1117.62/291.99 WORST_CASE(Omega(n^1), ?) 1117.62/292.04 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1117.62/292.04 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1117.62/292.04 1117.62/292.04 1117.62/292.04 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1117.62/292.04 1117.62/292.04 (0) CpxTRS 1117.62/292.04 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1117.62/292.04 (2) CpxTRS 1117.62/292.04 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1117.62/292.04 (4) typed CpxTrs 1117.62/292.04 (5) OrderProof [LOWER BOUND(ID), 0 ms] 1117.62/292.04 (6) typed CpxTrs 1117.62/292.04 (7) RewriteLemmaProof [LOWER BOUND(ID), 319 ms] 1117.62/292.04 (8) BEST 1117.62/292.04 (9) proven lower bound 1117.62/292.04 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 1117.62/292.04 (11) BOUNDS(n^1, INF) 1117.62/292.04 (12) typed CpxTrs 1117.62/292.04 (13) RewriteLemmaProof [LOWER BOUND(ID), 92 ms] 1117.62/292.04 (14) typed CpxTrs 1117.62/292.04 (15) RewriteLemmaProof [LOWER BOUND(ID), 77 ms] 1117.62/292.04 (16) typed CpxTrs 1117.62/292.04 (17) RewriteLemmaProof [LOWER BOUND(ID), 33 ms] 1117.62/292.04 (18) typed CpxTrs 1117.62/292.04 1117.62/292.04 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (0) 1117.62/292.04 Obligation: 1117.62/292.04 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1117.62/292.04 1117.62/292.04 1117.62/292.04 The TRS R consists of the following rules: 1117.62/292.04 1117.62/292.04 min(0, y) -> 0 1117.62/292.04 min(x, 0) -> 0 1117.62/292.04 min(s(x), s(y)) -> s(min(x, y)) 1117.62/292.04 max(0, y) -> y 1117.62/292.04 max(x, 0) -> x 1117.62/292.04 max(s(x), s(y)) -> s(max(x, y)) 1117.62/292.04 twice(0) -> 0 1117.62/292.04 twice(s(x)) -> s(s(twice(x))) 1117.62/292.04 -(x, 0) -> x 1117.62/292.04 -(s(x), s(y)) -> -(x, y) 1117.62/292.04 p(s(x)) -> x 1117.62/292.04 f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) 1117.62/292.04 1117.62/292.04 S is empty. 1117.62/292.04 Rewrite Strategy: INNERMOST 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 1117.62/292.04 Renamed function symbols to avoid clashes with predefined symbol. 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (2) 1117.62/292.04 Obligation: 1117.62/292.04 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1117.62/292.04 1117.62/292.04 1117.62/292.04 The TRS R consists of the following rules: 1117.62/292.04 1117.62/292.04 min(0', y) -> 0' 1117.62/292.04 min(x, 0') -> 0' 1117.62/292.04 min(s(x), s(y)) -> s(min(x, y)) 1117.62/292.04 max(0', y) -> y 1117.62/292.04 max(x, 0') -> x 1117.62/292.04 max(s(x), s(y)) -> s(max(x, y)) 1117.62/292.04 twice(0') -> 0' 1117.62/292.04 twice(s(x)) -> s(s(twice(x))) 1117.62/292.04 -(x, 0') -> x 1117.62/292.04 -(s(x), s(y)) -> -(x, y) 1117.62/292.04 p(s(x)) -> x 1117.62/292.04 f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) 1117.62/292.04 1117.62/292.04 S is empty. 1117.62/292.04 Rewrite Strategy: INNERMOST 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1117.62/292.04 Infered types. 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (4) 1117.62/292.04 Obligation: 1117.62/292.04 Innermost TRS: 1117.62/292.04 Rules: 1117.62/292.04 min(0', y) -> 0' 1117.62/292.04 min(x, 0') -> 0' 1117.62/292.04 min(s(x), s(y)) -> s(min(x, y)) 1117.62/292.04 max(0', y) -> y 1117.62/292.04 max(x, 0') -> x 1117.62/292.04 max(s(x), s(y)) -> s(max(x, y)) 1117.62/292.04 twice(0') -> 0' 1117.62/292.04 twice(s(x)) -> s(s(twice(x))) 1117.62/292.04 -(x, 0') -> x 1117.62/292.04 -(s(x), s(y)) -> -(x, y) 1117.62/292.04 p(s(x)) -> x 1117.62/292.04 f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) 1117.62/292.04 1117.62/292.04 Types: 1117.62/292.04 min :: 0':s -> 0':s -> 0':s 1117.62/292.04 0' :: 0':s 1117.62/292.04 s :: 0':s -> 0':s 1117.62/292.04 max :: 0':s -> 0':s -> 0':s 1117.62/292.04 twice :: 0':s -> 0':s 1117.62/292.04 - :: 0':s -> 0':s -> 0':s 1117.62/292.04 p :: 0':s -> 0':s 1117.62/292.04 f :: 0':s -> 0':s -> f 1117.62/292.04 hole_0':s1_0 :: 0':s 1117.62/292.04 hole_f2_0 :: f 1117.62/292.04 gen_0':s3_0 :: Nat -> 0':s 1117.62/292.04 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (5) OrderProof (LOWER BOUND(ID)) 1117.62/292.04 Heuristically decided to analyse the following defined symbols: 1117.62/292.04 min, max, twice, -, f 1117.62/292.04 1117.62/292.04 They will be analysed ascendingly in the following order: 1117.62/292.04 min < f 1117.62/292.04 max < f 1117.62/292.04 twice < f 1117.62/292.04 - < f 1117.62/292.04 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (6) 1117.62/292.04 Obligation: 1117.62/292.04 Innermost TRS: 1117.62/292.04 Rules: 1117.62/292.04 min(0', y) -> 0' 1117.62/292.04 min(x, 0') -> 0' 1117.62/292.04 min(s(x), s(y)) -> s(min(x, y)) 1117.62/292.04 max(0', y) -> y 1117.62/292.04 max(x, 0') -> x 1117.62/292.04 max(s(x), s(y)) -> s(max(x, y)) 1117.62/292.04 twice(0') -> 0' 1117.62/292.04 twice(s(x)) -> s(s(twice(x))) 1117.62/292.04 -(x, 0') -> x 1117.62/292.04 -(s(x), s(y)) -> -(x, y) 1117.62/292.04 p(s(x)) -> x 1117.62/292.04 f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) 1117.62/292.04 1117.62/292.04 Types: 1117.62/292.04 min :: 0':s -> 0':s -> 0':s 1117.62/292.04 0' :: 0':s 1117.62/292.04 s :: 0':s -> 0':s 1117.62/292.04 max :: 0':s -> 0':s -> 0':s 1117.62/292.04 twice :: 0':s -> 0':s 1117.62/292.04 - :: 0':s -> 0':s -> 0':s 1117.62/292.04 p :: 0':s -> 0':s 1117.62/292.04 f :: 0':s -> 0':s -> f 1117.62/292.04 hole_0':s1_0 :: 0':s 1117.62/292.04 hole_f2_0 :: f 1117.62/292.04 gen_0':s3_0 :: Nat -> 0':s 1117.62/292.04 1117.62/292.04 1117.62/292.04 Generator Equations: 1117.62/292.04 gen_0':s3_0(0) <=> 0' 1117.62/292.04 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1117.62/292.04 1117.62/292.04 1117.62/292.04 The following defined symbols remain to be analysed: 1117.62/292.04 min, max, twice, -, f 1117.62/292.04 1117.62/292.04 They will be analysed ascendingly in the following order: 1117.62/292.04 min < f 1117.62/292.04 max < f 1117.62/292.04 twice < f 1117.62/292.04 - < f 1117.62/292.04 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (7) RewriteLemmaProof (LOWER BOUND(ID)) 1117.62/292.04 Proved the following rewrite lemma: 1117.62/292.04 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) 1117.62/292.04 1117.62/292.04 Induction Base: 1117.62/292.04 min(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 1117.62/292.04 0' 1117.62/292.04 1117.62/292.04 Induction Step: 1117.62/292.04 min(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1) 1117.62/292.04 s(min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0))) ->_IH 1117.62/292.04 s(gen_0':s3_0(c6_0)) 1117.62/292.04 1117.62/292.04 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (8) 1117.62/292.04 Complex Obligation (BEST) 1117.62/292.04 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (9) 1117.62/292.04 Obligation: 1117.62/292.04 Proved the lower bound n^1 for the following obligation: 1117.62/292.04 1117.62/292.04 Innermost TRS: 1117.62/292.04 Rules: 1117.62/292.04 min(0', y) -> 0' 1117.62/292.04 min(x, 0') -> 0' 1117.62/292.04 min(s(x), s(y)) -> s(min(x, y)) 1117.62/292.04 max(0', y) -> y 1117.62/292.04 max(x, 0') -> x 1117.62/292.04 max(s(x), s(y)) -> s(max(x, y)) 1117.62/292.04 twice(0') -> 0' 1117.62/292.04 twice(s(x)) -> s(s(twice(x))) 1117.62/292.04 -(x, 0') -> x 1117.62/292.04 -(s(x), s(y)) -> -(x, y) 1117.62/292.04 p(s(x)) -> x 1117.62/292.04 f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) 1117.62/292.04 1117.62/292.04 Types: 1117.62/292.04 min :: 0':s -> 0':s -> 0':s 1117.62/292.04 0' :: 0':s 1117.62/292.04 s :: 0':s -> 0':s 1117.62/292.04 max :: 0':s -> 0':s -> 0':s 1117.62/292.04 twice :: 0':s -> 0':s 1117.62/292.04 - :: 0':s -> 0':s -> 0':s 1117.62/292.04 p :: 0':s -> 0':s 1117.62/292.04 f :: 0':s -> 0':s -> f 1117.62/292.04 hole_0':s1_0 :: 0':s 1117.62/292.04 hole_f2_0 :: f 1117.62/292.04 gen_0':s3_0 :: Nat -> 0':s 1117.62/292.04 1117.62/292.04 1117.62/292.04 Generator Equations: 1117.62/292.04 gen_0':s3_0(0) <=> 0' 1117.62/292.04 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1117.62/292.04 1117.62/292.04 1117.62/292.04 The following defined symbols remain to be analysed: 1117.62/292.04 min, max, twice, -, f 1117.62/292.04 1117.62/292.04 They will be analysed ascendingly in the following order: 1117.62/292.04 min < f 1117.62/292.04 max < f 1117.62/292.04 twice < f 1117.62/292.04 - < f 1117.62/292.04 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (10) LowerBoundPropagationProof (FINISHED) 1117.62/292.04 Propagated lower bound. 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (11) 1117.62/292.04 BOUNDS(n^1, INF) 1117.62/292.04 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (12) 1117.62/292.04 Obligation: 1117.62/292.04 Innermost TRS: 1117.62/292.04 Rules: 1117.62/292.04 min(0', y) -> 0' 1117.62/292.04 min(x, 0') -> 0' 1117.62/292.04 min(s(x), s(y)) -> s(min(x, y)) 1117.62/292.04 max(0', y) -> y 1117.62/292.04 max(x, 0') -> x 1117.62/292.04 max(s(x), s(y)) -> s(max(x, y)) 1117.62/292.04 twice(0') -> 0' 1117.62/292.04 twice(s(x)) -> s(s(twice(x))) 1117.62/292.04 -(x, 0') -> x 1117.62/292.04 -(s(x), s(y)) -> -(x, y) 1117.62/292.04 p(s(x)) -> x 1117.62/292.04 f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) 1117.62/292.04 1117.62/292.04 Types: 1117.62/292.04 min :: 0':s -> 0':s -> 0':s 1117.62/292.04 0' :: 0':s 1117.62/292.04 s :: 0':s -> 0':s 1117.62/292.04 max :: 0':s -> 0':s -> 0':s 1117.62/292.04 twice :: 0':s -> 0':s 1117.62/292.04 - :: 0':s -> 0':s -> 0':s 1117.62/292.04 p :: 0':s -> 0':s 1117.62/292.04 f :: 0':s -> 0':s -> f 1117.62/292.04 hole_0':s1_0 :: 0':s 1117.62/292.04 hole_f2_0 :: f 1117.62/292.04 gen_0':s3_0 :: Nat -> 0':s 1117.62/292.04 1117.62/292.04 1117.62/292.04 Lemmas: 1117.62/292.04 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) 1117.62/292.04 1117.62/292.04 1117.62/292.04 Generator Equations: 1117.62/292.04 gen_0':s3_0(0) <=> 0' 1117.62/292.04 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1117.62/292.04 1117.62/292.04 1117.62/292.04 The following defined symbols remain to be analysed: 1117.62/292.04 max, twice, -, f 1117.62/292.04 1117.62/292.04 They will be analysed ascendingly in the following order: 1117.62/292.04 max < f 1117.62/292.04 twice < f 1117.62/292.04 - < f 1117.62/292.04 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (13) RewriteLemmaProof (LOWER BOUND(ID)) 1117.62/292.04 Proved the following rewrite lemma: 1117.62/292.04 max(gen_0':s3_0(n385_0), gen_0':s3_0(n385_0)) -> gen_0':s3_0(n385_0), rt in Omega(1 + n385_0) 1117.62/292.04 1117.62/292.04 Induction Base: 1117.62/292.04 max(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 1117.62/292.04 gen_0':s3_0(0) 1117.62/292.04 1117.62/292.04 Induction Step: 1117.62/292.04 max(gen_0':s3_0(+(n385_0, 1)), gen_0':s3_0(+(n385_0, 1))) ->_R^Omega(1) 1117.62/292.04 s(max(gen_0':s3_0(n385_0), gen_0':s3_0(n385_0))) ->_IH 1117.62/292.04 s(gen_0':s3_0(c386_0)) 1117.62/292.04 1117.62/292.04 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (14) 1117.62/292.04 Obligation: 1117.62/292.04 Innermost TRS: 1117.62/292.04 Rules: 1117.62/292.04 min(0', y) -> 0' 1117.62/292.04 min(x, 0') -> 0' 1117.62/292.04 min(s(x), s(y)) -> s(min(x, y)) 1117.62/292.04 max(0', y) -> y 1117.62/292.04 max(x, 0') -> x 1117.62/292.04 max(s(x), s(y)) -> s(max(x, y)) 1117.62/292.04 twice(0') -> 0' 1117.62/292.04 twice(s(x)) -> s(s(twice(x))) 1117.62/292.04 -(x, 0') -> x 1117.62/292.04 -(s(x), s(y)) -> -(x, y) 1117.62/292.04 p(s(x)) -> x 1117.62/292.04 f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) 1117.62/292.04 1117.62/292.04 Types: 1117.62/292.04 min :: 0':s -> 0':s -> 0':s 1117.62/292.04 0' :: 0':s 1117.62/292.04 s :: 0':s -> 0':s 1117.62/292.04 max :: 0':s -> 0':s -> 0':s 1117.62/292.04 twice :: 0':s -> 0':s 1117.62/292.04 - :: 0':s -> 0':s -> 0':s 1117.62/292.04 p :: 0':s -> 0':s 1117.62/292.04 f :: 0':s -> 0':s -> f 1117.62/292.04 hole_0':s1_0 :: 0':s 1117.62/292.04 hole_f2_0 :: f 1117.62/292.04 gen_0':s3_0 :: Nat -> 0':s 1117.62/292.04 1117.62/292.04 1117.62/292.04 Lemmas: 1117.62/292.04 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) 1117.62/292.04 max(gen_0':s3_0(n385_0), gen_0':s3_0(n385_0)) -> gen_0':s3_0(n385_0), rt in Omega(1 + n385_0) 1117.62/292.04 1117.62/292.04 1117.62/292.04 Generator Equations: 1117.62/292.04 gen_0':s3_0(0) <=> 0' 1117.62/292.04 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1117.62/292.04 1117.62/292.04 1117.62/292.04 The following defined symbols remain to be analysed: 1117.62/292.04 twice, -, f 1117.62/292.04 1117.62/292.04 They will be analysed ascendingly in the following order: 1117.62/292.04 twice < f 1117.62/292.04 - < f 1117.62/292.04 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (15) RewriteLemmaProof (LOWER BOUND(ID)) 1117.62/292.04 Proved the following rewrite lemma: 1117.62/292.04 twice(gen_0':s3_0(n869_0)) -> gen_0':s3_0(*(2, n869_0)), rt in Omega(1 + n869_0) 1117.62/292.04 1117.62/292.04 Induction Base: 1117.62/292.04 twice(gen_0':s3_0(0)) ->_R^Omega(1) 1117.62/292.04 0' 1117.62/292.04 1117.62/292.04 Induction Step: 1117.62/292.04 twice(gen_0':s3_0(+(n869_0, 1))) ->_R^Omega(1) 1117.62/292.04 s(s(twice(gen_0':s3_0(n869_0)))) ->_IH 1117.62/292.04 s(s(gen_0':s3_0(*(2, c870_0)))) 1117.62/292.04 1117.62/292.04 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (16) 1117.62/292.04 Obligation: 1117.62/292.04 Innermost TRS: 1117.62/292.04 Rules: 1117.62/292.04 min(0', y) -> 0' 1117.62/292.04 min(x, 0') -> 0' 1117.62/292.04 min(s(x), s(y)) -> s(min(x, y)) 1117.62/292.04 max(0', y) -> y 1117.62/292.04 max(x, 0') -> x 1117.62/292.04 max(s(x), s(y)) -> s(max(x, y)) 1117.62/292.04 twice(0') -> 0' 1117.62/292.04 twice(s(x)) -> s(s(twice(x))) 1117.62/292.04 -(x, 0') -> x 1117.62/292.04 -(s(x), s(y)) -> -(x, y) 1117.62/292.04 p(s(x)) -> x 1117.62/292.04 f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) 1117.62/292.04 1117.62/292.04 Types: 1117.62/292.04 min :: 0':s -> 0':s -> 0':s 1117.62/292.04 0' :: 0':s 1117.62/292.04 s :: 0':s -> 0':s 1117.62/292.04 max :: 0':s -> 0':s -> 0':s 1117.62/292.04 twice :: 0':s -> 0':s 1117.62/292.04 - :: 0':s -> 0':s -> 0':s 1117.62/292.04 p :: 0':s -> 0':s 1117.62/292.04 f :: 0':s -> 0':s -> f 1117.62/292.04 hole_0':s1_0 :: 0':s 1117.62/292.04 hole_f2_0 :: f 1117.62/292.04 gen_0':s3_0 :: Nat -> 0':s 1117.62/292.04 1117.62/292.04 1117.62/292.04 Lemmas: 1117.62/292.04 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) 1117.62/292.04 max(gen_0':s3_0(n385_0), gen_0':s3_0(n385_0)) -> gen_0':s3_0(n385_0), rt in Omega(1 + n385_0) 1117.62/292.04 twice(gen_0':s3_0(n869_0)) -> gen_0':s3_0(*(2, n869_0)), rt in Omega(1 + n869_0) 1117.62/292.04 1117.62/292.04 1117.62/292.04 Generator Equations: 1117.62/292.04 gen_0':s3_0(0) <=> 0' 1117.62/292.04 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1117.62/292.04 1117.62/292.04 1117.62/292.04 The following defined symbols remain to be analysed: 1117.62/292.04 -, f 1117.62/292.04 1117.62/292.04 They will be analysed ascendingly in the following order: 1117.62/292.04 - < f 1117.62/292.04 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (17) RewriteLemmaProof (LOWER BOUND(ID)) 1117.62/292.04 Proved the following rewrite lemma: 1117.62/292.04 -(gen_0':s3_0(n1173_0), gen_0':s3_0(n1173_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1173_0) 1117.62/292.04 1117.62/292.04 Induction Base: 1117.62/292.04 -(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 1117.62/292.04 gen_0':s3_0(0) 1117.62/292.04 1117.62/292.04 Induction Step: 1117.62/292.04 -(gen_0':s3_0(+(n1173_0, 1)), gen_0':s3_0(+(n1173_0, 1))) ->_R^Omega(1) 1117.62/292.04 -(gen_0':s3_0(n1173_0), gen_0':s3_0(n1173_0)) ->_IH 1117.62/292.04 gen_0':s3_0(0) 1117.62/292.04 1117.62/292.04 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1117.62/292.04 ---------------------------------------- 1117.62/292.04 1117.62/292.04 (18) 1117.62/292.04 Obligation: 1117.62/292.04 Innermost TRS: 1117.62/292.04 Rules: 1117.62/292.04 min(0', y) -> 0' 1117.62/292.04 min(x, 0') -> 0' 1117.62/292.04 min(s(x), s(y)) -> s(min(x, y)) 1117.62/292.04 max(0', y) -> y 1117.62/292.04 max(x, 0') -> x 1117.62/292.04 max(s(x), s(y)) -> s(max(x, y)) 1117.62/292.04 twice(0') -> 0' 1117.62/292.04 twice(s(x)) -> s(s(twice(x))) 1117.62/292.04 -(x, 0') -> x 1117.62/292.04 -(s(x), s(y)) -> -(x, y) 1117.62/292.04 p(s(x)) -> x 1117.62/292.04 f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) 1117.62/292.04 1117.62/292.04 Types: 1117.62/292.04 min :: 0':s -> 0':s -> 0':s 1117.62/292.04 0' :: 0':s 1117.62/292.04 s :: 0':s -> 0':s 1117.62/292.04 max :: 0':s -> 0':s -> 0':s 1117.62/292.04 twice :: 0':s -> 0':s 1117.62/292.04 - :: 0':s -> 0':s -> 0':s 1117.62/292.04 p :: 0':s -> 0':s 1117.62/292.04 f :: 0':s -> 0':s -> f 1117.62/292.04 hole_0':s1_0 :: 0':s 1117.62/292.04 hole_f2_0 :: f 1117.62/292.04 gen_0':s3_0 :: Nat -> 0':s 1117.62/292.04 1117.62/292.04 1117.62/292.04 Lemmas: 1117.62/292.04 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) 1117.62/292.04 max(gen_0':s3_0(n385_0), gen_0':s3_0(n385_0)) -> gen_0':s3_0(n385_0), rt in Omega(1 + n385_0) 1117.62/292.04 twice(gen_0':s3_0(n869_0)) -> gen_0':s3_0(*(2, n869_0)), rt in Omega(1 + n869_0) 1117.62/292.04 -(gen_0':s3_0(n1173_0), gen_0':s3_0(n1173_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1173_0) 1117.62/292.04 1117.62/292.04 1117.62/292.04 Generator Equations: 1117.62/292.04 gen_0':s3_0(0) <=> 0' 1117.62/292.04 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1117.62/292.04 1117.62/292.04 1117.62/292.04 The following defined symbols remain to be analysed: 1117.62/292.04 f 1118.05/293.86 EOF