WORST_CASE(Omega(n^1), O(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 42 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), 0 ms] (12) typed CpxTrs (13) OrderProof [LOWER BOUND(ID), 0 ms] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 524 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), 260 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. "[45, 46, 47, 48, 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] {(45,46,[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]), (45,47,[c_1|1]), (45,48,[a_1|1]), (45,51,[d_1|1, b_1|1, c_1|1, a_1|1]), (45,52,[c_1|2]), (45,58,[a_1|2]), (45,61,[a_1|2]), (45,65,[a_1|2]), (45,85,[a_1|3]), (46,46,[d_1|0, cons_b_1|0, cons_c_1|0, cons_a_1|0]), (47,46,[c_1|1]), (47,48,[a_1|1]), (47,58,[a_1|2]), (48,49,[b_1|1]), (48,53,[c_1|2]), (48,54,[a_1|2]), (49,50,[c_1|1]), (50,46,[a_1|1]), (51,46,[encArg_1|1]), (51,51,[d_1|1, b_1|1, c_1|1, a_1|1]), (51,52,[c_1|2]), (51,61,[a_1|2]), (51,65,[a_1|2]), (51,58,[a_1|2]), (51,85,[a_1|3]), (52,51,[c_1|2]), (52,65,[a_1|2]), (52,85,[a_1|3]), (53,49,[c_1|2]), (54,55,[b_1|2]), (54,68,[c_1|3]), (55,56,[a_1|2]), (56,57,[b_1|2]), (56,69,[c_1|3]), (57,46,[c_1|2]), (57,48,[a_1|1]), (57,58,[a_1|2]), (58,59,[c_1|2]), (59,60,[b_1|2]), (59,70,[c_1|3]), (60,54,[a_1|2]), (60,51,[a_1|2]), (60,61,[a_1|2]), (60,65,[a_1|2]), (60,58,[a_1|2]), (60,71,[a_1|3]), (60,85,[a_1|2]), (61,62,[b_1|2]), (61,74,[c_1|3]), (62,63,[a_1|2]), (62,71,[a_1|3]), (63,64,[b_1|2]), (63,75,[c_1|3]), (63,61,[a_1|2]), (63,76,[a_1|3]), (64,51,[c_1|2]), (64,61,[c_1|2]), (64,65,[c_1|2, a_1|2]), (64,58,[c_1|2]), (64,85,[a_1|3, c_1|2]), (65,66,[b_1|2]), (65,80,[c_1|3]), (65,81,[a_1|3]), (66,67,[c_1|2]), (67,51,[a_1|2]), (67,58,[a_1|2]), (67,71,[a_1|3]), (68,55,[c_1|3]), (69,57,[c_1|3]), (70,60,[c_1|3]), (71,72,[c_1|3]), (72,73,[b_1|3]), (72,88,[c_1|4]), (73,61,[a_1|3]), (73,65,[a_1|3]), (73,58,[a_1|3]), (73,81,[a_1|3]), (73,76,[a_1|3]), (73,89,[a_1|4]), (73,85,[a_1|3]), (74,62,[c_1|3]), (75,64,[c_1|3]), (76,77,[b_1|3]), (76,92,[c_1|4]), (77,78,[a_1|3]), (77,100,[a_1|4]), (78,79,[b_1|3]), (78,93,[c_1|4]), (78,94,[a_1|4]), (79,61,[c_1|3]), (79,65,[c_1|3]), (79,58,[c_1|3]), (79,81,[c_1|3]), (79,85,[c_1|3]), (80,66,[c_1|3]), (81,82,[b_1|3]), (81,98,[c_1|4]), (82,83,[a_1|3]), (82,71,[a_1|3]), (82,103,[a_1|4]), (83,84,[b_1|3]), (83,99,[c_1|4]), (83,61,[a_1|2]), (83,76,[a_1|3]), (83,106,[a_1|4]), (84,51,[c_1|3]), (84,58,[c_1|3]), (84,71,[c_1|3]), (84,65,[a_1|2]), (84,85,[a_1|3]), (85,86,[c_1|3]), (86,87,[b_1|3]), (86,110,[c_1|4]), (87,81,[a_1|3]), (88,73,[c_1|4]), (89,90,[c_1|4]), (90,91,[b_1|4]), (90,111,[c_1|5]), (91,81,[a_1|4]), (92,77,[c_1|4]), (93,79,[c_1|4]), (94,95,[b_1|4]), (94,112,[c_1|5]), (95,96,[a_1|4]), (96,97,[b_1|4]), (96,113,[c_1|5]), (97,81,[c_1|4]), (98,82,[c_1|4]), (99,84,[c_1|4]), (100,101,[c_1|4]), (101,102,[b_1|4]), (101,114,[c_1|5]), (102,94,[a_1|4]), (103,104,[c_1|4]), (104,105,[b_1|4]), (104,115,[c_1|5]), (105,76,[a_1|4]), (105,106,[a_1|4]), (106,107,[b_1|4]), (106,116,[c_1|5]), (107,108,[a_1|4]), (108,109,[b_1|4]), (108,117,[c_1|5]), (109,85,[c_1|4]), (110,87,[c_1|4]), (111,91,[c_1|5]), (112,95,[c_1|5]), (113,97,[c_1|5]), (114,102,[c_1|5]), (115,105,[c_1|5]), (116,107,[c_1|5]), (117,109,[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)