1022.81/291.58 WORST_CASE(Omega(n^2), ?) 1022.81/291.61 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1022.81/291.61 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1022.81/291.61 1022.81/291.61 1022.81/291.61 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1022.81/291.61 1022.81/291.61 (0) CpxTRS 1022.81/291.61 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1022.81/291.61 (2) CpxTRS 1022.81/291.61 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1022.81/291.61 (4) typed CpxTrs 1022.81/291.61 (5) OrderProof [LOWER BOUND(ID), 0 ms] 1022.81/291.61 (6) typed CpxTrs 1022.81/291.61 (7) RewriteLemmaProof [LOWER BOUND(ID), 331 ms] 1022.81/291.61 (8) BEST 1022.81/291.61 (9) proven lower bound 1022.81/291.61 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 1022.81/291.61 (11) BOUNDS(n^1, INF) 1022.81/291.61 (12) typed CpxTrs 1022.81/291.61 (13) RewriteLemmaProof [LOWER BOUND(ID), 64 ms] 1022.81/291.61 (14) typed CpxTrs 1022.81/291.61 (15) RewriteLemmaProof [LOWER BOUND(ID), 29 ms] 1022.81/291.61 (16) typed CpxTrs 1022.81/291.61 (17) RewriteLemmaProof [LOWER BOUND(ID), 11 ms] 1022.81/291.61 (18) typed CpxTrs 1022.81/291.61 (19) RewriteLemmaProof [LOWER BOUND(ID), 49 ms] 1022.81/291.61 (20) BEST 1022.81/291.61 (21) proven lower bound 1022.81/291.61 (22) LowerBoundPropagationProof [FINISHED, 0 ms] 1022.81/291.61 (23) BOUNDS(n^2, INF) 1022.81/291.61 (24) typed CpxTrs 1022.81/291.61 1022.81/291.61 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (0) 1022.81/291.61 Obligation: 1022.81/291.61 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1022.81/291.61 1022.81/291.61 1022.81/291.61 The TRS R consists of the following rules: 1022.81/291.61 1022.81/291.61 min(0, y) -> 0 1022.81/291.61 min(x, 0) -> 0 1022.81/291.61 min(s(x), s(y)) -> s(min(x, y)) 1022.81/291.61 max(0, y) -> y 1022.81/291.61 max(x, 0) -> x 1022.81/291.61 max(s(x), s(y)) -> s(max(x, y)) 1022.81/291.61 +(0, y) -> y 1022.81/291.61 +(s(x), y) -> s(+(x, y)) 1022.81/291.61 -(x, 0) -> x 1022.81/291.61 -(s(x), s(y)) -> -(x, y) 1022.81/291.61 *(x, 0) -> 0 1022.81/291.61 *(x, s(y)) -> +(x, *(x, y)) 1022.81/291.61 f(s(x)) -> f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) 1022.81/291.61 1022.81/291.61 S is empty. 1022.81/291.61 Rewrite Strategy: INNERMOST 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 1022.81/291.61 Renamed function symbols to avoid clashes with predefined symbol. 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (2) 1022.81/291.61 Obligation: 1022.81/291.61 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1022.81/291.61 1022.81/291.61 1022.81/291.61 The TRS R consists of the following rules: 1022.81/291.61 1022.81/291.61 min(0', y) -> 0' 1022.81/291.61 min(x, 0') -> 0' 1022.81/291.61 min(s(x), s(y)) -> s(min(x, y)) 1022.81/291.61 max(0', y) -> y 1022.81/291.61 max(x, 0') -> x 1022.81/291.61 max(s(x), s(y)) -> s(max(x, y)) 1022.81/291.61 +'(0', y) -> y 1022.81/291.61 +'(s(x), y) -> s(+'(x, y)) 1022.81/291.61 -(x, 0') -> x 1022.81/291.61 -(s(x), s(y)) -> -(x, y) 1022.81/291.61 *'(x, 0') -> 0' 1022.81/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 1022.81/291.61 f(s(x)) -> f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0')))))))) 1022.81/291.61 1022.81/291.61 S is empty. 1022.81/291.61 Rewrite Strategy: INNERMOST 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1022.81/291.61 Infered types. 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (4) 1022.81/291.61 Obligation: 1022.81/291.61 Innermost TRS: 1022.81/291.61 Rules: 1022.81/291.61 min(0', y) -> 0' 1022.81/291.61 min(x, 0') -> 0' 1022.81/291.61 min(s(x), s(y)) -> s(min(x, y)) 1022.81/291.61 max(0', y) -> y 1022.81/291.61 max(x, 0') -> x 1022.81/291.61 max(s(x), s(y)) -> s(max(x, y)) 1022.81/291.61 +'(0', y) -> y 1022.81/291.61 +'(s(x), y) -> s(+'(x, y)) 1022.81/291.61 -(x, 0') -> x 1022.81/291.61 -(s(x), s(y)) -> -(x, y) 1022.81/291.61 *'(x, 0') -> 0' 1022.81/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 1022.81/291.61 f(s(x)) -> f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0')))))))) 1022.81/291.61 1022.81/291.61 Types: 1022.81/291.61 min :: 0':s -> 0':s -> 0':s 1022.81/291.61 0' :: 0':s 1022.81/291.61 s :: 0':s -> 0':s 1022.81/291.61 max :: 0':s -> 0':s -> 0':s 1022.81/291.61 +' :: 0':s -> 0':s -> 0':s 1022.81/291.61 - :: 0':s -> 0':s -> 0':s 1022.81/291.61 *' :: 0':s -> 0':s -> 0':s 1022.81/291.61 f :: 0':s -> f 1022.81/291.61 hole_0':s1_0 :: 0':s 1022.81/291.61 hole_f2_0 :: f 1022.81/291.61 gen_0':s3_0 :: Nat -> 0':s 1022.81/291.61 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (5) OrderProof (LOWER BOUND(ID)) 1022.81/291.61 Heuristically decided to analyse the following defined symbols: 1022.81/291.61 min, max, +', -, *', f 1022.81/291.61 1022.81/291.61 They will be analysed ascendingly in the following order: 1022.81/291.61 max < f 1022.81/291.61 +' < *' 1022.81/291.61 +' < f 1022.81/291.61 - < f 1022.81/291.61 *' < f 1022.81/291.61 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (6) 1022.81/291.61 Obligation: 1022.81/291.61 Innermost TRS: 1022.81/291.61 Rules: 1022.81/291.61 min(0', y) -> 0' 1022.81/291.61 min(x, 0') -> 0' 1022.81/291.61 min(s(x), s(y)) -> s(min(x, y)) 1022.81/291.61 max(0', y) -> y 1022.81/291.61 max(x, 0') -> x 1022.81/291.61 max(s(x), s(y)) -> s(max(x, y)) 1022.81/291.61 +'(0', y) -> y 1022.81/291.61 +'(s(x), y) -> s(+'(x, y)) 1022.81/291.61 -(x, 0') -> x 1022.81/291.61 -(s(x), s(y)) -> -(x, y) 1022.81/291.61 *'(x, 0') -> 0' 1022.81/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 1022.81/291.61 f(s(x)) -> f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0')))))))) 1022.81/291.61 1022.81/291.61 Types: 1022.81/291.61 min :: 0':s -> 0':s -> 0':s 1022.81/291.61 0' :: 0':s 1022.81/291.61 s :: 0':s -> 0':s 1022.81/291.61 max :: 0':s -> 0':s -> 0':s 1022.81/291.61 +' :: 0':s -> 0':s -> 0':s 1022.81/291.61 - :: 0':s -> 0':s -> 0':s 1022.81/291.61 *' :: 0':s -> 0':s -> 0':s 1022.81/291.61 f :: 0':s -> f 1022.81/291.61 hole_0':s1_0 :: 0':s 1022.81/291.61 hole_f2_0 :: f 1022.81/291.61 gen_0':s3_0 :: Nat -> 0':s 1022.81/291.61 1022.81/291.61 1022.81/291.61 Generator Equations: 1022.81/291.61 gen_0':s3_0(0) <=> 0' 1022.81/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1022.81/291.61 1022.81/291.61 1022.81/291.61 The following defined symbols remain to be analysed: 1022.81/291.61 min, max, +', -, *', f 1022.81/291.61 1022.81/291.61 They will be analysed ascendingly in the following order: 1022.81/291.61 max < f 1022.81/291.61 +' < *' 1022.81/291.61 +' < f 1022.81/291.61 - < f 1022.81/291.61 *' < f 1022.81/291.61 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (7) RewriteLemmaProof (LOWER BOUND(ID)) 1022.81/291.61 Proved the following rewrite lemma: 1022.81/291.61 min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0) 1022.81/291.61 1022.81/291.61 Induction Base: 1022.81/291.61 min(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 1022.81/291.61 0' 1022.81/291.61 1022.81/291.61 Induction Step: 1022.81/291.61 min(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1) 1022.81/291.61 s(min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0))) ->_IH 1022.81/291.61 s(gen_0':s3_0(c6_0)) 1022.81/291.61 1022.81/291.61 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (8) 1022.81/291.61 Complex Obligation (BEST) 1022.81/291.61 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (9) 1022.81/291.61 Obligation: 1022.81/291.61 Proved the lower bound n^1 for the following obligation: 1022.81/291.61 1022.81/291.61 Innermost TRS: 1022.81/291.61 Rules: 1022.81/291.61 min(0', y) -> 0' 1022.81/291.61 min(x, 0') -> 0' 1022.81/291.61 min(s(x), s(y)) -> s(min(x, y)) 1022.81/291.61 max(0', y) -> y 1022.81/291.61 max(x, 0') -> x 1022.81/291.61 max(s(x), s(y)) -> s(max(x, y)) 1022.81/291.61 +'(0', y) -> y 1022.81/291.61 +'(s(x), y) -> s(+'(x, y)) 1022.81/291.61 -(x, 0') -> x 1022.81/291.61 -(s(x), s(y)) -> -(x, y) 1022.81/291.61 *'(x, 0') -> 0' 1022.81/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 1022.81/291.61 f(s(x)) -> f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0')))))))) 1022.81/291.61 1022.81/291.61 Types: 1022.81/291.61 min :: 0':s -> 0':s -> 0':s 1022.81/291.61 0' :: 0':s 1022.81/291.61 s :: 0':s -> 0':s 1022.81/291.61 max :: 0':s -> 0':s -> 0':s 1022.81/291.61 +' :: 0':s -> 0':s -> 0':s 1022.81/291.61 - :: 0':s -> 0':s -> 0':s 1022.81/291.61 *' :: 0':s -> 0':s -> 0':s 1022.81/291.61 f :: 0':s -> f 1022.81/291.61 hole_0':s1_0 :: 0':s 1022.81/291.61 hole_f2_0 :: f 1022.81/291.61 gen_0':s3_0 :: Nat -> 0':s 1022.81/291.61 1022.81/291.61 1022.81/291.61 Generator Equations: 1022.81/291.61 gen_0':s3_0(0) <=> 0' 1022.81/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1022.81/291.61 1022.81/291.61 1022.81/291.61 The following defined symbols remain to be analysed: 1022.81/291.61 min, max, +', -, *', f 1022.81/291.61 1022.81/291.61 They will be analysed ascendingly in the following order: 1022.81/291.61 max < f 1022.81/291.61 +' < *' 1022.81/291.61 +' < f 1022.81/291.61 - < f 1022.81/291.61 *' < f 1022.81/291.61 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (10) LowerBoundPropagationProof (FINISHED) 1022.81/291.61 Propagated lower bound. 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (11) 1022.81/291.61 BOUNDS(n^1, INF) 1022.81/291.61 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (12) 1022.81/291.61 Obligation: 1022.81/291.61 Innermost TRS: 1022.81/291.61 Rules: 1022.81/291.61 min(0', y) -> 0' 1022.81/291.61 min(x, 0') -> 0' 1022.81/291.61 min(s(x), s(y)) -> s(min(x, y)) 1022.81/291.61 max(0', y) -> y 1022.81/291.61 max(x, 0') -> x 1022.81/291.61 max(s(x), s(y)) -> s(max(x, y)) 1022.81/291.61 +'(0', y) -> y 1022.81/291.61 +'(s(x), y) -> s(+'(x, y)) 1022.81/291.61 -(x, 0') -> x 1022.81/291.61 -(s(x), s(y)) -> -(x, y) 1022.81/291.61 *'(x, 0') -> 0' 1022.81/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 1022.81/291.61 f(s(x)) -> f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0')))))))) 1022.81/291.61 1022.81/291.61 Types: 1022.81/291.61 min :: 0':s -> 0':s -> 0':s 1022.81/291.61 0' :: 0':s 1022.81/291.61 s :: 0':s -> 0':s 1022.81/291.61 max :: 0':s -> 0':s -> 0':s 1022.81/291.61 +' :: 0':s -> 0':s -> 0':s 1022.81/291.61 - :: 0':s -> 0':s -> 0':s 1022.81/291.61 *' :: 0':s -> 0':s -> 0':s 1022.81/291.61 f :: 0':s -> f 1022.81/291.61 hole_0':s1_0 :: 0':s 1022.81/291.61 hole_f2_0 :: f 1022.81/291.61 gen_0':s3_0 :: Nat -> 0':s 1022.81/291.61 1022.81/291.61 1022.81/291.61 Lemmas: 1022.81/291.61 min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0) 1022.81/291.61 1022.81/291.61 1022.81/291.61 Generator Equations: 1022.81/291.61 gen_0':s3_0(0) <=> 0' 1022.81/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1022.81/291.61 1022.81/291.61 1022.81/291.61 The following defined symbols remain to be analysed: 1022.81/291.61 max, +', -, *', f 1022.81/291.61 1022.81/291.61 They will be analysed ascendingly in the following order: 1022.81/291.61 max < f 1022.81/291.61 +' < *' 1022.81/291.61 +' < f 1022.81/291.61 - < f 1022.81/291.61 *' < f 1022.81/291.61 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (13) RewriteLemmaProof (LOWER BOUND(ID)) 1022.81/291.61 Proved the following rewrite lemma: 1022.81/291.61 max(gen_0':s3_0(n399_0), gen_0':s3_0(n399_0)) -> gen_0':s3_0(n399_0), rt in Omega(1 + n399_0) 1022.81/291.61 1022.81/291.61 Induction Base: 1022.81/291.61 max(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 1022.81/291.61 gen_0':s3_0(0) 1022.81/291.61 1022.81/291.61 Induction Step: 1022.81/291.61 max(gen_0':s3_0(+(n399_0, 1)), gen_0':s3_0(+(n399_0, 1))) ->_R^Omega(1) 1022.81/291.61 s(max(gen_0':s3_0(n399_0), gen_0':s3_0(n399_0))) ->_IH 1022.81/291.61 s(gen_0':s3_0(c400_0)) 1022.81/291.61 1022.81/291.61 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (14) 1022.81/291.61 Obligation: 1022.81/291.61 Innermost TRS: 1022.81/291.61 Rules: 1022.81/291.61 min(0', y) -> 0' 1022.81/291.61 min(x, 0') -> 0' 1022.81/291.61 min(s(x), s(y)) -> s(min(x, y)) 1022.81/291.61 max(0', y) -> y 1022.81/291.61 max(x, 0') -> x 1022.81/291.61 max(s(x), s(y)) -> s(max(x, y)) 1022.81/291.61 +'(0', y) -> y 1022.81/291.61 +'(s(x), y) -> s(+'(x, y)) 1022.81/291.61 -(x, 0') -> x 1022.81/291.61 -(s(x), s(y)) -> -(x, y) 1022.81/291.61 *'(x, 0') -> 0' 1022.81/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 1022.81/291.61 f(s(x)) -> f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0')))))))) 1022.81/291.61 1022.81/291.61 Types: 1022.81/291.61 min :: 0':s -> 0':s -> 0':s 1022.81/291.61 0' :: 0':s 1022.81/291.61 s :: 0':s -> 0':s 1022.81/291.61 max :: 0':s -> 0':s -> 0':s 1022.81/291.61 +' :: 0':s -> 0':s -> 0':s 1022.81/291.61 - :: 0':s -> 0':s -> 0':s 1022.81/291.61 *' :: 0':s -> 0':s -> 0':s 1022.81/291.61 f :: 0':s -> f 1022.81/291.61 hole_0':s1_0 :: 0':s 1022.81/291.61 hole_f2_0 :: f 1022.81/291.61 gen_0':s3_0 :: Nat -> 0':s 1022.81/291.61 1022.81/291.61 1022.81/291.61 Lemmas: 1022.81/291.61 min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0) 1022.81/291.61 max(gen_0':s3_0(n399_0), gen_0':s3_0(n399_0)) -> gen_0':s3_0(n399_0), rt in Omega(1 + n399_0) 1022.81/291.61 1022.81/291.61 1022.81/291.61 Generator Equations: 1022.81/291.61 gen_0':s3_0(0) <=> 0' 1022.81/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1022.81/291.61 1022.81/291.61 1022.81/291.61 The following defined symbols remain to be analysed: 1022.81/291.61 +', -, *', f 1022.81/291.61 1022.81/291.61 They will be analysed ascendingly in the following order: 1022.81/291.61 +' < *' 1022.81/291.61 +' < f 1022.81/291.61 - < f 1022.81/291.61 *' < f 1022.81/291.61 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (15) RewriteLemmaProof (LOWER BOUND(ID)) 1022.81/291.61 Proved the following rewrite lemma: 1022.81/291.61 +'(gen_0':s3_0(n901_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n901_0, b)), rt in Omega(1 + n901_0) 1022.81/291.61 1022.81/291.61 Induction Base: 1022.81/291.61 +'(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1) 1022.81/291.61 gen_0':s3_0(b) 1022.81/291.61 1022.81/291.61 Induction Step: 1022.81/291.61 +'(gen_0':s3_0(+(n901_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1) 1022.81/291.61 s(+'(gen_0':s3_0(n901_0), gen_0':s3_0(b))) ->_IH 1022.81/291.61 s(gen_0':s3_0(+(b, c902_0))) 1022.81/291.61 1022.81/291.61 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (16) 1022.81/291.61 Obligation: 1022.81/291.61 Innermost TRS: 1022.81/291.61 Rules: 1022.81/291.61 min(0', y) -> 0' 1022.81/291.61 min(x, 0') -> 0' 1022.81/291.61 min(s(x), s(y)) -> s(min(x, y)) 1022.81/291.61 max(0', y) -> y 1022.81/291.61 max(x, 0') -> x 1022.81/291.61 max(s(x), s(y)) -> s(max(x, y)) 1022.81/291.61 +'(0', y) -> y 1022.81/291.61 +'(s(x), y) -> s(+'(x, y)) 1022.81/291.61 -(x, 0') -> x 1022.81/291.61 -(s(x), s(y)) -> -(x, y) 1022.81/291.61 *'(x, 0') -> 0' 1022.81/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 1022.81/291.61 f(s(x)) -> f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0')))))))) 1022.81/291.61 1022.81/291.61 Types: 1022.81/291.61 min :: 0':s -> 0':s -> 0':s 1022.81/291.61 0' :: 0':s 1022.81/291.61 s :: 0':s -> 0':s 1022.81/291.61 max :: 0':s -> 0':s -> 0':s 1022.81/291.61 +' :: 0':s -> 0':s -> 0':s 1022.81/291.61 - :: 0':s -> 0':s -> 0':s 1022.81/291.61 *' :: 0':s -> 0':s -> 0':s 1022.81/291.61 f :: 0':s -> f 1022.81/291.61 hole_0':s1_0 :: 0':s 1022.81/291.61 hole_f2_0 :: f 1022.81/291.61 gen_0':s3_0 :: Nat -> 0':s 1022.81/291.61 1022.81/291.61 1022.81/291.61 Lemmas: 1022.81/291.61 min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0) 1022.81/291.61 max(gen_0':s3_0(n399_0), gen_0':s3_0(n399_0)) -> gen_0':s3_0(n399_0), rt in Omega(1 + n399_0) 1022.81/291.61 +'(gen_0':s3_0(n901_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n901_0, b)), rt in Omega(1 + n901_0) 1022.81/291.61 1022.81/291.61 1022.81/291.61 Generator Equations: 1022.81/291.61 gen_0':s3_0(0) <=> 0' 1022.81/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1022.81/291.61 1022.81/291.61 1022.81/291.61 The following defined symbols remain to be analysed: 1022.81/291.61 -, *', f 1022.81/291.61 1022.81/291.61 They will be analysed ascendingly in the following order: 1022.81/291.61 - < f 1022.81/291.61 *' < f 1022.81/291.61 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (17) RewriteLemmaProof (LOWER BOUND(ID)) 1022.81/291.61 Proved the following rewrite lemma: 1022.81/291.61 -(gen_0':s3_0(n1594_0), gen_0':s3_0(n1594_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1594_0) 1022.81/291.61 1022.81/291.61 Induction Base: 1022.81/291.61 -(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 1022.81/291.61 gen_0':s3_0(0) 1022.81/291.61 1022.81/291.61 Induction Step: 1022.81/291.61 -(gen_0':s3_0(+(n1594_0, 1)), gen_0':s3_0(+(n1594_0, 1))) ->_R^Omega(1) 1022.81/291.61 -(gen_0':s3_0(n1594_0), gen_0':s3_0(n1594_0)) ->_IH 1022.81/291.61 gen_0':s3_0(0) 1022.81/291.61 1022.81/291.61 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (18) 1022.81/291.61 Obligation: 1022.81/291.61 Innermost TRS: 1022.81/291.61 Rules: 1022.81/291.61 min(0', y) -> 0' 1022.81/291.61 min(x, 0') -> 0' 1022.81/291.61 min(s(x), s(y)) -> s(min(x, y)) 1022.81/291.61 max(0', y) -> y 1022.81/291.61 max(x, 0') -> x 1022.81/291.61 max(s(x), s(y)) -> s(max(x, y)) 1022.81/291.61 +'(0', y) -> y 1022.81/291.61 +'(s(x), y) -> s(+'(x, y)) 1022.81/291.61 -(x, 0') -> x 1022.81/291.61 -(s(x), s(y)) -> -(x, y) 1022.81/291.61 *'(x, 0') -> 0' 1022.81/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 1022.81/291.61 f(s(x)) -> f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0')))))))) 1022.81/291.61 1022.81/291.61 Types: 1022.81/291.61 min :: 0':s -> 0':s -> 0':s 1022.81/291.61 0' :: 0':s 1022.81/291.61 s :: 0':s -> 0':s 1022.81/291.61 max :: 0':s -> 0':s -> 0':s 1022.81/291.61 +' :: 0':s -> 0':s -> 0':s 1022.81/291.61 - :: 0':s -> 0':s -> 0':s 1022.81/291.61 *' :: 0':s -> 0':s -> 0':s 1022.81/291.61 f :: 0':s -> f 1022.81/291.61 hole_0':s1_0 :: 0':s 1022.81/291.61 hole_f2_0 :: f 1022.81/291.61 gen_0':s3_0 :: Nat -> 0':s 1022.81/291.61 1022.81/291.61 1022.81/291.61 Lemmas: 1022.81/291.61 min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0) 1022.81/291.61 max(gen_0':s3_0(n399_0), gen_0':s3_0(n399_0)) -> gen_0':s3_0(n399_0), rt in Omega(1 + n399_0) 1022.81/291.61 +'(gen_0':s3_0(n901_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n901_0, b)), rt in Omega(1 + n901_0) 1022.81/291.61 -(gen_0':s3_0(n1594_0), gen_0':s3_0(n1594_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1594_0) 1022.81/291.61 1022.81/291.61 1022.81/291.61 Generator Equations: 1022.81/291.61 gen_0':s3_0(0) <=> 0' 1022.81/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1022.81/291.61 1022.81/291.61 1022.81/291.61 The following defined symbols remain to be analysed: 1022.81/291.61 *', f 1022.81/291.61 1022.81/291.61 They will be analysed ascendingly in the following order: 1022.81/291.61 *' < f 1022.81/291.61 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (19) RewriteLemmaProof (LOWER BOUND(ID)) 1022.81/291.61 Proved the following rewrite lemma: 1022.81/291.61 *'(gen_0':s3_0(a), gen_0':s3_0(n1982_0)) -> gen_0':s3_0(*(n1982_0, a)), rt in Omega(1 + a*n1982_0 + n1982_0) 1022.81/291.61 1022.81/291.61 Induction Base: 1022.81/291.61 *'(gen_0':s3_0(a), gen_0':s3_0(0)) ->_R^Omega(1) 1022.81/291.61 0' 1022.81/291.61 1022.81/291.61 Induction Step: 1022.81/291.61 *'(gen_0':s3_0(a), gen_0':s3_0(+(n1982_0, 1))) ->_R^Omega(1) 1022.81/291.61 +'(gen_0':s3_0(a), *'(gen_0':s3_0(a), gen_0':s3_0(n1982_0))) ->_IH 1022.81/291.61 +'(gen_0':s3_0(a), gen_0':s3_0(*(c1983_0, a))) ->_L^Omega(1 + a) 1022.81/291.61 gen_0':s3_0(+(a, *(n1982_0, a))) 1022.81/291.61 1022.81/291.61 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (20) 1022.81/291.61 Complex Obligation (BEST) 1022.81/291.61 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (21) 1022.81/291.61 Obligation: 1022.81/291.61 Proved the lower bound n^2 for the following obligation: 1022.81/291.61 1022.81/291.61 Innermost TRS: 1022.81/291.61 Rules: 1022.81/291.61 min(0', y) -> 0' 1022.81/291.61 min(x, 0') -> 0' 1022.81/291.61 min(s(x), s(y)) -> s(min(x, y)) 1022.81/291.61 max(0', y) -> y 1022.81/291.61 max(x, 0') -> x 1022.81/291.61 max(s(x), s(y)) -> s(max(x, y)) 1022.81/291.61 +'(0', y) -> y 1022.81/291.61 +'(s(x), y) -> s(+'(x, y)) 1022.81/291.61 -(x, 0') -> x 1022.81/291.61 -(s(x), s(y)) -> -(x, y) 1022.81/291.61 *'(x, 0') -> 0' 1022.81/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 1022.81/291.61 f(s(x)) -> f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0')))))))) 1022.81/291.61 1022.81/291.61 Types: 1022.81/291.61 min :: 0':s -> 0':s -> 0':s 1022.81/291.61 0' :: 0':s 1022.81/291.61 s :: 0':s -> 0':s 1022.81/291.61 max :: 0':s -> 0':s -> 0':s 1022.81/291.61 +' :: 0':s -> 0':s -> 0':s 1022.81/291.61 - :: 0':s -> 0':s -> 0':s 1022.81/291.61 *' :: 0':s -> 0':s -> 0':s 1022.81/291.61 f :: 0':s -> f 1022.81/291.61 hole_0':s1_0 :: 0':s 1022.81/291.61 hole_f2_0 :: f 1022.81/291.61 gen_0':s3_0 :: Nat -> 0':s 1022.81/291.61 1022.81/291.61 1022.81/291.61 Lemmas: 1022.81/291.61 min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0) 1022.81/291.61 max(gen_0':s3_0(n399_0), gen_0':s3_0(n399_0)) -> gen_0':s3_0(n399_0), rt in Omega(1 + n399_0) 1022.81/291.61 +'(gen_0':s3_0(n901_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n901_0, b)), rt in Omega(1 + n901_0) 1022.81/291.61 -(gen_0':s3_0(n1594_0), gen_0':s3_0(n1594_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1594_0) 1022.81/291.61 1022.81/291.61 1022.81/291.61 Generator Equations: 1022.81/291.61 gen_0':s3_0(0) <=> 0' 1022.81/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1022.81/291.61 1022.81/291.61 1022.81/291.61 The following defined symbols remain to be analysed: 1022.81/291.61 *', f 1022.81/291.61 1022.81/291.61 They will be analysed ascendingly in the following order: 1022.81/291.61 *' < f 1022.81/291.61 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (22) LowerBoundPropagationProof (FINISHED) 1022.81/291.61 Propagated lower bound. 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (23) 1022.81/291.61 BOUNDS(n^2, INF) 1022.81/291.61 1022.81/291.61 ---------------------------------------- 1022.81/291.61 1022.81/291.61 (24) 1022.81/291.61 Obligation: 1022.81/291.61 Innermost TRS: 1022.81/291.61 Rules: 1022.81/291.61 min(0', y) -> 0' 1022.81/291.61 min(x, 0') -> 0' 1022.81/291.61 min(s(x), s(y)) -> s(min(x, y)) 1022.81/291.61 max(0', y) -> y 1022.81/291.61 max(x, 0') -> x 1022.81/291.61 max(s(x), s(y)) -> s(max(x, y)) 1022.81/291.61 +'(0', y) -> y 1022.81/291.61 +'(s(x), y) -> s(+'(x, y)) 1022.81/291.61 -(x, 0') -> x 1022.81/291.61 -(s(x), s(y)) -> -(x, y) 1022.81/291.61 *'(x, 0') -> 0' 1022.81/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 1022.81/291.61 f(s(x)) -> f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0')))))))) 1022.81/291.61 1022.81/291.61 Types: 1022.81/291.61 min :: 0':s -> 0':s -> 0':s 1022.81/291.61 0' :: 0':s 1022.81/291.61 s :: 0':s -> 0':s 1022.81/291.61 max :: 0':s -> 0':s -> 0':s 1022.81/291.61 +' :: 0':s -> 0':s -> 0':s 1022.81/291.61 - :: 0':s -> 0':s -> 0':s 1022.81/291.61 *' :: 0':s -> 0':s -> 0':s 1022.81/291.61 f :: 0':s -> f 1022.81/291.61 hole_0':s1_0 :: 0':s 1022.81/291.61 hole_f2_0 :: f 1022.81/291.61 gen_0':s3_0 :: Nat -> 0':s 1022.81/291.61 1022.81/291.61 1022.81/291.61 Lemmas: 1022.81/291.61 min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0) 1022.81/291.61 max(gen_0':s3_0(n399_0), gen_0':s3_0(n399_0)) -> gen_0':s3_0(n399_0), rt in Omega(1 + n399_0) 1022.81/291.61 +'(gen_0':s3_0(n901_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n901_0, b)), rt in Omega(1 + n901_0) 1022.81/291.61 -(gen_0':s3_0(n1594_0), gen_0':s3_0(n1594_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1594_0) 1022.81/291.61 *'(gen_0':s3_0(a), gen_0':s3_0(n1982_0)) -> gen_0':s3_0(*(n1982_0, a)), rt in Omega(1 + a*n1982_0 + n1982_0) 1022.81/291.61 1022.81/291.61 1022.81/291.61 Generator Equations: 1022.81/291.61 gen_0':s3_0(0) <=> 0' 1022.81/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1022.81/291.61 1022.81/291.61 1022.81/291.61 The following defined symbols remain to be analysed: 1022.81/291.61 f 1023.21/291.69 EOF