311.11/291.60 WORST_CASE(Omega(n^2), ?) 311.11/291.61 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 311.11/291.61 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 311.11/291.61 311.11/291.61 311.11/291.61 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 311.11/291.61 311.11/291.61 (0) CpxTRS 311.11/291.61 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 311.11/291.61 (2) CpxTRS 311.11/291.61 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 311.11/291.61 (4) typed CpxTrs 311.11/291.61 (5) OrderProof [LOWER BOUND(ID), 0 ms] 311.11/291.61 (6) typed CpxTrs 311.11/291.61 (7) RewriteLemmaProof [LOWER BOUND(ID), 300 ms] 311.11/291.61 (8) BEST 311.11/291.61 (9) proven lower bound 311.11/291.61 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 311.11/291.61 (11) BOUNDS(n^1, INF) 311.11/291.61 (12) typed CpxTrs 311.11/291.61 (13) RewriteLemmaProof [LOWER BOUND(ID), 57 ms] 311.11/291.61 (14) typed CpxTrs 311.11/291.61 (15) RewriteLemmaProof [LOWER BOUND(ID), 48 ms] 311.11/291.61 (16) BEST 311.11/291.61 (17) proven lower bound 311.11/291.61 (18) LowerBoundPropagationProof [FINISHED, 0 ms] 311.11/291.61 (19) BOUNDS(n^2, INF) 311.11/291.61 (20) typed CpxTrs 311.11/291.61 311.11/291.61 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (0) 311.11/291.61 Obligation: 311.11/291.61 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 311.11/291.61 311.11/291.61 311.11/291.61 The TRS R consists of the following rules: 311.11/291.61 311.11/291.61 -(x, 0) -> x 311.11/291.61 -(s(x), s(y)) -> -(x, y) 311.11/291.61 +(0, y) -> y 311.11/291.61 +(s(x), y) -> s(+(x, y)) 311.11/291.61 *(x, 0) -> 0 311.11/291.61 *(x, s(y)) -> +(x, *(x, y)) 311.11/291.61 f(s(x)) -> f(-(+(*(s(x), s(x)), *(s(x), s(s(s(0))))), *(s(s(x)), s(s(x))))) 311.11/291.61 311.11/291.61 S is empty. 311.11/291.61 Rewrite Strategy: FULL 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 311.11/291.61 Renamed function symbols to avoid clashes with predefined symbol. 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (2) 311.11/291.61 Obligation: 311.11/291.61 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 311.11/291.61 311.11/291.61 311.11/291.61 The TRS R consists of the following rules: 311.11/291.61 311.11/291.61 -(x, 0') -> x 311.11/291.61 -(s(x), s(y)) -> -(x, y) 311.11/291.61 +'(0', y) -> y 311.11/291.61 +'(s(x), y) -> s(+'(x, y)) 311.11/291.61 *'(x, 0') -> 0' 311.11/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 311.11/291.61 f(s(x)) -> f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x))))) 311.11/291.61 311.11/291.61 S is empty. 311.11/291.61 Rewrite Strategy: FULL 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 311.11/291.61 Infered types. 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (4) 311.11/291.61 Obligation: 311.11/291.61 TRS: 311.11/291.61 Rules: 311.11/291.61 -(x, 0') -> x 311.11/291.61 -(s(x), s(y)) -> -(x, y) 311.11/291.61 +'(0', y) -> y 311.11/291.61 +'(s(x), y) -> s(+'(x, y)) 311.11/291.61 *'(x, 0') -> 0' 311.11/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 311.11/291.61 f(s(x)) -> f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x))))) 311.11/291.61 311.11/291.61 Types: 311.11/291.61 - :: 0':s -> 0':s -> 0':s 311.11/291.61 0' :: 0':s 311.11/291.61 s :: 0':s -> 0':s 311.11/291.61 +' :: 0':s -> 0':s -> 0':s 311.11/291.61 *' :: 0':s -> 0':s -> 0':s 311.11/291.61 f :: 0':s -> f 311.11/291.61 hole_0':s1_0 :: 0':s 311.11/291.61 hole_f2_0 :: f 311.11/291.61 gen_0':s3_0 :: Nat -> 0':s 311.11/291.61 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (5) OrderProof (LOWER BOUND(ID)) 311.11/291.61 Heuristically decided to analyse the following defined symbols: 311.11/291.61 -, +', *', f 311.11/291.61 311.11/291.61 They will be analysed ascendingly in the following order: 311.11/291.61 - < f 311.11/291.61 +' < *' 311.11/291.61 +' < f 311.11/291.61 *' < f 311.11/291.61 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (6) 311.11/291.61 Obligation: 311.11/291.61 TRS: 311.11/291.61 Rules: 311.11/291.61 -(x, 0') -> x 311.11/291.61 -(s(x), s(y)) -> -(x, y) 311.11/291.61 +'(0', y) -> y 311.11/291.61 +'(s(x), y) -> s(+'(x, y)) 311.11/291.61 *'(x, 0') -> 0' 311.11/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 311.11/291.61 f(s(x)) -> f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x))))) 311.11/291.61 311.11/291.61 Types: 311.11/291.61 - :: 0':s -> 0':s -> 0':s 311.11/291.61 0' :: 0':s 311.11/291.61 s :: 0':s -> 0':s 311.11/291.61 +' :: 0':s -> 0':s -> 0':s 311.11/291.61 *' :: 0':s -> 0':s -> 0':s 311.11/291.61 f :: 0':s -> f 311.11/291.61 hole_0':s1_0 :: 0':s 311.11/291.61 hole_f2_0 :: f 311.11/291.61 gen_0':s3_0 :: Nat -> 0':s 311.11/291.61 311.11/291.61 311.11/291.61 Generator Equations: 311.11/291.61 gen_0':s3_0(0) <=> 0' 311.11/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 311.11/291.61 311.11/291.61 311.11/291.61 The following defined symbols remain to be analysed: 311.11/291.61 -, +', *', f 311.11/291.61 311.11/291.61 They will be analysed ascendingly in the following order: 311.11/291.61 - < f 311.11/291.61 +' < *' 311.11/291.61 +' < f 311.11/291.61 *' < f 311.11/291.61 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (7) RewriteLemmaProof (LOWER BOUND(ID)) 311.11/291.61 Proved the following rewrite lemma: 311.11/291.61 -(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0) 311.11/291.61 311.11/291.61 Induction Base: 311.11/291.61 -(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 311.11/291.61 gen_0':s3_0(0) 311.11/291.61 311.11/291.61 Induction Step: 311.11/291.61 -(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1) 311.11/291.61 -(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH 311.11/291.61 gen_0':s3_0(0) 311.11/291.61 311.11/291.61 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (8) 311.11/291.61 Complex Obligation (BEST) 311.11/291.61 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (9) 311.11/291.61 Obligation: 311.11/291.61 Proved the lower bound n^1 for the following obligation: 311.11/291.61 311.11/291.61 TRS: 311.11/291.61 Rules: 311.11/291.61 -(x, 0') -> x 311.11/291.61 -(s(x), s(y)) -> -(x, y) 311.11/291.61 +'(0', y) -> y 311.11/291.61 +'(s(x), y) -> s(+'(x, y)) 311.11/291.61 *'(x, 0') -> 0' 311.11/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 311.11/291.61 f(s(x)) -> f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x))))) 311.11/291.61 311.11/291.61 Types: 311.11/291.61 - :: 0':s -> 0':s -> 0':s 311.11/291.61 0' :: 0':s 311.11/291.61 s :: 0':s -> 0':s 311.11/291.61 +' :: 0':s -> 0':s -> 0':s 311.11/291.61 *' :: 0':s -> 0':s -> 0':s 311.11/291.61 f :: 0':s -> f 311.11/291.61 hole_0':s1_0 :: 0':s 311.11/291.61 hole_f2_0 :: f 311.11/291.61 gen_0':s3_0 :: Nat -> 0':s 311.11/291.61 311.11/291.61 311.11/291.61 Generator Equations: 311.11/291.61 gen_0':s3_0(0) <=> 0' 311.11/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 311.11/291.61 311.11/291.61 311.11/291.61 The following defined symbols remain to be analysed: 311.11/291.61 -, +', *', f 311.11/291.61 311.11/291.61 They will be analysed ascendingly in the following order: 311.11/291.61 - < f 311.11/291.61 +' < *' 311.11/291.61 +' < f 311.11/291.61 *' < f 311.11/291.61 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (10) LowerBoundPropagationProof (FINISHED) 311.11/291.61 Propagated lower bound. 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (11) 311.11/291.61 BOUNDS(n^1, INF) 311.11/291.61 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (12) 311.11/291.61 Obligation: 311.11/291.61 TRS: 311.11/291.61 Rules: 311.11/291.61 -(x, 0') -> x 311.11/291.61 -(s(x), s(y)) -> -(x, y) 311.11/291.61 +'(0', y) -> y 311.11/291.61 +'(s(x), y) -> s(+'(x, y)) 311.11/291.61 *'(x, 0') -> 0' 311.11/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 311.11/291.61 f(s(x)) -> f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x))))) 311.11/291.61 311.11/291.61 Types: 311.11/291.61 - :: 0':s -> 0':s -> 0':s 311.11/291.61 0' :: 0':s 311.11/291.61 s :: 0':s -> 0':s 311.11/291.61 +' :: 0':s -> 0':s -> 0':s 311.11/291.61 *' :: 0':s -> 0':s -> 0':s 311.11/291.61 f :: 0':s -> f 311.11/291.61 hole_0':s1_0 :: 0':s 311.11/291.61 hole_f2_0 :: f 311.11/291.61 gen_0':s3_0 :: Nat -> 0':s 311.11/291.61 311.11/291.61 311.11/291.61 Lemmas: 311.11/291.61 -(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0) 311.11/291.61 311.11/291.61 311.11/291.61 Generator Equations: 311.11/291.61 gen_0':s3_0(0) <=> 0' 311.11/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 311.11/291.61 311.11/291.61 311.11/291.61 The following defined symbols remain to be analysed: 311.11/291.61 +', *', f 311.11/291.61 311.11/291.61 They will be analysed ascendingly in the following order: 311.11/291.61 +' < *' 311.11/291.61 +' < f 311.11/291.61 *' < f 311.11/291.61 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (13) RewriteLemmaProof (LOWER BOUND(ID)) 311.11/291.61 Proved the following rewrite lemma: 311.11/291.61 +'(gen_0':s3_0(n225_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n225_0, b)), rt in Omega(1 + n225_0) 311.11/291.61 311.11/291.61 Induction Base: 311.11/291.61 +'(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1) 311.11/291.61 gen_0':s3_0(b) 311.11/291.61 311.11/291.61 Induction Step: 311.11/291.61 +'(gen_0':s3_0(+(n225_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1) 311.11/291.61 s(+'(gen_0':s3_0(n225_0), gen_0':s3_0(b))) ->_IH 311.11/291.61 s(gen_0':s3_0(+(b, c226_0))) 311.11/291.61 311.11/291.61 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (14) 311.11/291.61 Obligation: 311.11/291.61 TRS: 311.11/291.61 Rules: 311.11/291.61 -(x, 0') -> x 311.11/291.61 -(s(x), s(y)) -> -(x, y) 311.11/291.61 +'(0', y) -> y 311.11/291.61 +'(s(x), y) -> s(+'(x, y)) 311.11/291.61 *'(x, 0') -> 0' 311.11/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 311.11/291.61 f(s(x)) -> f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x))))) 311.11/291.61 311.11/291.61 Types: 311.11/291.61 - :: 0':s -> 0':s -> 0':s 311.11/291.61 0' :: 0':s 311.11/291.61 s :: 0':s -> 0':s 311.11/291.61 +' :: 0':s -> 0':s -> 0':s 311.11/291.61 *' :: 0':s -> 0':s -> 0':s 311.11/291.61 f :: 0':s -> f 311.11/291.61 hole_0':s1_0 :: 0':s 311.11/291.61 hole_f2_0 :: f 311.11/291.61 gen_0':s3_0 :: Nat -> 0':s 311.11/291.61 311.11/291.61 311.11/291.61 Lemmas: 311.11/291.61 -(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0) 311.11/291.61 +'(gen_0':s3_0(n225_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n225_0, b)), rt in Omega(1 + n225_0) 311.11/291.61 311.11/291.61 311.11/291.61 Generator Equations: 311.11/291.61 gen_0':s3_0(0) <=> 0' 311.11/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 311.11/291.61 311.11/291.61 311.11/291.61 The following defined symbols remain to be analysed: 311.11/291.61 *', f 311.11/291.61 311.11/291.61 They will be analysed ascendingly in the following order: 311.11/291.61 *' < f 311.11/291.61 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (15) RewriteLemmaProof (LOWER BOUND(ID)) 311.11/291.61 Proved the following rewrite lemma: 311.11/291.61 *'(gen_0':s3_0(a), gen_0':s3_0(n696_0)) -> gen_0':s3_0(*(n696_0, a)), rt in Omega(1 + a*n696_0 + n696_0) 311.11/291.61 311.11/291.61 Induction Base: 311.11/291.61 *'(gen_0':s3_0(a), gen_0':s3_0(0)) ->_R^Omega(1) 311.11/291.61 0' 311.11/291.61 311.11/291.61 Induction Step: 311.11/291.61 *'(gen_0':s3_0(a), gen_0':s3_0(+(n696_0, 1))) ->_R^Omega(1) 311.11/291.61 +'(gen_0':s3_0(a), *'(gen_0':s3_0(a), gen_0':s3_0(n696_0))) ->_IH 311.11/291.61 +'(gen_0':s3_0(a), gen_0':s3_0(*(c697_0, a))) ->_L^Omega(1 + a) 311.11/291.61 gen_0':s3_0(+(a, *(n696_0, a))) 311.11/291.61 311.11/291.61 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (16) 311.11/291.61 Complex Obligation (BEST) 311.11/291.61 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (17) 311.11/291.61 Obligation: 311.11/291.61 Proved the lower bound n^2 for the following obligation: 311.11/291.61 311.11/291.61 TRS: 311.11/291.61 Rules: 311.11/291.61 -(x, 0') -> x 311.11/291.61 -(s(x), s(y)) -> -(x, y) 311.11/291.61 +'(0', y) -> y 311.11/291.61 +'(s(x), y) -> s(+'(x, y)) 311.11/291.61 *'(x, 0') -> 0' 311.11/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 311.11/291.61 f(s(x)) -> f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x))))) 311.11/291.61 311.11/291.61 Types: 311.11/291.61 - :: 0':s -> 0':s -> 0':s 311.11/291.61 0' :: 0':s 311.11/291.61 s :: 0':s -> 0':s 311.11/291.61 +' :: 0':s -> 0':s -> 0':s 311.11/291.61 *' :: 0':s -> 0':s -> 0':s 311.11/291.61 f :: 0':s -> f 311.11/291.61 hole_0':s1_0 :: 0':s 311.11/291.61 hole_f2_0 :: f 311.11/291.61 gen_0':s3_0 :: Nat -> 0':s 311.11/291.61 311.11/291.61 311.11/291.61 Lemmas: 311.11/291.61 -(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0) 311.11/291.61 +'(gen_0':s3_0(n225_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n225_0, b)), rt in Omega(1 + n225_0) 311.11/291.61 311.11/291.61 311.11/291.61 Generator Equations: 311.11/291.61 gen_0':s3_0(0) <=> 0' 311.11/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 311.11/291.61 311.11/291.61 311.11/291.61 The following defined symbols remain to be analysed: 311.11/291.61 *', f 311.11/291.61 311.11/291.61 They will be analysed ascendingly in the following order: 311.11/291.61 *' < f 311.11/291.61 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (18) LowerBoundPropagationProof (FINISHED) 311.11/291.61 Propagated lower bound. 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (19) 311.11/291.61 BOUNDS(n^2, INF) 311.11/291.61 311.11/291.61 ---------------------------------------- 311.11/291.61 311.11/291.61 (20) 311.11/291.61 Obligation: 311.11/291.61 TRS: 311.11/291.61 Rules: 311.11/291.61 -(x, 0') -> x 311.11/291.61 -(s(x), s(y)) -> -(x, y) 311.11/291.61 +'(0', y) -> y 311.11/291.61 +'(s(x), y) -> s(+'(x, y)) 311.11/291.61 *'(x, 0') -> 0' 311.11/291.61 *'(x, s(y)) -> +'(x, *'(x, y)) 311.11/291.61 f(s(x)) -> f(-(+'(*'(s(x), s(x)), *'(s(x), s(s(s(0'))))), *'(s(s(x)), s(s(x))))) 311.11/291.61 311.11/291.61 Types: 311.11/291.61 - :: 0':s -> 0':s -> 0':s 311.11/291.61 0' :: 0':s 311.11/291.61 s :: 0':s -> 0':s 311.11/291.61 +' :: 0':s -> 0':s -> 0':s 311.11/291.61 *' :: 0':s -> 0':s -> 0':s 311.11/291.61 f :: 0':s -> f 311.11/291.61 hole_0':s1_0 :: 0':s 311.11/291.61 hole_f2_0 :: f 311.11/291.61 gen_0':s3_0 :: Nat -> 0':s 311.11/291.61 311.11/291.61 311.11/291.61 Lemmas: 311.11/291.61 -(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0) 311.11/291.61 +'(gen_0':s3_0(n225_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n225_0, b)), rt in Omega(1 + n225_0) 311.11/291.61 *'(gen_0':s3_0(a), gen_0':s3_0(n696_0)) -> gen_0':s3_0(*(n696_0, a)), rt in Omega(1 + a*n696_0 + n696_0) 311.11/291.61 311.11/291.61 311.11/291.61 Generator Equations: 311.11/291.61 gen_0':s3_0(0) <=> 0' 311.11/291.61 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 311.11/291.61 311.11/291.61 311.11/291.61 The following defined symbols remain to be analysed: 311.11/291.61 f 311.20/291.64 EOF