/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 253 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 2 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 461 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), 133 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 75 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 210 ms] (22) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: tower(x) -> f(a, x, s(0)) f(a, 0, y) -> y f(a, s(x), y) -> f(b, y, s(x)) f(b, y, x) -> f(a, half(x), exp(y)) exp(0) -> s(0) exp(s(x)) -> double(exp(x)) double(0) -> 0 double(s(x)) -> s(s(double(x))) half(0) -> double(0) half(s(0)) -> half(0) half(s(s(x))) -> s(half(x)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (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(a) -> a encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(b) -> b encArg(cons_tower(x_1)) -> tower(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_exp(x_1)) -> exp(encArg(x_1)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_tower(x_1) -> tower(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a -> a encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_b -> b encode_half(x_1) -> half(encArg(x_1)) encode_exp(x_1) -> exp(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: tower(x) -> f(a, x, s(0)) f(a, 0, y) -> y f(a, s(x), y) -> f(b, y, s(x)) f(b, y, x) -> f(a, half(x), exp(y)) exp(0) -> s(0) exp(s(x)) -> double(exp(x)) double(0) -> 0 double(s(x)) -> s(s(double(x))) half(0) -> double(0) half(s(0)) -> half(0) half(s(s(x))) -> s(half(x)) The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(b) -> b encArg(cons_tower(x_1)) -> tower(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_exp(x_1)) -> exp(encArg(x_1)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_tower(x_1) -> tower(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a -> a encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_b -> b encode_half(x_1) -> half(encArg(x_1)) encode_exp(x_1) -> exp(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: tower(x) -> f(a, x, s(0)) f(a, 0, y) -> y f(a, s(x), y) -> f(b, y, s(x)) f(b, y, x) -> f(a, half(x), exp(y)) exp(0) -> s(0) exp(s(x)) -> double(exp(x)) double(0) -> 0 double(s(x)) -> s(s(double(x))) half(0) -> double(0) half(s(0)) -> half(0) half(s(s(x))) -> s(half(x)) The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(b) -> b encArg(cons_tower(x_1)) -> tower(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_exp(x_1)) -> exp(encArg(x_1)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_tower(x_1) -> tower(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a -> a encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_b -> b encode_half(x_1) -> half(encArg(x_1)) encode_exp(x_1) -> exp(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: tower(x) -> f(a, x, s(0')) f(a, 0', y) -> y f(a, s(x), y) -> f(b, y, s(x)) f(b, y, x) -> f(a, half(x), exp(y)) exp(0') -> s(0') exp(s(x)) -> double(exp(x)) double(0') -> 0' double(s(x)) -> s(s(double(x))) half(0') -> double(0') half(s(0')) -> half(0') half(s(s(x))) -> s(half(x)) The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(b) -> b encArg(cons_tower(x_1)) -> tower(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_exp(x_1)) -> exp(encArg(x_1)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_tower(x_1) -> tower(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a -> a encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_b -> b encode_half(x_1) -> half(encArg(x_1)) encode_exp(x_1) -> exp(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: TRS: Rules: tower(x) -> f(a, x, s(0')) f(a, 0', y) -> y f(a, s(x), y) -> f(b, y, s(x)) f(b, y, x) -> f(a, half(x), exp(y)) exp(0') -> s(0') exp(s(x)) -> double(exp(x)) double(0') -> 0' double(s(x)) -> s(s(double(x))) half(0') -> double(0') half(s(0')) -> half(0') half(s(s(x))) -> s(half(x)) encArg(a) -> a encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(b) -> b encArg(cons_tower(x_1)) -> tower(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_exp(x_1)) -> exp(encArg(x_1)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_tower(x_1) -> tower(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a -> a encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_b -> b encode_half(x_1) -> half(encArg(x_1)) encode_exp(x_1) -> exp(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) Types: tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half a :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half s :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half 0' :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half b :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encArg :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_a :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_s :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_0 :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_b :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half hole_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half1_4 :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4 :: Nat -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: f, half, exp, double, encArg They will be analysed ascendingly in the following order: half < f exp < f f < encArg double < half half < encArg double < exp exp < encArg double < encArg ---------------------------------------- (10) Obligation: TRS: Rules: tower(x) -> f(a, x, s(0')) f(a, 0', y) -> y f(a, s(x), y) -> f(b, y, s(x)) f(b, y, x) -> f(a, half(x), exp(y)) exp(0') -> s(0') exp(s(x)) -> double(exp(x)) double(0') -> 0' double(s(x)) -> s(s(double(x))) half(0') -> double(0') half(s(0')) -> half(0') half(s(s(x))) -> s(half(x)) encArg(a) -> a encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(b) -> b encArg(cons_tower(x_1)) -> tower(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_exp(x_1)) -> exp(encArg(x_1)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_tower(x_1) -> tower(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a -> a encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_b -> b encode_half(x_1) -> half(encArg(x_1)) encode_exp(x_1) -> exp(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) Types: tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half a :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half s :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half 0' :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half b :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encArg :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_a :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_s :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_0 :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_b :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half hole_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half1_4 :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4 :: Nat -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half Generator Equations: gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(0) <=> a gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(x, 1)) <=> s(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(x)) The following defined symbols remain to be analysed: double, f, half, exp, encArg They will be analysed ascendingly in the following order: half < f exp < f f < encArg double < half half < encArg double < exp exp < encArg double < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: double(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(1, n4_4))) -> *3_4, rt in Omega(n4_4) Induction Base: double(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(1, 0))) Induction Step: double(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(1, +(n4_4, 1)))) ->_R^Omega(1) s(s(double(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(1, n4_4))))) ->_IH s(s(*3_4)) 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: TRS: Rules: tower(x) -> f(a, x, s(0')) f(a, 0', y) -> y f(a, s(x), y) -> f(b, y, s(x)) f(b, y, x) -> f(a, half(x), exp(y)) exp(0') -> s(0') exp(s(x)) -> double(exp(x)) double(0') -> 0' double(s(x)) -> s(s(double(x))) half(0') -> double(0') half(s(0')) -> half(0') half(s(s(x))) -> s(half(x)) encArg(a) -> a encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(b) -> b encArg(cons_tower(x_1)) -> tower(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_exp(x_1)) -> exp(encArg(x_1)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_tower(x_1) -> tower(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a -> a encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_b -> b encode_half(x_1) -> half(encArg(x_1)) encode_exp(x_1) -> exp(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) Types: tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half a :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half s :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half 0' :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half b :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encArg :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_a :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_s :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_0 :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_b :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half hole_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half1_4 :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4 :: Nat -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half Generator Equations: gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(0) <=> a gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(x, 1)) <=> s(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(x)) The following defined symbols remain to be analysed: double, f, half, exp, encArg They will be analysed ascendingly in the following order: half < f exp < f f < encArg double < half half < encArg double < exp exp < encArg double < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: TRS: Rules: tower(x) -> f(a, x, s(0')) f(a, 0', y) -> y f(a, s(x), y) -> f(b, y, s(x)) f(b, y, x) -> f(a, half(x), exp(y)) exp(0') -> s(0') exp(s(x)) -> double(exp(x)) double(0') -> 0' double(s(x)) -> s(s(double(x))) half(0') -> double(0') half(s(0')) -> half(0') half(s(s(x))) -> s(half(x)) encArg(a) -> a encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(b) -> b encArg(cons_tower(x_1)) -> tower(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_exp(x_1)) -> exp(encArg(x_1)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_tower(x_1) -> tower(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a -> a encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_b -> b encode_half(x_1) -> half(encArg(x_1)) encode_exp(x_1) -> exp(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) Types: tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half a :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half s :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half 0' :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half b :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encArg :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_a :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_s :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_0 :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_b :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half hole_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half1_4 :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4 :: Nat -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half Lemmas: double(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(1, n4_4))) -> *3_4, rt in Omega(n4_4) Generator Equations: gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(0) <=> a gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(x, 1)) <=> s(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(x)) The following defined symbols remain to be analysed: half, f, exp, encArg They will be analysed ascendingly in the following order: half < f exp < f f < encArg half < encArg exp < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: half(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(2, *(2, n313_4)))) -> *3_4, rt in Omega(n313_4) Induction Base: half(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(2, *(2, 0)))) Induction Step: half(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(2, *(2, +(n313_4, 1))))) ->_R^Omega(1) s(half(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(2, *(2, n313_4))))) ->_IH s(*3_4) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: TRS: Rules: tower(x) -> f(a, x, s(0')) f(a, 0', y) -> y f(a, s(x), y) -> f(b, y, s(x)) f(b, y, x) -> f(a, half(x), exp(y)) exp(0') -> s(0') exp(s(x)) -> double(exp(x)) double(0') -> 0' double(s(x)) -> s(s(double(x))) half(0') -> double(0') half(s(0')) -> half(0') half(s(s(x))) -> s(half(x)) encArg(a) -> a encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(b) -> b encArg(cons_tower(x_1)) -> tower(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_exp(x_1)) -> exp(encArg(x_1)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_tower(x_1) -> tower(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a -> a encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_b -> b encode_half(x_1) -> half(encArg(x_1)) encode_exp(x_1) -> exp(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) Types: tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half a :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half s :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half 0' :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half b :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encArg :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_a :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_s :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_0 :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_b :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half hole_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half1_4 :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4 :: Nat -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half Lemmas: double(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(1, n4_4))) -> *3_4, rt in Omega(n4_4) half(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(2, *(2, n313_4)))) -> *3_4, rt in Omega(n313_4) Generator Equations: gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(0) <=> a gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(x, 1)) <=> s(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(x)) The following defined symbols remain to be analysed: exp, f, encArg They will be analysed ascendingly in the following order: exp < f f < encArg exp < encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: exp(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(1, n1022_4))) -> *3_4, rt in Omega(n1022_4) Induction Base: exp(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(1, 0))) Induction Step: exp(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(1, +(n1022_4, 1)))) ->_R^Omega(1) double(exp(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(1, n1022_4)))) ->_IH double(*3_4) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (20) Obligation: TRS: Rules: tower(x) -> f(a, x, s(0')) f(a, 0', y) -> y f(a, s(x), y) -> f(b, y, s(x)) f(b, y, x) -> f(a, half(x), exp(y)) exp(0') -> s(0') exp(s(x)) -> double(exp(x)) double(0') -> 0' double(s(x)) -> s(s(double(x))) half(0') -> double(0') half(s(0')) -> half(0') half(s(s(x))) -> s(half(x)) encArg(a) -> a encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(b) -> b encArg(cons_tower(x_1)) -> tower(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_exp(x_1)) -> exp(encArg(x_1)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encode_tower(x_1) -> tower(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a -> a encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_b -> b encode_half(x_1) -> half(encArg(x_1)) encode_exp(x_1) -> exp(encArg(x_1)) encode_double(x_1) -> double(encArg(x_1)) Types: tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half a :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half s :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half 0' :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half b :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encArg :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half cons_half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_tower :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_f :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_a :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_s :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_0 :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_b :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_half :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_exp :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half encode_double :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half hole_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half1_4 :: a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4 :: Nat -> a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half Lemmas: double(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(1, n4_4))) -> *3_4, rt in Omega(n4_4) half(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(2, *(2, n313_4)))) -> *3_4, rt in Omega(n313_4) exp(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(1, n1022_4))) -> *3_4, rt in Omega(n1022_4) Generator Equations: gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(0) <=> a gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(x, 1)) <=> s(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(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_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(n2582_4)) -> gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(n2582_4), rt in Omega(0) Induction Base: encArg(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(0)) ->_R^Omega(0) a Induction Step: encArg(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(+(n2582_4, 1))) ->_R^Omega(0) s(encArg(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(n2582_4))) ->_IH s(gen_a:0':s:b:cons_tower:cons_f:cons_exp:cons_double:cons_half2_4(c2583_4)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (22) BOUNDS(1, INF)