be analysed ascendingly in the following order: eq < f half < f double < f f < encArg eq < encArg half < encArg double < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: f(tt, x) -> f(eq(x, half(double(x))), s(x)) eq(s(x), s(y)) -> eq(x, y) eq(0', 0') -> tt double(s(x)) -> s(s(double(x))) double(0') -> 0' half(s(s(x))) -> s(half(x)) half(0') -> 0' encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_half(x_1) -> half(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half tt :: tt:s:0':cons_f:cons_eq:cons_double:cons_half eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half s :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half 0' :: tt:s:0':cons_f:cons_eq:cons_double:cons_half encArg :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_tt :: tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_s :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_0 :: tt:s:0':cons_f:cons_eq:cons_double:cons_half hole_tt:s:0':cons_f:cons_eq:cons_double:cons_half1_3 :: tt:s:0':cons_f:cons_eq:cons_double:cons_half gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3 :: Nat -> tt:s:0':cons_f:cons_eq:cons_double:cons_half Generator Equations: gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(0) <=> tt gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(x, 1)) <=> s(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(x)) The following defined symbols remain to be analysed: eq, f, half, double, encArg They will be analysed ascendingly in the following order: eq < f half < f double < f f < encArg eq < encArg half < encArg double < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: eq(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, n4_3)), gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3) Induction Base: eq(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, 0)), gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, 0))) Induction Step: eq(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, +(n4_3, 1))), gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, +(n4_3, 1)))) ->_R^Omega(1) eq(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, n4_3)), gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, n4_3))) ->_IH *3_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: f(tt, x) -> f(eq(x, half(double(x))), s(x)) eq(s(x), s(y)) -> eq(x, y) eq(0', 0') -> tt double(s(x)) -> s(s(double(x))) double(0') -> 0' half(s(s(x))) -> s(half(x)) half(0') -> 0' encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_half(x_1) -> half(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half tt :: tt:s:0':cons_f:cons_eq:cons_double:cons_half eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half s :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half 0' :: tt:s:0':cons_f:cons_eq:cons_double:cons_half encArg :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_tt :: tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_s :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_0 :: tt:s:0':cons_f:cons_eq:cons_double:cons_half hole_tt:s:0':cons_f:cons_eq:cons_double:cons_half1_3 :: tt:s:0':cons_f:cons_eq:cons_double:cons_half gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3 :: Nat -> tt:s:0':cons_f:cons_eq:cons_double:cons_half Generator Equations: gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(0) <=> tt gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(x, 1)) <=> s(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(x)) The following defined symbols remain to be analysed: eq, f, half, double, encArg They will be analysed ascendingly in the following order: eq < f half < f double < f f < encArg eq < encArg half < encArg double < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: f(tt, x) -> f(eq(x, half(double(x))), s(x)) eq(s(x), s(y)) -> eq(x, y) eq(0', 0') -> tt double(s(x)) -> s(s(double(x))) double(0') -> 0' half(s(s(x))) -> s(half(x)) half(0') -> 0' encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_half(x_1) -> half(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half tt :: tt:s:0':cons_f:cons_eq:cons_double:cons_half eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half s :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half 0' :: tt:s:0':cons_f:cons_eq:cons_double:cons_half encArg :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_tt :: tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_s :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_0 :: tt:s:0':cons_f:cons_eq:cons_double:cons_half hole_tt:s:0':cons_f:cons_eq:cons_double:cons_half1_3 :: tt:s:0':cons_f:cons_eq:cons_double:cons_half gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3 :: Nat -> tt:s:0':cons_f:cons_eq:cons_double:cons_half Lemmas: eq(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, n4_3)), gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3) Generator Equations: gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(0) <=> tt gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(x, 1)) <=> s(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(x)) The following defined symbols remain to be analysed: half, f, double, encArg They will be analysed ascendingly in the following order: half < f double < f f < encArg half < encArg double < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: half(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(2, *(2, n537_3)))) -> *3_3, rt in Omega(n537_3) Induction Base: half(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(2, *(2, 0)))) Induction Step: half(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(2, *(2, +(n537_3, 1))))) ->_R^Omega(1) s(half(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(2, *(2, n537_3))))) ->_IH s(*3_3) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: Innermost TRS: Rules: f(tt, x) -> f(eq(x, half(double(x))), s(x)) eq(s(x), s(y)) -> eq(x, y) eq(0', 0') -> tt double(s(x)) -> s(s(double(x))) double(0') -> 0' half(s(s(x))) -> s(half(x)) half(0') -> 0' encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_half(x_1) -> half(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half tt :: tt:s:0':cons_f:cons_eq:cons_double:cons_half eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half s :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half 0' :: tt:s:0':cons_f:cons_eq:cons_double:cons_half encArg :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_tt :: tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_s :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_0 :: tt:s:0':cons_f:cons_eq:cons_double:cons_half hole_tt:s:0':cons_f:cons_eq:cons_double:cons_half1_3 :: tt:s:0':cons_f:cons_eq:cons_double:cons_half gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3 :: Nat -> tt:s:0':cons_f:cons_eq:cons_double:cons_half Lemmas: eq(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, n4_3)), gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3) half(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(2, *(2, n537_3)))) -> *3_3, rt in Omega(n537_3) Generator Equations: gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(0) <=> tt gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(x, 1)) <=> s(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(x)) The following defined symbols remain to be analysed: double, f, encArg They will be analysed ascendingly in the following order: double < f f < encArg double < encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: double(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, n988_3))) -> *3_3, rt in Omega(n988_3) Induction Base: double(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, 0))) Induction Step: double(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, +(n988_3, 1)))) ->_R^Omega(1) s(s(double(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, n988_3))))) ->_IH s(s(*3_3)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (20) Obligation: Innermost TRS: Rules: f(tt, x) -> f(eq(x, half(double(x))), s(x)) eq(s(x), s(y)) -> eq(x, y) eq(0', 0') -> tt double(s(x)) -> s(s(double(x))) double(0') -> 0' half(s(s(x))) -> s(half(x)) half(0') -> 0' encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_half(x_1) -> half(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half tt :: tt:s:0':cons_f:cons_eq:cons_double:cons_half eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half s :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half 0' :: tt:s:0':cons_f:cons_eq:cons_double:cons_half encArg :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half cons_half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_f :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_tt :: tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_eq :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_half :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_double :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_s :: tt:s:0':cons_f:cons_eq:cons_double:cons_half -> tt:s:0':cons_f:cons_eq:cons_double:cons_half encode_0 :: tt:s:0':cons_f:cons_eq:cons_double:cons_half hole_tt:s:0':cons_f:cons_eq:cons_double:cons_half1_3 :: tt:s:0':cons_f:cons_eq:cons_double:cons_half gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3 :: Nat -> tt:s:0':cons_f:cons_eq:cons_double:cons_half Lemmas: eq(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, n4_3)), gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3) half(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(2, *(2, n537_3)))) -> *3_3, rt in Omega(n537_3) double(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(1, n988_3))) -> *3_3, rt in Omega(n988_3) Generator Equations: gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(0) <=> tt gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(x, 1)) <=> s(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(x)) The following defined symbols remain to be analysed: f, encArg They will be analysed ascendingly in the following order: f < encArg ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(n2541_3)) -> gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(n2541_3), rt in Omega(0) Induction Base: encArg(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(0)) ->_R^Omega(0) tt Induction Step: encArg(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(+(n2541_3, 1))) ->_R^Omega(0) s(encArg(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(n2541_3))) ->_IH s(gen_tt:s:0':cons_f:cons_eq:cons_double:cons_half2_3(c2542_3)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (22) BOUNDS(1, INF)