1117.86/291.62 WORST_CASE(Omega(n^2), ?) 1117.86/291.65 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1117.86/291.65 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1117.86/291.65 1117.86/291.65 1117.86/291.65 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1117.86/291.65 1117.86/291.65 (0) CpxTRS 1117.86/291.65 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1117.86/291.65 (2) CpxTRS 1117.86/291.65 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1117.86/291.65 (4) typed CpxTrs 1117.86/291.65 (5) OrderProof [LOWER BOUND(ID), 0 ms] 1117.86/291.65 (6) typed CpxTrs 1117.86/291.65 (7) RewriteLemmaProof [LOWER BOUND(ID), 321 ms] 1117.86/291.65 (8) BEST 1117.86/291.65 (9) proven lower bound 1117.86/291.65 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 1117.86/291.65 (11) BOUNDS(n^1, INF) 1117.86/291.65 (12) typed CpxTrs 1117.86/291.65 (13) RewriteLemmaProof [LOWER BOUND(ID), 73 ms] 1117.86/291.65 (14) typed CpxTrs 1117.86/291.65 (15) RewriteLemmaProof [LOWER BOUND(ID), 64 ms] 1117.86/291.65 (16) typed CpxTrs 1117.86/291.65 (17) RewriteLemmaProof [LOWER BOUND(ID), 0 ms] 1117.86/291.65 (18) typed CpxTrs 1117.86/291.65 (19) RewriteLemmaProof [LOWER BOUND(ID), 60 ms] 1117.86/291.65 (20) BEST 1117.86/291.65 (21) proven lower bound 1117.86/291.65 (22) LowerBoundPropagationProof [FINISHED, 0 ms] 1117.86/291.65 (23) BOUNDS(n^2, INF) 1117.86/291.65 (24) typed CpxTrs 1117.86/291.65 1117.86/291.65 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (0) 1117.86/291.65 Obligation: 1117.86/291.65 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1117.86/291.65 1117.86/291.65 1117.86/291.65 The TRS R consists of the following rules: 1117.86/291.65 1117.86/291.65 min(0, y) -> 0 1117.86/291.65 min(x, 0) -> 0 1117.86/291.65 min(s(x), s(y)) -> s(min(x, y)) 1117.86/291.65 max(0, y) -> y 1117.86/291.65 max(x, 0) -> x 1117.86/291.65 max(s(x), s(y)) -> s(max(x, y)) 1117.86/291.65 +(0, y) -> y 1117.86/291.65 +(s(x), y) -> s(+(x, y)) 1117.86/291.65 -(x, 0) -> x 1117.86/291.65 -(s(x), s(y)) -> -(x, y) 1117.86/291.65 *(x, 0) -> 0 1117.86/291.65 *(x, s(y)) -> +(x, *(x, y)) 1117.86/291.65 p(s(x)) -> x 1117.86/291.65 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *(s(x), s(y))) 1117.86/291.65 1117.86/291.65 S is empty. 1117.86/291.65 Rewrite Strategy: INNERMOST 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 1117.86/291.65 Renamed function symbols to avoid clashes with predefined symbol. 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (2) 1117.86/291.65 Obligation: 1117.86/291.65 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1117.86/291.65 1117.86/291.65 1117.86/291.65 The TRS R consists of the following rules: 1117.86/291.65 1117.86/291.65 min(0', y) -> 0' 1117.86/291.65 min(x, 0') -> 0' 1117.86/291.65 min(s(x), s(y)) -> s(min(x, y)) 1117.86/291.65 max(0', y) -> y 1117.86/291.65 max(x, 0') -> x 1117.86/291.65 max(s(x), s(y)) -> s(max(x, y)) 1117.86/291.65 +'(0', y) -> y 1117.86/291.65 +'(s(x), y) -> s(+'(x, y)) 1117.86/291.65 -(x, 0') -> x 1117.86/291.65 -(s(x), s(y)) -> -(x, y) 1117.86/291.65 *'(x, 0') -> 0' 1117.86/291.65 *'(x, s(y)) -> +'(x, *'(x, y)) 1117.86/291.65 p(s(x)) -> x 1117.86/291.65 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 1117.86/291.65 1117.86/291.65 S is empty. 1117.86/291.65 Rewrite Strategy: INNERMOST 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1117.86/291.65 Infered types. 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (4) 1117.86/291.65 Obligation: 1117.86/291.65 Innermost TRS: 1117.86/291.65 Rules: 1117.86/291.65 min(0', y) -> 0' 1117.86/291.65 min(x, 0') -> 0' 1117.86/291.65 min(s(x), s(y)) -> s(min(x, y)) 1117.86/291.65 max(0', y) -> y 1117.86/291.65 max(x, 0') -> x 1117.86/291.65 max(s(x), s(y)) -> s(max(x, y)) 1117.86/291.65 +'(0', y) -> y 1117.86/291.65 +'(s(x), y) -> s(+'(x, y)) 1117.86/291.65 -(x, 0') -> x 1117.86/291.65 -(s(x), s(y)) -> -(x, y) 1117.86/291.65 *'(x, 0') -> 0' 1117.86/291.65 *'(x, s(y)) -> +'(x, *'(x, y)) 1117.86/291.65 p(s(x)) -> x 1117.86/291.65 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 1117.86/291.65 1117.86/291.65 Types: 1117.86/291.65 min :: 0':s -> 0':s -> 0':s 1117.86/291.65 0' :: 0':s 1117.86/291.65 s :: 0':s -> 0':s 1117.86/291.65 max :: 0':s -> 0':s -> 0':s 1117.86/291.65 +' :: 0':s -> 0':s -> 0':s 1117.86/291.65 - :: 0':s -> 0':s -> 0':s 1117.86/291.65 *' :: 0':s -> 0':s -> 0':s 1117.86/291.65 p :: 0':s -> 0':s 1117.86/291.65 f :: 0':s -> 0':s -> f 1117.86/291.65 hole_0':s1_0 :: 0':s 1117.86/291.65 hole_f2_0 :: f 1117.86/291.65 gen_0':s3_0 :: Nat -> 0':s 1117.86/291.65 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (5) OrderProof (LOWER BOUND(ID)) 1117.86/291.65 Heuristically decided to analyse the following defined symbols: 1117.86/291.65 min, max, +', -, *', f 1117.86/291.65 1117.86/291.65 They will be analysed ascendingly in the following order: 1117.86/291.65 min < f 1117.86/291.65 max < f 1117.86/291.65 +' < *' 1117.86/291.65 - < f 1117.86/291.65 *' < f 1117.86/291.65 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (6) 1117.86/291.65 Obligation: 1117.86/291.65 Innermost TRS: 1117.86/291.65 Rules: 1117.86/291.65 min(0', y) -> 0' 1117.86/291.65 min(x, 0') -> 0' 1117.86/291.65 min(s(x), s(y)) -> s(min(x, y)) 1117.86/291.65 max(0', y) -> y 1117.86/291.65 max(x, 0') -> x 1117.86/291.65 max(s(x), s(y)) -> s(max(x, y)) 1117.86/291.65 +'(0', y) -> y 1117.86/291.65 +'(s(x), y) -> s(+'(x, y)) 1117.86/291.65 -(x, 0') -> x 1117.86/291.65 -(s(x), s(y)) -> -(x, y) 1117.86/291.65 *'(x, 0') -> 0' 1117.86/291.65 *'(x, s(y)) -> +'(x, *'(x, y)) 1117.86/291.65 p(s(x)) -> x 1117.86/291.65 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 1117.86/291.65 1117.86/291.65 Types: 1117.86/291.65 min :: 0':s -> 0':s -> 0':s 1117.86/291.65 0' :: 0':s 1117.86/291.65 s :: 0':s -> 0':s 1117.86/291.65 max :: 0':s -> 0':s -> 0':s 1117.86/291.65 +' :: 0':s -> 0':s -> 0':s 1117.86/291.65 - :: 0':s -> 0':s -> 0':s 1117.86/291.65 *' :: 0':s -> 0':s -> 0':s 1117.86/291.65 p :: 0':s -> 0':s 1117.86/291.65 f :: 0':s -> 0':s -> f 1117.86/291.65 hole_0':s1_0 :: 0':s 1117.86/291.65 hole_f2_0 :: f 1117.86/291.65 gen_0':s3_0 :: Nat -> 0':s 1117.86/291.65 1117.86/291.65 1117.86/291.65 Generator Equations: 1117.86/291.65 gen_0':s3_0(0) <=> 0' 1117.86/291.65 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1117.86/291.65 1117.86/291.65 1117.86/291.65 The following defined symbols remain to be analysed: 1117.86/291.65 min, max, +', -, *', f 1117.86/291.65 1117.86/291.65 They will be analysed ascendingly in the following order: 1117.86/291.65 min < f 1117.86/291.65 max < f 1117.86/291.65 +' < *' 1117.86/291.65 - < f 1117.86/291.65 *' < f 1117.86/291.65 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (7) RewriteLemmaProof (LOWER BOUND(ID)) 1117.86/291.65 Proved the following rewrite lemma: 1117.86/291.65 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) 1117.86/291.65 1117.86/291.65 Induction Base: 1117.86/291.65 min(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 1117.86/291.65 0' 1117.86/291.65 1117.86/291.65 Induction Step: 1117.86/291.65 min(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1) 1117.86/291.65 s(min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0))) ->_IH 1117.86/291.65 s(gen_0':s3_0(c6_0)) 1117.86/291.65 1117.86/291.65 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (8) 1117.86/291.65 Complex Obligation (BEST) 1117.86/291.65 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (9) 1117.86/291.65 Obligation: 1117.86/291.65 Proved the lower bound n^1 for the following obligation: 1117.86/291.65 1117.86/291.65 Innermost TRS: 1117.86/291.65 Rules: 1117.86/291.65 min(0', y) -> 0' 1117.86/291.65 min(x, 0') -> 0' 1117.86/291.65 min(s(x), s(y)) -> s(min(x, y)) 1117.86/291.65 max(0', y) -> y 1117.86/291.65 max(x, 0') -> x 1117.86/291.65 max(s(x), s(y)) -> s(max(x, y)) 1117.86/291.65 +'(0', y) -> y 1117.86/291.65 +'(s(x), y) -> s(+'(x, y)) 1117.86/291.65 -(x, 0') -> x 1117.86/291.65 -(s(x), s(y)) -> -(x, y) 1117.86/291.65 *'(x, 0') -> 0' 1117.86/291.65 *'(x, s(y)) -> +'(x, *'(x, y)) 1117.86/291.65 p(s(x)) -> x 1117.86/291.65 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 1117.86/291.65 1117.86/291.65 Types: 1117.86/291.65 min :: 0':s -> 0':s -> 0':s 1117.86/291.65 0' :: 0':s 1117.86/291.65 s :: 0':s -> 0':s 1117.86/291.65 max :: 0':s -> 0':s -> 0':s 1117.86/291.65 +' :: 0':s -> 0':s -> 0':s 1117.86/291.65 - :: 0':s -> 0':s -> 0':s 1117.86/291.65 *' :: 0':s -> 0':s -> 0':s 1117.86/291.65 p :: 0':s -> 0':s 1117.86/291.65 f :: 0':s -> 0':s -> f 1117.86/291.65 hole_0':s1_0 :: 0':s 1117.86/291.65 hole_f2_0 :: f 1117.86/291.65 gen_0':s3_0 :: Nat -> 0':s 1117.86/291.65 1117.86/291.65 1117.86/291.65 Generator Equations: 1117.86/291.65 gen_0':s3_0(0) <=> 0' 1117.86/291.65 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1117.86/291.65 1117.86/291.65 1117.86/291.65 The following defined symbols remain to be analysed: 1117.86/291.65 min, max, +', -, *', f 1117.86/291.65 1117.86/291.65 They will be analysed ascendingly in the following order: 1117.86/291.65 min < f 1117.86/291.65 max < f 1117.86/291.65 +' < *' 1117.86/291.65 - < f 1117.86/291.65 *' < f 1117.86/291.65 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (10) LowerBoundPropagationProof (FINISHED) 1117.86/291.65 Propagated lower bound. 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (11) 1117.86/291.65 BOUNDS(n^1, INF) 1117.86/291.65 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (12) 1117.86/291.65 Obligation: 1117.86/291.65 Innermost TRS: 1117.86/291.65 Rules: 1117.86/291.65 min(0', y) -> 0' 1117.86/291.65 min(x, 0') -> 0' 1117.86/291.65 min(s(x), s(y)) -> s(min(x, y)) 1117.86/291.65 max(0', y) -> y 1117.86/291.65 max(x, 0') -> x 1117.86/291.65 max(s(x), s(y)) -> s(max(x, y)) 1117.86/291.65 +'(0', y) -> y 1117.86/291.65 +'(s(x), y) -> s(+'(x, y)) 1117.86/291.65 -(x, 0') -> x 1117.86/291.65 -(s(x), s(y)) -> -(x, y) 1117.86/291.65 *'(x, 0') -> 0' 1117.86/291.65 *'(x, s(y)) -> +'(x, *'(x, y)) 1117.86/291.65 p(s(x)) -> x 1117.86/291.65 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 1117.86/291.65 1117.86/291.65 Types: 1117.86/291.65 min :: 0':s -> 0':s -> 0':s 1117.86/291.65 0' :: 0':s 1117.86/291.65 s :: 0':s -> 0':s 1117.86/291.65 max :: 0':s -> 0':s -> 0':s 1117.86/291.65 +' :: 0':s -> 0':s -> 0':s 1117.86/291.65 - :: 0':s -> 0':s -> 0':s 1117.86/291.65 *' :: 0':s -> 0':s -> 0':s 1117.86/291.65 p :: 0':s -> 0':s 1117.86/291.65 f :: 0':s -> 0':s -> f 1117.86/291.65 hole_0':s1_0 :: 0':s 1117.86/291.65 hole_f2_0 :: f 1117.86/291.65 gen_0':s3_0 :: Nat -> 0':s 1117.86/291.65 1117.86/291.65 1117.86/291.65 Lemmas: 1117.86/291.65 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) 1117.86/291.65 1117.86/291.65 1117.86/291.65 Generator Equations: 1117.86/291.65 gen_0':s3_0(0) <=> 0' 1117.86/291.65 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1117.86/291.65 1117.86/291.65 1117.86/291.65 The following defined symbols remain to be analysed: 1117.86/291.65 max, +', -, *', f 1117.86/291.65 1117.86/291.65 They will be analysed ascendingly in the following order: 1117.86/291.65 max < f 1117.86/291.65 +' < *' 1117.86/291.65 - < f 1117.86/291.65 *' < f 1117.86/291.65 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (13) RewriteLemmaProof (LOWER BOUND(ID)) 1117.86/291.65 Proved the following rewrite lemma: 1117.86/291.65 max(gen_0':s3_0(n413_0), gen_0':s3_0(n413_0)) -> gen_0':s3_0(n413_0), rt in Omega(1 + n413_0) 1117.86/291.65 1117.86/291.65 Induction Base: 1117.86/291.65 max(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 1117.86/291.65 gen_0':s3_0(0) 1117.86/291.65 1117.86/291.65 Induction Step: 1117.86/291.65 max(gen_0':s3_0(+(n413_0, 1)), gen_0':s3_0(+(n413_0, 1))) ->_R^Omega(1) 1117.86/291.65 s(max(gen_0':s3_0(n413_0), gen_0':s3_0(n413_0))) ->_IH 1117.86/291.65 s(gen_0':s3_0(c414_0)) 1117.86/291.65 1117.86/291.65 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (14) 1117.86/291.65 Obligation: 1117.86/291.65 Innermost TRS: 1117.86/291.65 Rules: 1117.86/291.65 min(0', y) -> 0' 1117.86/291.65 min(x, 0') -> 0' 1117.86/291.65 min(s(x), s(y)) -> s(min(x, y)) 1117.86/291.65 max(0', y) -> y 1117.86/291.65 max(x, 0') -> x 1117.86/291.65 max(s(x), s(y)) -> s(max(x, y)) 1117.86/291.65 +'(0', y) -> y 1117.86/291.65 +'(s(x), y) -> s(+'(x, y)) 1117.86/291.65 -(x, 0') -> x 1117.86/291.65 -(s(x), s(y)) -> -(x, y) 1117.86/291.65 *'(x, 0') -> 0' 1117.86/291.65 *'(x, s(y)) -> +'(x, *'(x, y)) 1117.86/291.65 p(s(x)) -> x 1117.86/291.65 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 1117.86/291.65 1117.86/291.65 Types: 1117.86/291.65 min :: 0':s -> 0':s -> 0':s 1117.86/291.65 0' :: 0':s 1117.86/291.65 s :: 0':s -> 0':s 1117.86/291.65 max :: 0':s -> 0':s -> 0':s 1117.86/291.65 +' :: 0':s -> 0':s -> 0':s 1117.86/291.65 - :: 0':s -> 0':s -> 0':s 1117.86/291.65 *' :: 0':s -> 0':s -> 0':s 1117.86/291.65 p :: 0':s -> 0':s 1117.86/291.65 f :: 0':s -> 0':s -> f 1117.86/291.65 hole_0':s1_0 :: 0':s 1117.86/291.65 hole_f2_0 :: f 1117.86/291.65 gen_0':s3_0 :: Nat -> 0':s 1117.86/291.65 1117.86/291.65 1117.86/291.65 Lemmas: 1117.86/291.65 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) 1117.86/291.65 max(gen_0':s3_0(n413_0), gen_0':s3_0(n413_0)) -> gen_0':s3_0(n413_0), rt in Omega(1 + n413_0) 1117.86/291.65 1117.86/291.65 1117.86/291.65 Generator Equations: 1117.86/291.65 gen_0':s3_0(0) <=> 0' 1117.86/291.65 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1117.86/291.65 1117.86/291.65 1117.86/291.65 The following defined symbols remain to be analysed: 1117.86/291.65 +', -, *', f 1117.86/291.65 1117.86/291.65 They will be analysed ascendingly in the following order: 1117.86/291.65 +' < *' 1117.86/291.65 - < f 1117.86/291.65 *' < f 1117.86/291.65 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (15) RewriteLemmaProof (LOWER BOUND(ID)) 1117.86/291.65 Proved the following rewrite lemma: 1117.86/291.65 +'(gen_0':s3_0(n933_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n933_0, b)), rt in Omega(1 + n933_0) 1117.86/291.65 1117.86/291.65 Induction Base: 1117.86/291.65 +'(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1) 1117.86/291.65 gen_0':s3_0(b) 1117.86/291.65 1117.86/291.65 Induction Step: 1117.86/291.65 +'(gen_0':s3_0(+(n933_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1) 1117.86/291.65 s(+'(gen_0':s3_0(n933_0), gen_0':s3_0(b))) ->_IH 1117.86/291.65 s(gen_0':s3_0(+(b, c934_0))) 1117.86/291.65 1117.86/291.65 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (16) 1117.86/291.65 Obligation: 1117.86/291.65 Innermost TRS: 1117.86/291.65 Rules: 1117.86/291.65 min(0', y) -> 0' 1117.86/291.65 min(x, 0') -> 0' 1117.86/291.65 min(s(x), s(y)) -> s(min(x, y)) 1117.86/291.65 max(0', y) -> y 1117.86/291.65 max(x, 0') -> x 1117.86/291.65 max(s(x), s(y)) -> s(max(x, y)) 1117.86/291.65 +'(0', y) -> y 1117.86/291.65 +'(s(x), y) -> s(+'(x, y)) 1117.86/291.65 -(x, 0') -> x 1117.86/291.65 -(s(x), s(y)) -> -(x, y) 1117.86/291.65 *'(x, 0') -> 0' 1117.86/291.65 *'(x, s(y)) -> +'(x, *'(x, y)) 1117.86/291.65 p(s(x)) -> x 1117.86/291.65 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 1117.86/291.65 1117.86/291.65 Types: 1117.86/291.65 min :: 0':s -> 0':s -> 0':s 1117.86/291.65 0' :: 0':s 1117.86/291.65 s :: 0':s -> 0':s 1117.86/291.65 max :: 0':s -> 0':s -> 0':s 1117.86/291.65 +' :: 0':s -> 0':s -> 0':s 1117.86/291.65 - :: 0':s -> 0':s -> 0':s 1117.86/291.65 *' :: 0':s -> 0':s -> 0':s 1117.86/291.65 p :: 0':s -> 0':s 1117.86/291.65 f :: 0':s -> 0':s -> f 1117.86/291.65 hole_0':s1_0 :: 0':s 1117.86/291.65 hole_f2_0 :: f 1117.86/291.65 gen_0':s3_0 :: Nat -> 0':s 1117.86/291.65 1117.86/291.65 1117.86/291.65 Lemmas: 1117.86/291.65 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) 1117.86/291.65 max(gen_0':s3_0(n413_0), gen_0':s3_0(n413_0)) -> gen_0':s3_0(n413_0), rt in Omega(1 + n413_0) 1117.86/291.65 +'(gen_0':s3_0(n933_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n933_0, b)), rt in Omega(1 + n933_0) 1117.86/291.65 1117.86/291.65 1117.86/291.65 Generator Equations: 1117.86/291.65 gen_0':s3_0(0) <=> 0' 1117.86/291.65 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1117.86/291.65 1117.86/291.65 1117.86/291.65 The following defined symbols remain to be analysed: 1117.86/291.65 -, *', f 1117.86/291.65 1117.86/291.65 They will be analysed ascendingly in the following order: 1117.86/291.65 - < f 1117.86/291.65 *' < f 1117.86/291.65 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (17) RewriteLemmaProof (LOWER BOUND(ID)) 1117.86/291.65 Proved the following rewrite lemma: 1117.86/291.65 -(gen_0':s3_0(n1652_0), gen_0':s3_0(n1652_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1652_0) 1117.86/291.65 1117.86/291.65 Induction Base: 1117.86/291.65 -(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 1117.86/291.65 gen_0':s3_0(0) 1117.86/291.65 1117.86/291.65 Induction Step: 1117.86/291.65 -(gen_0':s3_0(+(n1652_0, 1)), gen_0':s3_0(+(n1652_0, 1))) ->_R^Omega(1) 1117.86/291.65 -(gen_0':s3_0(n1652_0), gen_0':s3_0(n1652_0)) ->_IH 1117.86/291.65 gen_0':s3_0(0) 1117.86/291.65 1117.86/291.65 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (18) 1117.86/291.65 Obligation: 1117.86/291.65 Innermost TRS: 1117.86/291.65 Rules: 1117.86/291.65 min(0', y) -> 0' 1117.86/291.65 min(x, 0') -> 0' 1117.86/291.65 min(s(x), s(y)) -> s(min(x, y)) 1117.86/291.65 max(0', y) -> y 1117.86/291.65 max(x, 0') -> x 1117.86/291.65 max(s(x), s(y)) -> s(max(x, y)) 1117.86/291.65 +'(0', y) -> y 1117.86/291.65 +'(s(x), y) -> s(+'(x, y)) 1117.86/291.65 -(x, 0') -> x 1117.86/291.65 -(s(x), s(y)) -> -(x, y) 1117.86/291.65 *'(x, 0') -> 0' 1117.86/291.65 *'(x, s(y)) -> +'(x, *'(x, y)) 1117.86/291.65 p(s(x)) -> x 1117.86/291.65 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 1117.86/291.65 1117.86/291.65 Types: 1117.86/291.65 min :: 0':s -> 0':s -> 0':s 1117.86/291.65 0' :: 0':s 1117.86/291.65 s :: 0':s -> 0':s 1117.86/291.65 max :: 0':s -> 0':s -> 0':s 1117.86/291.65 +' :: 0':s -> 0':s -> 0':s 1117.86/291.65 - :: 0':s -> 0':s -> 0':s 1117.86/291.65 *' :: 0':s -> 0':s -> 0':s 1117.86/291.65 p :: 0':s -> 0':s 1117.86/291.65 f :: 0':s -> 0':s -> f 1117.86/291.65 hole_0':s1_0 :: 0':s 1117.86/291.65 hole_f2_0 :: f 1117.86/291.65 gen_0':s3_0 :: Nat -> 0':s 1117.86/291.65 1117.86/291.65 1117.86/291.65 Lemmas: 1117.86/291.65 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) 1117.86/291.65 max(gen_0':s3_0(n413_0), gen_0':s3_0(n413_0)) -> gen_0':s3_0(n413_0), rt in Omega(1 + n413_0) 1117.86/291.65 +'(gen_0':s3_0(n933_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n933_0, b)), rt in Omega(1 + n933_0) 1117.86/291.65 -(gen_0':s3_0(n1652_0), gen_0':s3_0(n1652_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1652_0) 1117.86/291.65 1117.86/291.65 1117.86/291.65 Generator Equations: 1117.86/291.65 gen_0':s3_0(0) <=> 0' 1117.86/291.65 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1117.86/291.65 1117.86/291.65 1117.86/291.65 The following defined symbols remain to be analysed: 1117.86/291.65 *', f 1117.86/291.65 1117.86/291.65 They will be analysed ascendingly in the following order: 1117.86/291.65 *' < f 1117.86/291.65 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (19) RewriteLemmaProof (LOWER BOUND(ID)) 1117.86/291.65 Proved the following rewrite lemma: 1117.86/291.65 *'(gen_0':s3_0(a), gen_0':s3_0(n2056_0)) -> gen_0':s3_0(*(n2056_0, a)), rt in Omega(1 + a*n2056_0 + n2056_0) 1117.86/291.65 1117.86/291.65 Induction Base: 1117.86/291.65 *'(gen_0':s3_0(a), gen_0':s3_0(0)) ->_R^Omega(1) 1117.86/291.65 0' 1117.86/291.65 1117.86/291.65 Induction Step: 1117.86/291.65 *'(gen_0':s3_0(a), gen_0':s3_0(+(n2056_0, 1))) ->_R^Omega(1) 1117.86/291.65 +'(gen_0':s3_0(a), *'(gen_0':s3_0(a), gen_0':s3_0(n2056_0))) ->_IH 1117.86/291.65 +'(gen_0':s3_0(a), gen_0':s3_0(*(c2057_0, a))) ->_L^Omega(1 + a) 1117.86/291.65 gen_0':s3_0(+(a, *(n2056_0, a))) 1117.86/291.65 1117.86/291.65 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (20) 1117.86/291.65 Complex Obligation (BEST) 1117.86/291.65 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (21) 1117.86/291.65 Obligation: 1117.86/291.65 Proved the lower bound n^2 for the following obligation: 1117.86/291.65 1117.86/291.65 Innermost TRS: 1117.86/291.65 Rules: 1117.86/291.65 min(0', y) -> 0' 1117.86/291.65 min(x, 0') -> 0' 1117.86/291.65 min(s(x), s(y)) -> s(min(x, y)) 1117.86/291.65 max(0', y) -> y 1117.86/291.65 max(x, 0') -> x 1117.86/291.65 max(s(x), s(y)) -> s(max(x, y)) 1117.86/291.65 +'(0', y) -> y 1117.86/291.65 +'(s(x), y) -> s(+'(x, y)) 1117.86/291.65 -(x, 0') -> x 1117.86/291.65 -(s(x), s(y)) -> -(x, y) 1117.86/291.65 *'(x, 0') -> 0' 1117.86/291.65 *'(x, s(y)) -> +'(x, *'(x, y)) 1117.86/291.65 p(s(x)) -> x 1117.86/291.65 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 1117.86/291.65 1117.86/291.65 Types: 1117.86/291.65 min :: 0':s -> 0':s -> 0':s 1117.86/291.65 0' :: 0':s 1117.86/291.65 s :: 0':s -> 0':s 1117.86/291.65 max :: 0':s -> 0':s -> 0':s 1117.86/291.65 +' :: 0':s -> 0':s -> 0':s 1117.86/291.65 - :: 0':s -> 0':s -> 0':s 1117.86/291.65 *' :: 0':s -> 0':s -> 0':s 1117.86/291.65 p :: 0':s -> 0':s 1117.86/291.65 f :: 0':s -> 0':s -> f 1117.86/291.65 hole_0':s1_0 :: 0':s 1117.86/291.65 hole_f2_0 :: f 1117.86/291.65 gen_0':s3_0 :: Nat -> 0':s 1117.86/291.65 1117.86/291.65 1117.86/291.65 Lemmas: 1117.86/291.65 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) 1117.86/291.65 max(gen_0':s3_0(n413_0), gen_0':s3_0(n413_0)) -> gen_0':s3_0(n413_0), rt in Omega(1 + n413_0) 1117.86/291.65 +'(gen_0':s3_0(n933_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n933_0, b)), rt in Omega(1 + n933_0) 1117.86/291.65 -(gen_0':s3_0(n1652_0), gen_0':s3_0(n1652_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1652_0) 1117.86/291.65 1117.86/291.65 1117.86/291.65 Generator Equations: 1117.86/291.65 gen_0':s3_0(0) <=> 0' 1117.86/291.65 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1117.86/291.65 1117.86/291.65 1117.86/291.65 The following defined symbols remain to be analysed: 1117.86/291.65 *', f 1117.86/291.65 1117.86/291.65 They will be analysed ascendingly in the following order: 1117.86/291.65 *' < f 1117.86/291.65 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (22) LowerBoundPropagationProof (FINISHED) 1117.86/291.65 Propagated lower bound. 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (23) 1117.86/291.65 BOUNDS(n^2, INF) 1117.86/291.65 1117.86/291.65 ---------------------------------------- 1117.86/291.65 1117.86/291.65 (24) 1117.86/291.65 Obligation: 1117.86/291.65 Innermost TRS: 1117.86/291.65 Rules: 1117.86/291.65 min(0', y) -> 0' 1117.86/291.65 min(x, 0') -> 0' 1117.86/291.65 min(s(x), s(y)) -> s(min(x, y)) 1117.86/291.65 max(0', y) -> y 1117.86/291.65 max(x, 0') -> x 1117.86/291.65 max(s(x), s(y)) -> s(max(x, y)) 1117.86/291.65 +'(0', y) -> y 1117.86/291.65 +'(s(x), y) -> s(+'(x, y)) 1117.86/291.65 -(x, 0') -> x 1117.86/291.65 -(s(x), s(y)) -> -(x, y) 1117.86/291.65 *'(x, 0') -> 0' 1117.86/291.65 *'(x, s(y)) -> +'(x, *'(x, y)) 1117.86/291.65 p(s(x)) -> x 1117.86/291.65 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 1117.86/291.65 1117.86/291.65 Types: 1117.86/291.65 min :: 0':s -> 0':s -> 0':s 1117.86/291.65 0' :: 0':s 1117.86/291.65 s :: 0':s -> 0':s 1117.86/291.65 max :: 0':s -> 0':s -> 0':s 1117.86/291.65 +' :: 0':s -> 0':s -> 0':s 1117.86/291.65 - :: 0':s -> 0':s -> 0':s 1117.86/291.65 *' :: 0':s -> 0':s -> 0':s 1117.86/291.65 p :: 0':s -> 0':s 1117.86/291.65 f :: 0':s -> 0':s -> f 1117.86/291.65 hole_0':s1_0 :: 0':s 1117.86/291.65 hole_f2_0 :: f 1117.86/291.65 gen_0':s3_0 :: Nat -> 0':s 1117.86/291.65 1117.86/291.65 1117.86/291.65 Lemmas: 1117.86/291.65 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) 1117.86/291.65 max(gen_0':s3_0(n413_0), gen_0':s3_0(n413_0)) -> gen_0':s3_0(n413_0), rt in Omega(1 + n413_0) 1117.86/291.65 +'(gen_0':s3_0(n933_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n933_0, b)), rt in Omega(1 + n933_0) 1117.86/291.65 -(gen_0':s3_0(n1652_0), gen_0':s3_0(n1652_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1652_0) 1117.86/291.65 *'(gen_0':s3_0(a), gen_0':s3_0(n2056_0)) -> gen_0':s3_0(*(n2056_0, a)), rt in Omega(1 + a*n2056_0 + n2056_0) 1117.86/291.65 1117.86/291.65 1117.86/291.65 Generator Equations: 1117.86/291.65 gen_0':s3_0(0) <=> 0' 1117.86/291.65 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1117.86/291.65 1117.86/291.65 1117.86/291.65 The following defined symbols remain to be analysed: 1117.86/291.65 f 1118.22/291.77 EOF