/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 43 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 0 ms] (8) BOUNDS(1, n^1) (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxRelTRS (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 8 ms] (12) typed CpxTrs (13) OrderProof [LOWER BOUND(ID), 0 ms] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 493 ms] (16) BEST (17) proven lower bound (18) LowerBoundPropagationProof [FINISHED, 0 ms] (19) BOUNDS(n^1, INF) (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 225 ms] (22) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: b(c(a(x1))) -> a(b(a(b(c(x1))))) b(x1) -> c(c(x1)) c(d(x1)) -> a(b(c(a(x1)))) a(a(x1)) -> a(c(b(a(x1)))) 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(d(x_1)) -> d(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: b(c(a(x1))) -> a(b(a(b(c(x1))))) b(x1) -> c(c(x1)) c(d(x1)) -> a(b(c(a(x1)))) a(a(x1)) -> a(c(b(a(x1)))) The (relative) TRS S consists of the following rules: encArg(d(x_1)) -> d(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_d(x_1) -> d(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^1, n^1). The TRS R consists of the following rules: b(c(a(x1))) -> a(b(a(b(c(x1))))) b(x1) -> c(c(x1)) c(d(x1)) -> a(b(c(a(x1)))) a(a(x1)) -> a(c(b(a(x1)))) The (relative) TRS S consists of the following rules: encArg(d(x_1)) -> d(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: b(c(a(x1))) -> a(b(a(b(c(x1))))) b(x1) -> c(c(x1)) c(d(x1)) -> a(b(c(a(x1)))) a(a(x1)) -> a(c(b(a(x1)))) encArg(d(x_1)) -> d(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 5. The certificate found is represented by the following graph. "[49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121] {(49,50,[b_1|0, c_1|0, a_1|0, encArg_1|0, encode_b_1|0, encode_c_1|0, encode_a_1|0, encode_d_1|0]), (49,51,[c_1|1]), (49,52,[a_1|1]), (49,55,[d_1|1, b_1|1, c_1|1, a_1|1]), (49,56,[c_1|2]), (49,62,[a_1|2]), (49,65,[a_1|2]), (49,69,[a_1|2]), (49,89,[a_1|3]), (50,50,[d_1|0, cons_b_1|0, cons_c_1|0, cons_a_1|0]), (51,50,[c_1|1]), (51,52,[a_1|1]), (51,62,[a_1|2]), (52,53,[b_1|1]), (52,57,[c_1|2]), (52,58,[a_1|2]), (53,54,[c_1|1]), (54,50,[a_1|1]), (55,50,[encArg_1|1]), (55,55,[d_1|1, b_1|1, c_1|1, a_1|1]), (55,56,[c_1|2]), (55,65,[a_1|2]), (55,69,[a_1|2]), (55,62,[a_1|2]), (55,89,[a_1|3]), (56,55,[c_1|2]), (56,69,[a_1|2]), (56,89,[a_1|3]), (57,53,[c_1|2]), (58,59,[b_1|2]), (58,72,[c_1|3]), (59,60,[a_1|2]), (60,61,[b_1|2]), (60,73,[c_1|3]), (61,50,[c_1|2]), (61,52,[a_1|1]), (61,62,[a_1|2]), (62,63,[c_1|2]), (63,64,[b_1|2]), (63,74,[c_1|3]), (64,58,[a_1|2]), (64,55,[a_1|2]), (64,65,[a_1|2]), (64,69,[a_1|2]), (64,62,[a_1|2]), (64,75,[a_1|3]), (64,89,[a_1|2]), (65,66,[b_1|2]), (65,78,[c_1|3]), (66,67,[a_1|2]), (66,75,[a_1|3]), (67,68,[b_1|2]), (67,79,[c_1|3]), (67,65,[a_1|2]), (67,80,[a_1|3]), (68,55,[c_1|2]), (68,65,[c_1|2]), (68,69,[c_1|2, a_1|2]), (68,62,[c_1|2]), (68,89,[a_1|3, c_1|2]), (69,70,[b_1|2]), (69,84,[c_1|3]), (69,85,[a_1|3]), (70,71,[c_1|2]), (71,55,[a_1|2]), (71,62,[a_1|2]), (71,75,[a_1|3]), (72,59,[c_1|3]), (73,61,[c_1|3]), (74,64,[c_1|3]), (75,76,[c_1|3]), (76,77,[b_1|3]), (76,92,[c_1|4]), (77,65,[a_1|3]), (77,69,[a_1|3]), (77,62,[a_1|3]), (77,85,[a_1|3]), (77,80,[a_1|3]), (77,93,[a_1|4]), (77,89,[a_1|3]), (78,66,[c_1|3]), (79,68,[c_1|3]), (80,81,[b_1|3]), (80,96,[c_1|4]), (81,82,[a_1|3]), (81,104,[a_1|4]), (82,83,[b_1|3]), (82,97,[c_1|4]), (82,98,[a_1|4]), (83,65,[c_1|3]), (83,69,[c_1|3]), (83,62,[c_1|3]), (83,85,[c_1|3]), (83,89,[c_1|3]), (84,70,[c_1|3]), (85,86,[b_1|3]), (85,102,[c_1|4]), (86,87,[a_1|3]), (86,75,[a_1|3]), (86,107,[a_1|4]), (87,88,[b_1|3]), (87,103,[c_1|4]), (87,65,[a_1|2]), (87,80,[a_1|3]), (87,110,[a_1|4]), (88,55,[c_1|3]), (88,62,[c_1|3]), (88,75,[c_1|3]), (88,69,[a_1|2]), (88,89,[a_1|3]), (89,90,[c_1|3]), (90,91,[b_1|3]), (90,114,[c_1|4]), (91,85,[a_1|3]), (92,77,[c_1|4]), (93,94,[c_1|4]), (94,95,[b_1|4]), (94,115,[c_1|5]), (95,85,[a_1|4]), (96,81,[c_1|4]), (97,83,[c_1|4]), (98,99,[b_1|4]), (98,116,[c_1|5]), (99,100,[a_1|4]), (100,101,[b_1|4]), (100,117,[c_1|5]), (101,85,[c_1|4]), (102,86,[c_1|4]), (103,88,[c_1|4]), (104,105,[c_1|4]), (105,106,[b_1|4]), (105,118,[c_1|5]), (106,98,[a_1|4]), (107,108,[c_1|4]), (108,109,[b_1|4]), (108,119,[c_1|5]), (109,80,[a_1|4]), (109,110,[a_1|4]), (110,111,[b_1|4]), (110,120,[c_1|5]), (111,112,[a_1|4]), (112,113,[b_1|4]), (112,121,[c_1|5]), (113,89,[c_1|4]), (114,91,[c_1|4]), (115,95,[c_1|5]), (116,99,[c_1|5]), (117,101,[c_1|5]), (118,106,[c_1|5]), (119,109,[c_1|5]), (120,111,[c_1|5]), (121,113,[c_1|5])}" ---------------------------------------- (8) BOUNDS(1, n^1) ---------------------------------------- (9) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (10) 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: b(c(a(x1))) -> a(b(a(b(c(x1))))) b(x1) -> c(c(x1)) c(d(x1)) -> a(b(c(a(x1)))) a(a(x1)) -> a(c(b(a(x1)))) The (relative) TRS S consists of the following rules: encArg(d(x_1)) -> d(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (11) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (12) Obligation: Innermost TRS: Rules: b(c(a(x1))) -> a(b(a(b(c(x1))))) b(x1) -> c(c(x1)) c(d(x1)) -> a(b(c(a(x1)))) a(a(x1)) -> a(c(b(a(x1)))) encArg(d(x_1)) -> d(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) Types: b :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a c :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a a :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a d :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encArg :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a cons_b :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a cons_c :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a cons_a :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encode_b :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encode_c :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encode_a :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encode_d :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a hole_d:cons_b:cons_c:cons_a1_0 :: d:cons_b:cons_c:cons_a gen_d:cons_b:cons_c:cons_a2_0 :: Nat -> d:cons_b:cons_c:cons_a ---------------------------------------- (13) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: b, a, c, encArg They will be analysed ascendingly in the following order: b = a b = c b < encArg a = c a < encArg c < encArg ---------------------------------------- (14) Obligation: Innermost TRS: Rules: b(c(a(x1))) -> a(b(a(b(c(x1))))) b(x1) -> c(c(x1)) c(d(x1)) -> a(b(c(a(x1)))) a(a(x1)) -> a(c(b(a(x1)))) encArg(d(x_1)) -> d(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) Types: b :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a c :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a a :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a d :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encArg :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a cons_b :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a cons_c :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a cons_a :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encode_b :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encode_c :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encode_a :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encode_d :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a hole_d:cons_b:cons_c:cons_a1_0 :: d:cons_b:cons_c:cons_a gen_d:cons_b:cons_c:cons_a2_0 :: Nat -> d:cons_b:cons_c:cons_a Generator Equations: gen_d:cons_b:cons_c:cons_a2_0(0) <=> hole_d:cons_b:cons_c:cons_a1_0 gen_d:cons_b:cons_c:cons_a2_0(+(x, 1)) <=> d(gen_d:cons_b:cons_c:cons_a2_0(x)) The following defined symbols remain to be analysed: a, b, c, encArg They will be analysed ascendingly in the following order: b = a b = c b < encArg a = c a < encArg c < encArg ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: c(gen_d:cons_b:cons_c:cons_a2_0(+(1, n8_0))) -> *3_0, rt in Omega(n8_0) Induction Base: c(gen_d:cons_b:cons_c:cons_a2_0(+(1, 0))) Induction Step: c(gen_d:cons_b:cons_c:cons_a2_0(+(1, +(n8_0, 1)))) ->_R^Omega(1) a(b(c(a(gen_d:cons_b:cons_c:cons_a2_0(+(1, n8_0)))))) ->_R^Omega(1) a(a(b(a(b(c(gen_d:cons_b:cons_c:cons_a2_0(+(1, n8_0)))))))) ->_IH a(a(b(a(b(*3_0))))) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (16) Complex Obligation (BEST) ---------------------------------------- (17) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: b(c(a(x1))) -> a(b(a(b(c(x1))))) b(x1) -> c(c(x1)) c(d(x1)) -> a(b(c(a(x1)))) a(a(x1)) -> a(c(b(a(x1)))) encArg(d(x_1)) -> d(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) Types: b :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a c :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a a :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a d :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encArg :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a cons_b :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a cons_c :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a cons_a :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encode_b :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encode_c :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encode_a :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encode_d :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a hole_d:cons_b:cons_c:cons_a1_0 :: d:cons_b:cons_c:cons_a gen_d:cons_b:cons_c:cons_a2_0 :: Nat -> d:cons_b:cons_c:cons_a Generator Equations: gen_d:cons_b:cons_c:cons_a2_0(0) <=> hole_d:cons_b:cons_c:cons_a1_0 gen_d:cons_b:cons_c:cons_a2_0(+(x, 1)) <=> d(gen_d:cons_b:cons_c:cons_a2_0(x)) The following defined symbols remain to be analysed: c, b, encArg They will be analysed ascendingly in the following order: b = a b = c b < encArg a = c a < encArg c < encArg ---------------------------------------- (18) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (19) BOUNDS(n^1, INF) ---------------------------------------- (20) Obligation: Innermost TRS: Rules: b(c(a(x1))) -> a(b(a(b(c(x1))))) b(x1) -> c(c(x1)) c(d(x1)) -> a(b(c(a(x1)))) a(a(x1)) -> a(c(b(a(x1)))) encArg(d(x_1)) -> d(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) Types: b :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a c :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a a :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a d :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encArg :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a cons_b :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a cons_c :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a cons_a :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encode_b :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encode_c :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encode_a :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a encode_d :: d:cons_b:cons_c:cons_a -> d:cons_b:cons_c:cons_a hole_d:cons_b:cons_c:cons_a1_0 :: d:cons_b:cons_c:cons_a gen_d:cons_b:cons_c:cons_a2_0 :: Nat -> d:cons_b:cons_c:cons_a Lemmas: c(gen_d:cons_b:cons_c:cons_a2_0(+(1, n8_0))) -> *3_0, rt in Omega(n8_0) Generator Equations: gen_d:cons_b:cons_c:cons_a2_0(0) <=> hole_d:cons_b:cons_c:cons_a1_0 gen_d:cons_b:cons_c:cons_a2_0(+(x, 1)) <=> d(gen_d:cons_b:cons_c:cons_a2_0(x)) The following defined symbols remain to be analysed: b, a, encArg They will be analysed ascendingly in the following order: b = a b = c b < encArg a = c a < encArg c < encArg ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_d:cons_b:cons_c:cons_a2_0(+(1, n1488_0))) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_d:cons_b:cons_c:cons_a2_0(+(1, 0))) Induction Step: encArg(gen_d:cons_b:cons_c:cons_a2_0(+(1, +(n1488_0, 1)))) ->_R^Omega(0) d(encArg(gen_d:cons_b:cons_c:cons_a2_0(+(1, n1488_0)))) ->_IH d(*3_0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (22) BOUNDS(1, INF)