WORST_CASE(NON_POLY,?) proof of input_mXDEJhQn29.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). (0) CpxTRS (1) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CdtProblem (3) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 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), 10 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 1895 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) typed CpxTrs (17) RewriteLemmaProof [FINISHED, 327 ms] (18) BOUNDS(EXP, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: fac(0) -> 1 fac(s(x)) -> *(s(x), fac(x)) floop(0, y) -> y floop(s(x), y) -> floop(x, *(s(x), y)) *(x, 0) -> 0 *(x, s(y)) -> +(*(x, y), x) +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) 1 -> s(0) fac(0) -> s(0) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (2) Obligation: Complexity Dependency Tuples Problem Rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) floop(0, z0) -> z0 floop(s(z0), z1) -> floop(z0, *(s(z0), z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) 1 -> s(0) Tuples: FAC(0) -> c(1') FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FAC(0) -> c2 FLOOP(0, z0) -> c3 FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, 0) -> c5 *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, 0) -> c7 +'(z0, s(z1)) -> c8(+'(z0, z1)) 1' -> c9 S tuples: FAC(0) -> c(1') FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FAC(0) -> c2 FLOOP(0, z0) -> c3 FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, 0) -> c5 *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, 0) -> c7 +'(z0, s(z1)) -> c8(+'(z0, z1)) 1' -> c9 K tuples:none Defined Rule Symbols: fac_1, floop_2, *_2, +_2, 1 Defined Pair Symbols: FAC_1, FLOOP_2, *'_2, +'_2, 1' Compound Symbols: c_1, c1_2, c2, c3, c4_2, c5, c6_2, c7, c8_1, c9 ---------------------------------------- (3) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: FAC(0) -> c(1') FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FAC(0) -> c2 FLOOP(0, z0) -> c3 FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, 0) -> c5 *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, 0) -> c7 +'(z0, s(z1)) -> c8(+'(z0, z1)) 1' -> c9 The (relative) TRS S consists of the following rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) floop(0, z0) -> z0 floop(s(z0), z1) -> floop(z0, *(s(z0), z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) 1 -> s(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(EXP, INF). The TRS R consists of the following rules: FAC(0') -> c(1') FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FAC(0') -> c2 FLOOP(0', z0) -> c3 FLOOP(s(z0), z1) -> c4(FLOOP(z0, *'(s(z0), z1)), *'(s(z0), z1)) *'(z0, 0') -> c5 *'(z0, s(z1)) -> c6(+'(*'(z0, z1), z0), *'(z0, z1)) +'(z0, 0') -> c7 +'(z0, s(z1)) -> c8(+'(z0, z1)) 1' -> c9 The (relative) TRS S consists of the following rules: fac(0') -> 1' fac(s(z0)) -> *'(s(z0), fac(z0)) fac(0') -> s(0') floop(0', z0) -> z0 floop(s(z0), z1) -> floop(z0, *'(s(z0), z1)) *'(z0, 0') -> 0' *'(z0, s(z1)) -> +'(*'(z0, z1), z0) +'(z0, 0') -> z0 +'(z0, s(z1)) -> s(+'(z0, z1)) 1' -> s(0') Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: FAC(0') -> c(1') FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FAC(0') -> c2 FLOOP(0', z0) -> c3 FLOOP(s(z0), z1) -> c4(FLOOP(z0, *'(s(z0), z1)), *'(s(z0), z1)) *'(z0, 0') -> c5 *'(z0, s(z1)) -> c6(+'(*'(z0, z1), z0), *'(z0, z1)) +'(z0, 0') -> c7 +'(z0, s(z1)) -> c8(+'(z0, z1)) 1' -> c9 fac(0') -> 1' fac(s(z0)) -> *'(s(z0), fac(z0)) fac(0') -> s(0') floop(0', z0) -> z0 floop(s(z0), z1) -> floop(z0, *'(s(z0), z1)) *'(z0, 0') -> 0' *'(z0, s(z1)) -> +'(*'(z0, z1), z0) +'(z0, 0') -> z0 +'(z0, s(z1)) -> s(+'(z0, z1)) 1' -> s(0') Types: FAC :: 0':s:c5:c6:c7:c8:c9 -> c:c1:c2 0' :: 0':s:c5:c6:c7:c8:c9 c :: 0':s:c5:c6:c7:c8:c9 -> c:c1:c2 1' :: 0':s:c5:c6:c7:c8:c9 s :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 c1 :: 0':s:c5:c6:c7:c8:c9 -> c:c1:c2 -> c:c1:c2 *' :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 fac :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 c2 :: c:c1:c2 FLOOP :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> c3:c4 c3 :: c3:c4 c4 :: c3:c4 -> 0':s:c5:c6:c7:c8:c9 -> c3:c4 c5 :: 0':s:c5:c6:c7:c8:c9 c6 :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 +' :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 c7 :: 0':s:c5:c6:c7:c8:c9 c8 :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 c9 :: 0':s:c5:c6:c7:c8:c9 floop :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 hole_c:c1:c21_10 :: c:c1:c2 hole_0':s:c5:c6:c7:c8:c92_10 :: 0':s:c5:c6:c7:c8:c9 hole_c3:c43_10 :: c3:c4 gen_c:c1:c24_10 :: Nat -> c:c1:c2 gen_0':s:c5:c6:c7:c8:c95_10 :: Nat -> 0':s:c5:c6:c7:c8:c9 gen_c3:c46_10 :: Nat -> c3:c4 ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: FAC, *', fac, FLOOP, +', floop They will be analysed ascendingly in the following order: *' < FAC fac < FAC *' < fac *' < FLOOP +' < *' *' < floop ---------------------------------------- (10) Obligation: Innermost TRS: Rules: FAC(0') -> c(1') FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FAC(0') -> c2 FLOOP(0', z0) -> c3 FLOOP(s(z0), z1) -> c4(FLOOP(z0, *'(s(z0), z1)), *'(s(z0), z1)) *'(z0, 0') -> c5 *'(z0, s(z1)) -> c6(+'(*'(z0, z1), z0), *'(z0, z1)) +'(z0, 0') -> c7 +'(z0, s(z1)) -> c8(+'(z0, z1)) 1' -> c9 fac(0') -> 1' fac(s(z0)) -> *'(s(z0), fac(z0)) fac(0') -> s(0') floop(0', z0) -> z0 floop(s(z0), z1) -> floop(z0, *'(s(z0), z1)) *'(z0, 0') -> 0' *'(z0, s(z1)) -> +'(*'(z0, z1), z0) +'(z0, 0') -> z0 +'(z0, s(z1)) -> s(+'(z0, z1)) 1' -> s(0') Types: FAC :: 0':s:c5:c6:c7:c8:c9 -> c:c1:c2 0' :: 0':s:c5:c6:c7:c8:c9 c :: 0':s:c5:c6:c7:c8:c9 -> c:c1:c2 1' :: 0':s:c5:c6:c7:c8:c9 s :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 c1 :: 0':s:c5:c6:c7:c8:c9 -> c:c1:c2 -> c:c1:c2 *' :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 fac :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 c2 :: c:c1:c2 FLOOP :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> c3:c4 c3 :: c3:c4 c4 :: c3:c4 -> 0':s:c5:c6:c7:c8:c9 -> c3:c4 c5 :: 0':s:c5:c6:c7:c8:c9 c6 :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 +' :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 c7 :: 0':s:c5:c6:c7:c8:c9 c8 :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 c9 :: 0':s:c5:c6:c7:c8:c9 floop :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 hole_c:c1:c21_10 :: c:c1:c2 hole_0':s:c5:c6:c7:c8:c92_10 :: 0':s:c5:c6:c7:c8:c9 hole_c3:c43_10 :: c3:c4 gen_c:c1:c24_10 :: Nat -> c:c1:c2 gen_0':s:c5:c6:c7:c8:c95_10 :: Nat -> 0':s:c5:c6:c7:c8:c9 gen_c3:c46_10 :: Nat -> c3:c4 Generator Equations: gen_c:c1:c24_10(0) <=> c(0') gen_c:c1:c24_10(+(x, 1)) <=> c1(0', gen_c:c1:c24_10(x)) gen_0':s:c5:c6:c7:c8:c95_10(0) <=> 0' gen_0':s:c5:c6:c7:c8:c95_10(+(x, 1)) <=> s(gen_0':s:c5:c6:c7:c8:c95_10(x)) gen_c3:c46_10(0) <=> c3 gen_c3:c46_10(+(x, 1)) <=> c4(gen_c3:c46_10(x), 0') The following defined symbols remain to be analysed: +', FAC, *', fac, FLOOP, floop They will be analysed ascendingly in the following order: *' < FAC fac < FAC *' < fac *' < FLOOP +' < *' *' < floop ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: +'(gen_0':s:c5:c6:c7:c8:c95_10(a), gen_0':s:c5:c6:c7:c8:c95_10(+(1, n8_10))) -> *7_10, rt in Omega(n8_10) Induction Base: +'(gen_0':s:c5:c6:c7:c8:c95_10(a), gen_0':s:c5:c6:c7:c8:c95_10(+(1, 0))) Induction Step: +'(gen_0':s:c5:c6:c7:c8:c95_10(a), gen_0':s:c5:c6:c7:c8:c95_10(+(1, +(n8_10, 1)))) ->_R^Omega(1) c8(+'(gen_0':s:c5:c6:c7:c8:c95_10(a), gen_0':s:c5:c6:c7:c8:c95_10(+(1, n8_10)))) ->_IH c8(*7_10) 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: FAC(0') -> c(1') FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FAC(0') -> c2 FLOOP(0', z0) -> c3 FLOOP(s(z0), z1) -> c4(FLOOP(z0, *'(s(z0), z1)), *'(s(z0), z1)) *'(z0, 0') -> c5 *'(z0, s(z1)) -> c6(+'(*'(z0, z1), z0), *'(z0, z1)) +'(z0, 0') -> c7 +'(z0, s(z1)) -> c8(+'(z0, z1)) 1' -> c9 fac(0') -> 1' fac(s(z0)) -> *'(s(z0), fac(z0)) fac(0') -> s(0') floop(0', z0) -> z0 floop(s(z0), z1) -> floop(z0, *'(s(z0), z1)) *'(z0, 0') -> 0' *'(z0, s(z1)) -> +'(*'(z0, z1), z0) +'(z0, 0') -> z0 +'(z0, s(z1)) -> s(+'(z0, z1)) 1' -> s(0') Types: FAC :: 0':s:c5:c6:c7:c8:c9 -> c:c1:c2 0' :: 0':s:c5:c6:c7:c8:c9 c :: 0':s:c5:c6:c7:c8:c9 -> c:c1:c2 1' :: 0':s:c5:c6:c7:c8:c9 s :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 c1 :: 0':s:c5:c6:c7:c8:c9 -> c:c1:c2 -> c:c1:c2 *' :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 fac :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 c2 :: c:c1:c2 FLOOP :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> c3:c4 c3 :: c3:c4 c4 :: c3:c4 -> 0':s:c5:c6:c7:c8:c9 -> c3:c4 c5 :: 0':s:c5:c6:c7:c8:c9 c6 :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 +' :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 c7 :: 0':s:c5:c6:c7:c8:c9 c8 :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 c9 :: 0':s:c5:c6:c7:c8:c9 floop :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 hole_c:c1:c21_10 :: c:c1:c2 hole_0':s:c5:c6:c7:c8:c92_10 :: 0':s:c5:c6:c7:c8:c9 hole_c3:c43_10 :: c3:c4 gen_c:c1:c24_10 :: Nat -> c:c1:c2 gen_0':s:c5:c6:c7:c8:c95_10 :: Nat -> 0':s:c5:c6:c7:c8:c9 gen_c3:c46_10 :: Nat -> c3:c4 Generator Equations: gen_c:c1:c24_10(0) <=> c(0') gen_c:c1:c24_10(+(x, 1)) <=> c1(0', gen_c:c1:c24_10(x)) gen_0':s:c5:c6:c7:c8:c95_10(0) <=> 0' gen_0':s:c5:c6:c7:c8:c95_10(+(x, 1)) <=> s(gen_0':s:c5:c6:c7:c8:c95_10(x)) gen_c3:c46_10(0) <=> c3 gen_c3:c46_10(+(x, 1)) <=> c4(gen_c3:c46_10(x), 0') The following defined symbols remain to be analysed: +', FAC, *', fac, FLOOP, floop They will be analysed ascendingly in the following order: *' < FAC fac < FAC *' < fac *' < FLOOP +' < *' *' < floop ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: FAC(0') -> c(1') FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FAC(0') -> c2 FLOOP(0', z0) -> c3 FLOOP(s(z0), z1) -> c4(FLOOP(z0, *'(s(z0), z1)), *'(s(z0), z1)) *'(z0, 0') -> c5 *'(z0, s(z1)) -> c6(+'(*'(z0, z1), z0), *'(z0, z1)) +'(z0, 0') -> c7 +'(z0, s(z1)) -> c8(+'(z0, z1)) 1' -> c9 fac(0') -> 1' fac(s(z0)) -> *'(s(z0), fac(z0)) fac(0') -> s(0') floop(0', z0) -> z0 floop(s(z0), z1) -> floop(z0, *'(s(z0), z1)) *'(z0, 0') -> 0' *'(z0, s(z1)) -> +'(*'(z0, z1), z0) +'(z0, 0') -> z0 +'(z0, s(z1)) -> s(+'(z0, z1)) 1' -> s(0') Types: FAC :: 0':s:c5:c6:c7:c8:c9 -> c:c1:c2 0' :: 0':s:c5:c6:c7:c8:c9 c :: 0':s:c5:c6:c7:c8:c9 -> c:c1:c2 1' :: 0':s:c5:c6:c7:c8:c9 s :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 c1 :: 0':s:c5:c6:c7:c8:c9 -> c:c1:c2 -> c:c1:c2 *' :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 fac :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 c2 :: c:c1:c2 FLOOP :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> c3:c4 c3 :: c3:c4 c4 :: c3:c4 -> 0':s:c5:c6:c7:c8:c9 -> c3:c4 c5 :: 0':s:c5:c6:c7:c8:c9 c6 :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 +' :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 c7 :: 0':s:c5:c6:c7:c8:c9 c8 :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 c9 :: 0':s:c5:c6:c7:c8:c9 floop :: 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 -> 0':s:c5:c6:c7:c8:c9 hole_c:c1:c21_10 :: c:c1:c2 hole_0':s:c5:c6:c7:c8:c92_10 :: 0':s:c5:c6:c7:c8:c9 hole_c3:c43_10 :: c3:c4 gen_c:c1:c24_10 :: Nat -> c:c1:c2 gen_0':s:c5:c6:c7:c8:c95_10 :: Nat -> 0':s:c5:c6:c7:c8:c9 gen_c3:c46_10 :: Nat -> c3:c4 Lemmas: +'(gen_0':s:c5:c6:c7:c8:c95_10(a), gen_0':s:c5:c6:c7:c8:c95_10(+(1, n8_10))) -> *7_10, rt in Omega(n8_10) Generator Equations: gen_c:c1:c24_10(0) <=> c(0') gen_c:c1:c24_10(+(x, 1)) <=> c1(0', gen_c:c1:c24_10(x)) gen_0':s:c5:c6:c7:c8:c95_10(0) <=> 0' gen_0':s:c5:c6:c7:c8:c95_10(+(x, 1)) <=> s(gen_0':s:c5:c6:c7:c8:c95_10(x)) gen_c3:c46_10(0) <=> c3 gen_c3:c46_10(+(x, 1)) <=> c4(gen_c3:c46_10(x), 0') The following defined symbols remain to be analysed: *', FAC, fac, FLOOP, floop They will be analysed ascendingly in the following order: *' < FAC fac < FAC *' < fac *' < FLOOP *' < floop ---------------------------------------- (17) RewriteLemmaProof (FINISHED) Proved the following rewrite lemma: *'(gen_0':s:c5:c6:c7:c8:c95_10(a), gen_0':s:c5:c6:c7:c8:c95_10(+(1, n2158_10))) -> *7_10, rt in Omega(EXP) Induction Base: *'(gen_0':s:c5:c6:c7:c8:c95_10(a), gen_0':s:c5:c6:c7:c8:c95_10(+(1, 0))) Induction Step: *'(gen_0':s:c5:c6:c7:c8:c95_10(a), gen_0':s:c5:c6:c7:c8:c95_10(+(1, +(n2158_10, 1)))) ->_R^Omega(1) c6(+'(*'(gen_0':s:c5:c6:c7:c8:c95_10(a), gen_0':s:c5:c6:c7:c8:c95_10(+(1, n2158_10))), gen_0':s:c5:c6:c7:c8:c95_10(a)), *'(gen_0':s:c5:c6:c7:c8:c95_10(a), gen_0':s:c5:c6:c7:c8:c95_10(+(1, n2158_10)))) ->_IH c6(+'(*7_10, gen_0':s:c5:c6:c7:c8:c95_10(a)), *'(gen_0':s:c5:c6:c7:c8:c95_10(a), gen_0':s:c5:c6:c7:c8:c95_10(+(1, n2158_10)))) ->_IH c6(+'(*7_10, gen_0':s:c5:c6:c7:c8:c95_10(a)), *7_10) We have rt in EXP and sz in O(n). Thus, we have irc_R in EXP ---------------------------------------- (18) BOUNDS(EXP, INF)