312.08/291.60 WORST_CASE(Omega(n^2), ?) 312.08/291.61 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 312.08/291.61 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 312.08/291.61 312.08/291.61 312.08/291.61 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 312.08/291.61 312.08/291.61 (0) CpxTRS 312.08/291.61 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 312.08/291.61 (2) CpxTRS 312.08/291.61 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 312.08/291.61 (4) typed CpxTrs 312.08/291.61 (5) OrderProof [LOWER BOUND(ID), 0 ms] 312.08/291.61 (6) typed CpxTrs 312.08/291.61 (7) RewriteLemmaProof [LOWER BOUND(ID), 334 ms] 312.08/291.61 (8) BEST 312.08/291.61 (9) proven lower bound 312.08/291.61 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 312.08/291.61 (11) BOUNDS(n^1, INF) 312.08/291.61 (12) typed CpxTrs 312.08/291.61 (13) RewriteLemmaProof [LOWER BOUND(ID), 36 ms] 312.08/291.61 (14) typed CpxTrs 312.08/291.61 (15) RewriteLemmaProof [LOWER BOUND(ID), 40 ms] 312.08/291.61 (16) typed CpxTrs 312.08/291.61 (17) RewriteLemmaProof [LOWER BOUND(ID), 0 ms] 312.08/291.61 (18) typed CpxTrs 312.08/291.61 (19) RewriteLemmaProof [LOWER BOUND(ID), 38 ms] 312.08/291.61 (20) BEST 312.08/291.61 (21) proven lower bound 312.08/291.61 (22) LowerBoundPropagationProof [FINISHED, 0 ms] 312.08/291.61 (23) BOUNDS(n^2, INF) 312.08/291.61 (24) typed CpxTrs 312.08/291.61 312.08/291.61 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (0) 312.08/291.61 Obligation: 312.08/291.61 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 312.08/291.61 312.08/291.61 312.08/291.61 The TRS R consists of the following rules: 312.08/291.61 312.08/291.61 min(0, y) -> 0 312.08/291.61 min(x, 0) -> 0 312.08/291.61 min(s(x), s(y)) -> s(min(x, y)) 312.08/291.61 max(0, y) -> y 312.08/291.61 max(x, 0) -> x 312.08/291.61 max(s(x), s(y)) -> s(max(x, y)) 312.08/291.61 +(0, y) -> y 312.08/291.61 +(s(x), y) -> s(+(x, y)) 312.08/291.61 -(x, 0) -> x 312.08/291.61 -(s(x), s(y)) -> -(x, y) 312.08/291.61 *(x, 0) -> 0 312.08/291.61 *(x, s(y)) -> +(x, *(x, y)) 312.08/291.61 p(s(x)) -> x 312.08/291.61 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *(s(x), s(y))) 312.08/291.61 312.08/291.61 S is empty. 312.08/291.61 Rewrite Strategy: FULL 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 312.08/291.61 Renamed function symbols to avoid clashes with predefined symbol. 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (2) 312.08/291.61 Obligation: 312.08/291.61 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 312.08/291.61 312.08/291.61 312.08/291.61 The TRS R consists of the following rules: 312.08/291.61 312.08/291.61 min(0', y) -> 0' 312.08/291.61 min(x, 0') -> 0' 312.08/291.61 min(s(x), s(y)) -> s(min(x, y)) 312.08/291.61 max(0', y) -> y 312.08/291.61 max(x, 0') -> x 312.08/291.61 max(s(x), s(y)) -> s(max(x, y)) 312.08/291.61 +'(0', y) -> y 312.08/291.61 +'(s(x), y) -> s(+'(x, y)) 312.08/291.61 -(x, 0') -> x 312.08/291.61 -(s(x), s(y)) -> -(x, y) 312.08/291.61 *'(x, 0') -> 0' 312.08/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 312.08/291.61 p(s(x)) -> x 312.08/291.61 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 312.08/291.61 312.08/291.61 S is empty. 312.08/291.61 Rewrite Strategy: FULL 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 312.08/291.61 Infered types. 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (4) 312.08/291.61 Obligation: 312.08/291.61 TRS: 312.08/291.61 Rules: 312.08/291.61 min(0', y) -> 0' 312.08/291.61 min(x, 0') -> 0' 312.08/291.61 min(s(x), s(y)) -> s(min(x, y)) 312.08/291.61 max(0', y) -> y 312.08/291.61 max(x, 0') -> x 312.08/291.61 max(s(x), s(y)) -> s(max(x, y)) 312.08/291.61 +'(0', y) -> y 312.08/291.61 +'(s(x), y) -> s(+'(x, y)) 312.08/291.61 -(x, 0') -> x 312.08/291.61 -(s(x), s(y)) -> -(x, y) 312.08/291.61 *'(x, 0') -> 0' 312.08/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 312.08/291.61 p(s(x)) -> x 312.08/291.61 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 312.08/291.61 312.08/291.61 Types: 312.08/291.61 min :: 0':s -> 0':s -> 0':s 312.08/291.61 0' :: 0':s 312.08/291.61 s :: 0':s -> 0':s 312.08/291.61 max :: 0':s -> 0':s -> 0':s 312.08/291.61 +' :: 0':s -> 0':s -> 0':s 312.08/291.61 - :: 0':s -> 0':s -> 0':s 312.08/291.61 *' :: 0':s -> 0':s -> 0':s 312.08/291.61 p :: 0':s -> 0':s 312.08/291.61 f :: 0':s -> 0':s -> f 312.08/291.61 hole_0':s1_0 :: 0':s 312.08/291.61 hole_f2_0 :: f 312.08/291.61 gen_0':s3_0 :: Nat -> 0':s 312.08/291.61 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (5) OrderProof (LOWER BOUND(ID)) 312.08/291.61 Heuristically decided to analyse the following defined symbols: 312.08/291.61 min, max, +', -, *', f 312.08/291.61 312.08/291.61 They will be analysed ascendingly in the following order: 312.08/291.61 min < f 312.08/291.61 max < f 312.08/291.61 +' < *' 312.08/291.61 - < f 312.08/291.61 *' < f 312.08/291.61 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (6) 312.08/291.61 Obligation: 312.08/291.61 TRS: 312.08/291.61 Rules: 312.08/291.61 min(0', y) -> 0' 312.08/291.61 min(x, 0') -> 0' 312.08/291.61 min(s(x), s(y)) -> s(min(x, y)) 312.08/291.61 max(0', y) -> y 312.08/291.61 max(x, 0') -> x 312.08/291.61 max(s(x), s(y)) -> s(max(x, y)) 312.08/291.61 +'(0', y) -> y 312.08/291.61 +'(s(x), y) -> s(+'(x, y)) 312.08/291.61 -(x, 0') -> x 312.08/291.61 -(s(x), s(y)) -> -(x, y) 312.08/291.61 *'(x, 0') -> 0' 312.08/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 312.08/291.61 p(s(x)) -> x 312.08/291.61 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 312.08/291.61 312.08/291.61 Types: 312.08/291.61 min :: 0':s -> 0':s -> 0':s 312.08/291.61 0' :: 0':s 312.08/291.61 s :: 0':s -> 0':s 312.08/291.61 max :: 0':s -> 0':s -> 0':s 312.08/291.61 +' :: 0':s -> 0':s -> 0':s 312.08/291.61 - :: 0':s -> 0':s -> 0':s 312.08/291.61 *' :: 0':s -> 0':s -> 0':s 312.08/291.61 p :: 0':s -> 0':s 312.08/291.61 f :: 0':s -> 0':s -> f 312.08/291.61 hole_0':s1_0 :: 0':s 312.08/291.61 hole_f2_0 :: f 312.08/291.61 gen_0':s3_0 :: Nat -> 0':s 312.08/291.61 312.08/291.61 312.08/291.61 Generator Equations: 312.08/291.61 gen_0':s3_0(0) <=> 0' 312.08/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 312.08/291.61 312.08/291.61 312.08/291.61 The following defined symbols remain to be analysed: 312.08/291.61 min, max, +', -, *', f 312.08/291.61 312.08/291.61 They will be analysed ascendingly in the following order: 312.08/291.61 min < f 312.08/291.61 max < f 312.08/291.61 +' < *' 312.08/291.61 - < f 312.08/291.61 *' < f 312.08/291.61 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (7) RewriteLemmaProof (LOWER BOUND(ID)) 312.08/291.61 Proved the following rewrite lemma: 312.08/291.61 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) 312.08/291.61 312.08/291.61 Induction Base: 312.08/291.61 min(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 312.08/291.61 0' 312.08/291.61 312.08/291.61 Induction Step: 312.08/291.61 min(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1) 312.08/291.61 s(min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0))) ->_IH 312.08/291.61 s(gen_0':s3_0(c6_0)) 312.08/291.61 312.08/291.61 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (8) 312.08/291.61 Complex Obligation (BEST) 312.08/291.61 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (9) 312.08/291.61 Obligation: 312.08/291.61 Proved the lower bound n^1 for the following obligation: 312.08/291.61 312.08/291.61 TRS: 312.08/291.61 Rules: 312.08/291.61 min(0', y) -> 0' 312.08/291.61 min(x, 0') -> 0' 312.08/291.61 min(s(x), s(y)) -> s(min(x, y)) 312.08/291.61 max(0', y) -> y 312.08/291.61 max(x, 0') -> x 312.08/291.61 max(s(x), s(y)) -> s(max(x, y)) 312.08/291.61 +'(0', y) -> y 312.08/291.61 +'(s(x), y) -> s(+'(x, y)) 312.08/291.61 -(x, 0') -> x 312.08/291.61 -(s(x), s(y)) -> -(x, y) 312.08/291.61 *'(x, 0') -> 0' 312.08/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 312.08/291.61 p(s(x)) -> x 312.08/291.61 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 312.08/291.61 312.08/291.61 Types: 312.08/291.61 min :: 0':s -> 0':s -> 0':s 312.08/291.61 0' :: 0':s 312.08/291.61 s :: 0':s -> 0':s 312.08/291.61 max :: 0':s -> 0':s -> 0':s 312.08/291.61 +' :: 0':s -> 0':s -> 0':s 312.08/291.61 - :: 0':s -> 0':s -> 0':s 312.08/291.61 *' :: 0':s -> 0':s -> 0':s 312.08/291.61 p :: 0':s -> 0':s 312.08/291.61 f :: 0':s -> 0':s -> f 312.08/291.61 hole_0':s1_0 :: 0':s 312.08/291.61 hole_f2_0 :: f 312.08/291.61 gen_0':s3_0 :: Nat -> 0':s 312.08/291.61 312.08/291.61 312.08/291.61 Generator Equations: 312.08/291.61 gen_0':s3_0(0) <=> 0' 312.08/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 312.08/291.61 312.08/291.61 312.08/291.61 The following defined symbols remain to be analysed: 312.08/291.61 min, max, +', -, *', f 312.08/291.61 312.08/291.61 They will be analysed ascendingly in the following order: 312.08/291.61 min < f 312.08/291.61 max < f 312.08/291.61 +' < *' 312.08/291.61 - < f 312.08/291.61 *' < f 312.08/291.61 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (10) LowerBoundPropagationProof (FINISHED) 312.08/291.61 Propagated lower bound. 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (11) 312.08/291.61 BOUNDS(n^1, INF) 312.08/291.61 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (12) 312.08/291.61 Obligation: 312.08/291.61 TRS: 312.08/291.61 Rules: 312.08/291.61 min(0', y) -> 0' 312.08/291.61 min(x, 0') -> 0' 312.08/291.61 min(s(x), s(y)) -> s(min(x, y)) 312.08/291.61 max(0', y) -> y 312.08/291.61 max(x, 0') -> x 312.08/291.61 max(s(x), s(y)) -> s(max(x, y)) 312.08/291.61 +'(0', y) -> y 312.08/291.61 +'(s(x), y) -> s(+'(x, y)) 312.08/291.61 -(x, 0') -> x 312.08/291.61 -(s(x), s(y)) -> -(x, y) 312.08/291.61 *'(x, 0') -> 0' 312.08/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 312.08/291.61 p(s(x)) -> x 312.08/291.61 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 312.08/291.61 312.08/291.61 Types: 312.08/291.61 min :: 0':s -> 0':s -> 0':s 312.08/291.61 0' :: 0':s 312.08/291.61 s :: 0':s -> 0':s 312.08/291.61 max :: 0':s -> 0':s -> 0':s 312.08/291.61 +' :: 0':s -> 0':s -> 0':s 312.08/291.61 - :: 0':s -> 0':s -> 0':s 312.08/291.61 *' :: 0':s -> 0':s -> 0':s 312.08/291.61 p :: 0':s -> 0':s 312.08/291.61 f :: 0':s -> 0':s -> f 312.08/291.61 hole_0':s1_0 :: 0':s 312.08/291.61 hole_f2_0 :: f 312.08/291.61 gen_0':s3_0 :: Nat -> 0':s 312.08/291.61 312.08/291.61 312.08/291.61 Lemmas: 312.08/291.61 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) 312.08/291.61 312.08/291.61 312.08/291.61 Generator Equations: 312.08/291.61 gen_0':s3_0(0) <=> 0' 312.08/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 312.08/291.61 312.08/291.61 312.08/291.61 The following defined symbols remain to be analysed: 312.08/291.61 max, +', -, *', f 312.08/291.61 312.08/291.61 They will be analysed ascendingly in the following order: 312.08/291.61 max < f 312.08/291.61 +' < *' 312.08/291.61 - < f 312.08/291.61 *' < f 312.08/291.61 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (13) RewriteLemmaProof (LOWER BOUND(ID)) 312.08/291.61 Proved the following rewrite lemma: 312.08/291.61 max(gen_0':s3_0(n322_0), gen_0':s3_0(n322_0)) -> gen_0':s3_0(n322_0), rt in Omega(1 + n322_0) 312.08/291.61 312.08/291.61 Induction Base: 312.08/291.61 max(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 312.08/291.61 gen_0':s3_0(0) 312.08/291.61 312.08/291.61 Induction Step: 312.08/291.61 max(gen_0':s3_0(+(n322_0, 1)), gen_0':s3_0(+(n322_0, 1))) ->_R^Omega(1) 312.08/291.61 s(max(gen_0':s3_0(n322_0), gen_0':s3_0(n322_0))) ->_IH 312.08/291.61 s(gen_0':s3_0(c323_0)) 312.08/291.61 312.08/291.61 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (14) 312.08/291.61 Obligation: 312.08/291.61 TRS: 312.08/291.61 Rules: 312.08/291.61 min(0', y) -> 0' 312.08/291.61 min(x, 0') -> 0' 312.08/291.61 min(s(x), s(y)) -> s(min(x, y)) 312.08/291.61 max(0', y) -> y 312.08/291.61 max(x, 0') -> x 312.08/291.61 max(s(x), s(y)) -> s(max(x, y)) 312.08/291.61 +'(0', y) -> y 312.08/291.61 +'(s(x), y) -> s(+'(x, y)) 312.08/291.61 -(x, 0') -> x 312.08/291.61 -(s(x), s(y)) -> -(x, y) 312.08/291.61 *'(x, 0') -> 0' 312.08/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 312.08/291.61 p(s(x)) -> x 312.08/291.61 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 312.08/291.61 312.08/291.61 Types: 312.08/291.61 min :: 0':s -> 0':s -> 0':s 312.08/291.61 0' :: 0':s 312.08/291.61 s :: 0':s -> 0':s 312.08/291.61 max :: 0':s -> 0':s -> 0':s 312.08/291.61 +' :: 0':s -> 0':s -> 0':s 312.08/291.61 - :: 0':s -> 0':s -> 0':s 312.08/291.61 *' :: 0':s -> 0':s -> 0':s 312.08/291.61 p :: 0':s -> 0':s 312.08/291.61 f :: 0':s -> 0':s -> f 312.08/291.61 hole_0':s1_0 :: 0':s 312.08/291.61 hole_f2_0 :: f 312.08/291.61 gen_0':s3_0 :: Nat -> 0':s 312.08/291.61 312.08/291.61 312.08/291.61 Lemmas: 312.08/291.61 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) 312.08/291.61 max(gen_0':s3_0(n322_0), gen_0':s3_0(n322_0)) -> gen_0':s3_0(n322_0), rt in Omega(1 + n322_0) 312.08/291.61 312.08/291.61 312.08/291.61 Generator Equations: 312.08/291.61 gen_0':s3_0(0) <=> 0' 312.08/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 312.08/291.61 312.08/291.61 312.08/291.61 The following defined symbols remain to be analysed: 312.08/291.61 +', -, *', f 312.08/291.61 312.08/291.61 They will be analysed ascendingly in the following order: 312.08/291.61 +' < *' 312.08/291.61 - < f 312.08/291.61 *' < f 312.08/291.61 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (15) RewriteLemmaProof (LOWER BOUND(ID)) 312.08/291.61 Proved the following rewrite lemma: 312.08/291.61 +'(gen_0':s3_0(n725_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n725_0, b)), rt in Omega(1 + n725_0) 312.08/291.61 312.08/291.61 Induction Base: 312.08/291.61 +'(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1) 312.08/291.61 gen_0':s3_0(b) 312.08/291.61 312.08/291.61 Induction Step: 312.08/291.61 +'(gen_0':s3_0(+(n725_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1) 312.08/291.61 s(+'(gen_0':s3_0(n725_0), gen_0':s3_0(b))) ->_IH 312.08/291.61 s(gen_0':s3_0(+(b, c726_0))) 312.08/291.61 312.08/291.61 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (16) 312.08/291.61 Obligation: 312.08/291.61 TRS: 312.08/291.61 Rules: 312.08/291.61 min(0', y) -> 0' 312.08/291.61 min(x, 0') -> 0' 312.08/291.61 min(s(x), s(y)) -> s(min(x, y)) 312.08/291.61 max(0', y) -> y 312.08/291.61 max(x, 0') -> x 312.08/291.61 max(s(x), s(y)) -> s(max(x, y)) 312.08/291.61 +'(0', y) -> y 312.08/291.61 +'(s(x), y) -> s(+'(x, y)) 312.08/291.61 -(x, 0') -> x 312.08/291.61 -(s(x), s(y)) -> -(x, y) 312.08/291.61 *'(x, 0') -> 0' 312.08/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 312.08/291.61 p(s(x)) -> x 312.08/291.61 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 312.08/291.61 312.08/291.61 Types: 312.08/291.61 min :: 0':s -> 0':s -> 0':s 312.08/291.61 0' :: 0':s 312.08/291.61 s :: 0':s -> 0':s 312.08/291.61 max :: 0':s -> 0':s -> 0':s 312.08/291.61 +' :: 0':s -> 0':s -> 0':s 312.08/291.61 - :: 0':s -> 0':s -> 0':s 312.08/291.61 *' :: 0':s -> 0':s -> 0':s 312.08/291.61 p :: 0':s -> 0':s 312.08/291.61 f :: 0':s -> 0':s -> f 312.08/291.61 hole_0':s1_0 :: 0':s 312.08/291.61 hole_f2_0 :: f 312.08/291.61 gen_0':s3_0 :: Nat -> 0':s 312.08/291.61 312.08/291.61 312.08/291.61 Lemmas: 312.08/291.61 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) 312.08/291.61 max(gen_0':s3_0(n322_0), gen_0':s3_0(n322_0)) -> gen_0':s3_0(n322_0), rt in Omega(1 + n322_0) 312.08/291.61 +'(gen_0':s3_0(n725_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n725_0, b)), rt in Omega(1 + n725_0) 312.08/291.61 312.08/291.61 312.08/291.61 Generator Equations: 312.08/291.61 gen_0':s3_0(0) <=> 0' 312.08/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 312.08/291.61 312.08/291.61 312.08/291.61 The following defined symbols remain to be analysed: 312.08/291.61 -, *', f 312.08/291.61 312.08/291.61 They will be analysed ascendingly in the following order: 312.08/291.61 - < f 312.08/291.61 *' < f 312.08/291.61 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (17) RewriteLemmaProof (LOWER BOUND(ID)) 312.08/291.61 Proved the following rewrite lemma: 312.08/291.61 -(gen_0':s3_0(n1340_0), gen_0':s3_0(n1340_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1340_0) 312.08/291.61 312.08/291.61 Induction Base: 312.08/291.61 -(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 312.08/291.61 gen_0':s3_0(0) 312.08/291.61 312.08/291.61 Induction Step: 312.08/291.61 -(gen_0':s3_0(+(n1340_0, 1)), gen_0':s3_0(+(n1340_0, 1))) ->_R^Omega(1) 312.08/291.61 -(gen_0':s3_0(n1340_0), gen_0':s3_0(n1340_0)) ->_IH 312.08/291.61 gen_0':s3_0(0) 312.08/291.61 312.08/291.61 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 312.08/291.61 ---------------------------------------- 312.08/291.61 312.08/291.61 (18) 312.08/291.61 Obligation: 312.08/291.61 TRS: 312.08/291.61 Rules: 312.08/291.61 min(0', y) -> 0' 312.08/291.62 min(x, 0') -> 0' 312.08/291.62 min(s(x), s(y)) -> s(min(x, y)) 312.08/291.62 max(0', y) -> y 312.08/291.62 max(x, 0') -> x 312.08/291.62 max(s(x), s(y)) -> s(max(x, y)) 312.08/291.62 +'(0', y) -> y 312.08/291.62 +'(s(x), y) -> s(+'(x, y)) 312.08/291.62 -(x, 0') -> x 312.08/291.62 -(s(x), s(y)) -> -(x, y) 312.08/291.62 *'(x, 0') -> 0' 312.08/291.62 *'(x, s(y)) -> +'(x, *'(x, y)) 312.08/291.62 p(s(x)) -> x 312.08/291.62 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 312.08/291.62 312.08/291.62 Types: 312.08/291.62 min :: 0':s -> 0':s -> 0':s 312.08/291.62 0' :: 0':s 312.08/291.62 s :: 0':s -> 0':s 312.08/291.62 max :: 0':s -> 0':s -> 0':s 312.08/291.62 +' :: 0':s -> 0':s -> 0':s 312.08/291.62 - :: 0':s -> 0':s -> 0':s 312.08/291.62 *' :: 0':s -> 0':s -> 0':s 312.08/291.62 p :: 0':s -> 0':s 312.08/291.62 f :: 0':s -> 0':s -> f 312.08/291.62 hole_0':s1_0 :: 0':s 312.08/291.62 hole_f2_0 :: f 312.08/291.62 gen_0':s3_0 :: Nat -> 0':s 312.08/291.62 312.08/291.62 312.08/291.62 Lemmas: 312.08/291.62 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) 312.08/291.62 max(gen_0':s3_0(n322_0), gen_0':s3_0(n322_0)) -> gen_0':s3_0(n322_0), rt in Omega(1 + n322_0) 312.08/291.62 +'(gen_0':s3_0(n725_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n725_0, b)), rt in Omega(1 + n725_0) 312.08/291.62 -(gen_0':s3_0(n1340_0), gen_0':s3_0(n1340_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1340_0) 312.08/291.62 312.08/291.62 312.08/291.62 Generator Equations: 312.08/291.62 gen_0':s3_0(0) <=> 0' 312.08/291.62 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 312.08/291.62 312.08/291.62 312.08/291.62 The following defined symbols remain to be analysed: 312.08/291.62 *', f 312.08/291.62 312.08/291.62 They will be analysed ascendingly in the following order: 312.08/291.62 *' < f 312.08/291.62 312.08/291.62 ---------------------------------------- 312.08/291.62 312.08/291.62 (19) RewriteLemmaProof (LOWER BOUND(ID)) 312.08/291.62 Proved the following rewrite lemma: 312.08/291.62 *'(gen_0':s3_0(a), gen_0':s3_0(n1640_0)) -> gen_0':s3_0(*(n1640_0, a)), rt in Omega(1 + a*n1640_0 + n1640_0) 312.08/291.62 312.08/291.62 Induction Base: 312.08/291.62 *'(gen_0':s3_0(a), gen_0':s3_0(0)) ->_R^Omega(1) 312.08/291.62 0' 312.08/291.62 312.08/291.62 Induction Step: 312.08/291.62 *'(gen_0':s3_0(a), gen_0':s3_0(+(n1640_0, 1))) ->_R^Omega(1) 312.08/291.62 +'(gen_0':s3_0(a), *'(gen_0':s3_0(a), gen_0':s3_0(n1640_0))) ->_IH 312.08/291.62 +'(gen_0':s3_0(a), gen_0':s3_0(*(c1641_0, a))) ->_L^Omega(1 + a) 312.08/291.62 gen_0':s3_0(+(a, *(n1640_0, a))) 312.08/291.62 312.08/291.62 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 312.08/291.62 ---------------------------------------- 312.08/291.62 312.08/291.62 (20) 312.08/291.62 Complex Obligation (BEST) 312.08/291.62 312.08/291.62 ---------------------------------------- 312.08/291.62 312.08/291.62 (21) 312.08/291.62 Obligation: 312.08/291.62 Proved the lower bound n^2 for the following obligation: 312.08/291.62 312.08/291.62 TRS: 312.08/291.62 Rules: 312.08/291.62 min(0', y) -> 0' 312.08/291.62 min(x, 0') -> 0' 312.08/291.62 min(s(x), s(y)) -> s(min(x, y)) 312.08/291.62 max(0', y) -> y 312.08/291.62 max(x, 0') -> x 312.08/291.62 max(s(x), s(y)) -> s(max(x, y)) 312.08/291.62 +'(0', y) -> y 312.08/291.62 +'(s(x), y) -> s(+'(x, y)) 312.08/291.62 -(x, 0') -> x 312.08/291.62 -(s(x), s(y)) -> -(x, y) 312.08/291.62 *'(x, 0') -> 0' 312.08/291.62 *'(x, s(y)) -> +'(x, *'(x, y)) 312.08/291.62 p(s(x)) -> x 312.08/291.62 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 312.08/291.62 312.08/291.62 Types: 312.08/291.62 min :: 0':s -> 0':s -> 0':s 312.08/291.62 0' :: 0':s 312.08/291.62 s :: 0':s -> 0':s 312.08/291.62 max :: 0':s -> 0':s -> 0':s 312.08/291.62 +' :: 0':s -> 0':s -> 0':s 312.08/291.62 - :: 0':s -> 0':s -> 0':s 312.08/291.62 *' :: 0':s -> 0':s -> 0':s 312.08/291.62 p :: 0':s -> 0':s 312.08/291.62 f :: 0':s -> 0':s -> f 312.08/291.62 hole_0':s1_0 :: 0':s 312.08/291.62 hole_f2_0 :: f 312.08/291.62 gen_0':s3_0 :: Nat -> 0':s 312.08/291.62 312.08/291.62 312.08/291.62 Lemmas: 312.08/291.62 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) 312.08/291.62 max(gen_0':s3_0(n322_0), gen_0':s3_0(n322_0)) -> gen_0':s3_0(n322_0), rt in Omega(1 + n322_0) 312.08/291.62 +'(gen_0':s3_0(n725_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n725_0, b)), rt in Omega(1 + n725_0) 312.08/291.62 -(gen_0':s3_0(n1340_0), gen_0':s3_0(n1340_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1340_0) 312.08/291.62 312.08/291.62 312.08/291.62 Generator Equations: 312.08/291.62 gen_0':s3_0(0) <=> 0' 312.08/291.62 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 312.08/291.62 312.08/291.62 312.08/291.62 The following defined symbols remain to be analysed: 312.08/291.62 *', f 312.08/291.62 312.08/291.62 They will be analysed ascendingly in the following order: 312.08/291.62 *' < f 312.08/291.62 312.08/291.62 ---------------------------------------- 312.08/291.62 312.08/291.62 (22) LowerBoundPropagationProof (FINISHED) 312.08/291.62 Propagated lower bound. 312.08/291.62 ---------------------------------------- 312.08/291.62 312.08/291.62 (23) 312.08/291.62 BOUNDS(n^2, INF) 312.08/291.62 312.08/291.62 ---------------------------------------- 312.08/291.62 312.08/291.62 (24) 312.08/291.62 Obligation: 312.08/291.62 TRS: 312.08/291.62 Rules: 312.08/291.62 min(0', y) -> 0' 312.08/291.62 min(x, 0') -> 0' 312.08/291.62 min(s(x), s(y)) -> s(min(x, y)) 312.08/291.62 max(0', y) -> y 312.08/291.62 max(x, 0') -> x 312.08/291.62 max(s(x), s(y)) -> s(max(x, y)) 312.08/291.62 +'(0', y) -> y 312.08/291.62 +'(s(x), y) -> s(+'(x, y)) 312.08/291.62 -(x, 0') -> x 312.08/291.62 -(s(x), s(y)) -> -(x, y) 312.08/291.62 *'(x, 0') -> 0' 312.08/291.62 *'(x, s(y)) -> +'(x, *'(x, y)) 312.08/291.62 p(s(x)) -> x 312.08/291.62 f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 312.08/291.62 312.08/291.62 Types: 312.08/291.62 min :: 0':s -> 0':s -> 0':s 312.08/291.62 0' :: 0':s 312.08/291.62 s :: 0':s -> 0':s 312.08/291.62 max :: 0':s -> 0':s -> 0':s 312.08/291.62 +' :: 0':s -> 0':s -> 0':s 312.08/291.62 - :: 0':s -> 0':s -> 0':s 312.08/291.62 *' :: 0':s -> 0':s -> 0':s 312.08/291.62 p :: 0':s -> 0':s 312.08/291.62 f :: 0':s -> 0':s -> f 312.08/291.62 hole_0':s1_0 :: 0':s 312.08/291.62 hole_f2_0 :: f 312.08/291.62 gen_0':s3_0 :: Nat -> 0':s 312.08/291.62 312.08/291.62 312.08/291.62 Lemmas: 312.08/291.62 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) 312.08/291.62 max(gen_0':s3_0(n322_0), gen_0':s3_0(n322_0)) -> gen_0':s3_0(n322_0), rt in Omega(1 + n322_0) 312.08/291.62 +'(gen_0':s3_0(n725_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n725_0, b)), rt in Omega(1 + n725_0) 312.08/291.62 -(gen_0':s3_0(n1340_0), gen_0':s3_0(n1340_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1340_0) 312.08/291.62 *'(gen_0':s3_0(a), gen_0':s3_0(n1640_0)) -> gen_0':s3_0(*(n1640_0, a)), rt in Omega(1 + a*n1640_0 + n1640_0) 312.08/291.62 312.08/291.62 312.08/291.62 Generator Equations: 312.08/291.62 gen_0':s3_0(0) <=> 0' 312.08/291.62 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 312.08/291.62 312.08/291.62 312.08/291.62 The following defined symbols remain to be analysed: 312.08/291.62 f 312.15/291.64 EOF