1109.77/291.54 WORST_CASE(Omega(n^2), ?) 1110.26/291.60 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1110.26/291.60 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1110.26/291.60 1110.26/291.60 1110.26/291.60 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1110.26/291.60 1110.26/291.60 (0) CpxTRS 1110.26/291.60 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1110.26/291.60 (2) CpxTRS 1110.26/291.60 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1110.26/291.60 (4) typed CpxTrs 1110.26/291.60 (5) OrderProof [LOWER BOUND(ID), 0 ms] 1110.26/291.60 (6) typed CpxTrs 1110.26/291.60 (7) RewriteLemmaProof [LOWER BOUND(ID), 240 ms] 1110.26/291.60 (8) BEST 1110.26/291.60 (9) proven lower bound 1110.26/291.60 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 1110.26/291.60 (11) BOUNDS(n^1, INF) 1110.26/291.60 (12) typed CpxTrs 1110.26/291.60 (13) RewriteLemmaProof [LOWER BOUND(ID), 51 ms] 1110.26/291.60 (14) typed CpxTrs 1110.26/291.60 (15) RewriteLemmaProof [LOWER BOUND(ID), 84 ms] 1110.26/291.60 (16) BEST 1110.26/291.60 (17) proven lower bound 1110.26/291.60 (18) LowerBoundPropagationProof [FINISHED, 0 ms] 1110.26/291.60 (19) BOUNDS(n^2, INF) 1110.26/291.60 (20) typed CpxTrs 1110.26/291.60 1110.26/291.60 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (0) 1110.26/291.60 Obligation: 1110.26/291.60 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1110.26/291.60 1110.26/291.60 1110.26/291.60 The TRS R consists of the following rules: 1110.26/291.60 1110.26/291.60 -(x, 0) -> x 1110.26/291.60 -(s(x), s(y)) -> -(x, y) 1110.26/291.60 +(0, y) -> y 1110.26/291.60 +(s(x), y) -> s(+(x, y)) 1110.26/291.60 *(x, 0) -> 0 1110.26/291.60 *(x, s(y)) -> +(x, *(x, y)) 1110.26/291.60 p(s(x)) -> x 1110.26/291.60 f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x)))) 1110.26/291.60 1110.26/291.60 S is empty. 1110.26/291.60 Rewrite Strategy: INNERMOST 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 1110.26/291.60 Renamed function symbols to avoid clashes with predefined symbol. 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (2) 1110.26/291.60 Obligation: 1110.26/291.60 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1110.26/291.60 1110.26/291.60 1110.26/291.60 The TRS R consists of the following rules: 1110.26/291.60 1110.26/291.60 -(x, 0') -> x 1110.26/291.60 -(s(x), s(y)) -> -(x, y) 1110.26/291.60 +'(0', y) -> y 1110.26/291.60 +'(s(x), y) -> s(+'(x, y)) 1110.26/291.60 *'(x, 0') -> 0' 1110.26/291.60 *'(x, s(y)) -> +'(x, *'(x, y)) 1110.26/291.60 p(s(x)) -> x 1110.26/291.60 f(s(x)) -> f(-(p(*'(s(x), s(x))), *'(s(x), s(x)))) 1110.26/291.60 1110.26/291.60 S is empty. 1110.26/291.60 Rewrite Strategy: INNERMOST 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1110.26/291.60 Infered types. 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (4) 1110.26/291.60 Obligation: 1110.26/291.60 Innermost TRS: 1110.26/291.60 Rules: 1110.26/291.60 -(x, 0') -> x 1110.26/291.60 -(s(x), s(y)) -> -(x, y) 1110.26/291.60 +'(0', y) -> y 1110.26/291.60 +'(s(x), y) -> s(+'(x, y)) 1110.26/291.60 *'(x, 0') -> 0' 1110.26/291.60 *'(x, s(y)) -> +'(x, *'(x, y)) 1110.26/291.60 p(s(x)) -> x 1110.26/291.60 f(s(x)) -> f(-(p(*'(s(x), s(x))), *'(s(x), s(x)))) 1110.26/291.60 1110.26/291.60 Types: 1110.26/291.60 - :: 0':s -> 0':s -> 0':s 1110.26/291.60 0' :: 0':s 1110.26/291.60 s :: 0':s -> 0':s 1110.26/291.60 +' :: 0':s -> 0':s -> 0':s 1110.26/291.60 *' :: 0':s -> 0':s -> 0':s 1110.26/291.60 p :: 0':s -> 0':s 1110.26/291.60 f :: 0':s -> f 1110.26/291.60 hole_0':s1_0 :: 0':s 1110.26/291.60 hole_f2_0 :: f 1110.26/291.60 gen_0':s3_0 :: Nat -> 0':s 1110.26/291.60 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (5) OrderProof (LOWER BOUND(ID)) 1110.26/291.60 Heuristically decided to analyse the following defined symbols: 1110.26/291.60 -, +', *', f 1110.26/291.60 1110.26/291.60 They will be analysed ascendingly in the following order: 1110.26/291.60 - < f 1110.26/291.60 +' < *' 1110.26/291.60 *' < f 1110.26/291.60 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (6) 1110.26/291.60 Obligation: 1110.26/291.60 Innermost TRS: 1110.26/291.60 Rules: 1110.26/291.60 -(x, 0') -> x 1110.26/291.60 -(s(x), s(y)) -> -(x, y) 1110.26/291.60 +'(0', y) -> y 1110.26/291.60 +'(s(x), y) -> s(+'(x, y)) 1110.26/291.60 *'(x, 0') -> 0' 1110.26/291.60 *'(x, s(y)) -> +'(x, *'(x, y)) 1110.26/291.60 p(s(x)) -> x 1110.26/291.60 f(s(x)) -> f(-(p(*'(s(x), s(x))), *'(s(x), s(x)))) 1110.26/291.60 1110.26/291.60 Types: 1110.26/291.60 - :: 0':s -> 0':s -> 0':s 1110.26/291.60 0' :: 0':s 1110.26/291.60 s :: 0':s -> 0':s 1110.26/291.60 +' :: 0':s -> 0':s -> 0':s 1110.26/291.60 *' :: 0':s -> 0':s -> 0':s 1110.26/291.60 p :: 0':s -> 0':s 1110.26/291.60 f :: 0':s -> f 1110.26/291.60 hole_0':s1_0 :: 0':s 1110.26/291.60 hole_f2_0 :: f 1110.26/291.60 gen_0':s3_0 :: Nat -> 0':s 1110.26/291.60 1110.26/291.60 1110.26/291.60 Generator Equations: 1110.26/291.60 gen_0':s3_0(0) <=> 0' 1110.26/291.60 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1110.26/291.60 1110.26/291.60 1110.26/291.60 The following defined symbols remain to be analysed: 1110.26/291.60 -, +', *', f 1110.26/291.60 1110.26/291.60 They will be analysed ascendingly in the following order: 1110.26/291.60 - < f 1110.26/291.60 +' < *' 1110.26/291.60 *' < f 1110.26/291.60 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (7) RewriteLemmaProof (LOWER BOUND(ID)) 1110.26/291.60 Proved the following rewrite lemma: 1110.26/291.60 -(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0) 1110.26/291.60 1110.26/291.60 Induction Base: 1110.26/291.60 -(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 1110.26/291.60 gen_0':s3_0(0) 1110.26/291.60 1110.26/291.60 Induction Step: 1110.26/291.60 -(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1) 1110.26/291.60 -(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH 1110.26/291.60 gen_0':s3_0(0) 1110.26/291.60 1110.26/291.60 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (8) 1110.26/291.60 Complex Obligation (BEST) 1110.26/291.60 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (9) 1110.26/291.60 Obligation: 1110.26/291.60 Proved the lower bound n^1 for the following obligation: 1110.26/291.60 1110.26/291.60 Innermost TRS: 1110.26/291.60 Rules: 1110.26/291.60 -(x, 0') -> x 1110.26/291.60 -(s(x), s(y)) -> -(x, y) 1110.26/291.60 +'(0', y) -> y 1110.26/291.60 +'(s(x), y) -> s(+'(x, y)) 1110.26/291.60 *'(x, 0') -> 0' 1110.26/291.60 *'(x, s(y)) -> +'(x, *'(x, y)) 1110.26/291.60 p(s(x)) -> x 1110.26/291.60 f(s(x)) -> f(-(p(*'(s(x), s(x))), *'(s(x), s(x)))) 1110.26/291.60 1110.26/291.60 Types: 1110.26/291.60 - :: 0':s -> 0':s -> 0':s 1110.26/291.60 0' :: 0':s 1110.26/291.60 s :: 0':s -> 0':s 1110.26/291.60 +' :: 0':s -> 0':s -> 0':s 1110.26/291.60 *' :: 0':s -> 0':s -> 0':s 1110.26/291.60 p :: 0':s -> 0':s 1110.26/291.60 f :: 0':s -> f 1110.26/291.60 hole_0':s1_0 :: 0':s 1110.26/291.60 hole_f2_0 :: f 1110.26/291.60 gen_0':s3_0 :: Nat -> 0':s 1110.26/291.60 1110.26/291.60 1110.26/291.60 Generator Equations: 1110.26/291.60 gen_0':s3_0(0) <=> 0' 1110.26/291.60 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1110.26/291.60 1110.26/291.60 1110.26/291.60 The following defined symbols remain to be analysed: 1110.26/291.60 -, +', *', f 1110.26/291.60 1110.26/291.60 They will be analysed ascendingly in the following order: 1110.26/291.60 - < f 1110.26/291.60 +' < *' 1110.26/291.60 *' < f 1110.26/291.60 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (10) LowerBoundPropagationProof (FINISHED) 1110.26/291.60 Propagated lower bound. 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (11) 1110.26/291.60 BOUNDS(n^1, INF) 1110.26/291.60 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (12) 1110.26/291.60 Obligation: 1110.26/291.60 Innermost TRS: 1110.26/291.60 Rules: 1110.26/291.60 -(x, 0') -> x 1110.26/291.60 -(s(x), s(y)) -> -(x, y) 1110.26/291.60 +'(0', y) -> y 1110.26/291.60 +'(s(x), y) -> s(+'(x, y)) 1110.26/291.60 *'(x, 0') -> 0' 1110.26/291.60 *'(x, s(y)) -> +'(x, *'(x, y)) 1110.26/291.60 p(s(x)) -> x 1110.26/291.60 f(s(x)) -> f(-(p(*'(s(x), s(x))), *'(s(x), s(x)))) 1110.26/291.60 1110.26/291.60 Types: 1110.26/291.60 - :: 0':s -> 0':s -> 0':s 1110.26/291.60 0' :: 0':s 1110.26/291.60 s :: 0':s -> 0':s 1110.26/291.60 +' :: 0':s -> 0':s -> 0':s 1110.26/291.60 *' :: 0':s -> 0':s -> 0':s 1110.26/291.60 p :: 0':s -> 0':s 1110.26/291.60 f :: 0':s -> f 1110.26/291.60 hole_0':s1_0 :: 0':s 1110.26/291.60 hole_f2_0 :: f 1110.26/291.60 gen_0':s3_0 :: Nat -> 0':s 1110.26/291.60 1110.26/291.60 1110.26/291.60 Lemmas: 1110.26/291.60 -(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0) 1110.26/291.60 1110.26/291.60 1110.26/291.60 Generator Equations: 1110.26/291.60 gen_0':s3_0(0) <=> 0' 1110.26/291.60 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1110.26/291.60 1110.26/291.60 1110.26/291.60 The following defined symbols remain to be analysed: 1110.26/291.60 +', *', f 1110.26/291.60 1110.26/291.60 They will be analysed ascendingly in the following order: 1110.26/291.60 +' < *' 1110.26/291.60 *' < f 1110.26/291.60 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (13) RewriteLemmaProof (LOWER BOUND(ID)) 1110.26/291.60 Proved the following rewrite lemma: 1110.26/291.60 +'(gen_0':s3_0(n289_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n289_0, b)), rt in Omega(1 + n289_0) 1110.26/291.60 1110.26/291.60 Induction Base: 1110.26/291.60 +'(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1) 1110.26/291.60 gen_0':s3_0(b) 1110.26/291.60 1110.26/291.60 Induction Step: 1110.26/291.60 +'(gen_0':s3_0(+(n289_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1) 1110.26/291.60 s(+'(gen_0':s3_0(n289_0), gen_0':s3_0(b))) ->_IH 1110.26/291.60 s(gen_0':s3_0(+(b, c290_0))) 1110.26/291.60 1110.26/291.60 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (14) 1110.26/291.60 Obligation: 1110.26/291.60 Innermost TRS: 1110.26/291.60 Rules: 1110.26/291.60 -(x, 0') -> x 1110.26/291.60 -(s(x), s(y)) -> -(x, y) 1110.26/291.60 +'(0', y) -> y 1110.26/291.60 +'(s(x), y) -> s(+'(x, y)) 1110.26/291.60 *'(x, 0') -> 0' 1110.26/291.60 *'(x, s(y)) -> +'(x, *'(x, y)) 1110.26/291.60 p(s(x)) -> x 1110.26/291.60 f(s(x)) -> f(-(p(*'(s(x), s(x))), *'(s(x), s(x)))) 1110.26/291.60 1110.26/291.60 Types: 1110.26/291.60 - :: 0':s -> 0':s -> 0':s 1110.26/291.60 0' :: 0':s 1110.26/291.60 s :: 0':s -> 0':s 1110.26/291.60 +' :: 0':s -> 0':s -> 0':s 1110.26/291.60 *' :: 0':s -> 0':s -> 0':s 1110.26/291.60 p :: 0':s -> 0':s 1110.26/291.60 f :: 0':s -> f 1110.26/291.60 hole_0':s1_0 :: 0':s 1110.26/291.60 hole_f2_0 :: f 1110.26/291.60 gen_0':s3_0 :: Nat -> 0':s 1110.26/291.60 1110.26/291.60 1110.26/291.60 Lemmas: 1110.26/291.60 -(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0) 1110.26/291.60 +'(gen_0':s3_0(n289_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n289_0, b)), rt in Omega(1 + n289_0) 1110.26/291.60 1110.26/291.60 1110.26/291.60 Generator Equations: 1110.26/291.60 gen_0':s3_0(0) <=> 0' 1110.26/291.60 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1110.26/291.60 1110.26/291.60 1110.26/291.60 The following defined symbols remain to be analysed: 1110.26/291.60 *', f 1110.26/291.60 1110.26/291.60 They will be analysed ascendingly in the following order: 1110.26/291.60 *' < f 1110.26/291.60 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (15) RewriteLemmaProof (LOWER BOUND(ID)) 1110.26/291.60 Proved the following rewrite lemma: 1110.26/291.60 *'(gen_0':s3_0(a), gen_0':s3_0(n834_0)) -> gen_0':s3_0(*(n834_0, a)), rt in Omega(1 + a*n834_0 + n834_0) 1110.26/291.60 1110.26/291.60 Induction Base: 1110.26/291.60 *'(gen_0':s3_0(a), gen_0':s3_0(0)) ->_R^Omega(1) 1110.26/291.60 0' 1110.26/291.60 1110.26/291.60 Induction Step: 1110.26/291.60 *'(gen_0':s3_0(a), gen_0':s3_0(+(n834_0, 1))) ->_R^Omega(1) 1110.26/291.60 +'(gen_0':s3_0(a), *'(gen_0':s3_0(a), gen_0':s3_0(n834_0))) ->_IH 1110.26/291.60 +'(gen_0':s3_0(a), gen_0':s3_0(*(c835_0, a))) ->_L^Omega(1 + a) 1110.26/291.60 gen_0':s3_0(+(a, *(n834_0, a))) 1110.26/291.60 1110.26/291.60 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (16) 1110.26/291.60 Complex Obligation (BEST) 1110.26/291.60 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (17) 1110.26/291.60 Obligation: 1110.26/291.60 Proved the lower bound n^2 for the following obligation: 1110.26/291.60 1110.26/291.60 Innermost TRS: 1110.26/291.60 Rules: 1110.26/291.60 -(x, 0') -> x 1110.26/291.60 -(s(x), s(y)) -> -(x, y) 1110.26/291.60 +'(0', y) -> y 1110.26/291.60 +'(s(x), y) -> s(+'(x, y)) 1110.26/291.60 *'(x, 0') -> 0' 1110.26/291.60 *'(x, s(y)) -> +'(x, *'(x, y)) 1110.26/291.60 p(s(x)) -> x 1110.26/291.60 f(s(x)) -> f(-(p(*'(s(x), s(x))), *'(s(x), s(x)))) 1110.26/291.60 1110.26/291.60 Types: 1110.26/291.60 - :: 0':s -> 0':s -> 0':s 1110.26/291.60 0' :: 0':s 1110.26/291.60 s :: 0':s -> 0':s 1110.26/291.60 +' :: 0':s -> 0':s -> 0':s 1110.26/291.60 *' :: 0':s -> 0':s -> 0':s 1110.26/291.60 p :: 0':s -> 0':s 1110.26/291.60 f :: 0':s -> f 1110.26/291.60 hole_0':s1_0 :: 0':s 1110.26/291.60 hole_f2_0 :: f 1110.26/291.60 gen_0':s3_0 :: Nat -> 0':s 1110.26/291.60 1110.26/291.60 1110.26/291.60 Lemmas: 1110.26/291.60 -(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0) 1110.26/291.60 +'(gen_0':s3_0(n289_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n289_0, b)), rt in Omega(1 + n289_0) 1110.26/291.60 1110.26/291.60 1110.26/291.60 Generator Equations: 1110.26/291.60 gen_0':s3_0(0) <=> 0' 1110.26/291.60 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1110.26/291.60 1110.26/291.60 1110.26/291.60 The following defined symbols remain to be analysed: 1110.26/291.60 *', f 1110.26/291.60 1110.26/291.60 They will be analysed ascendingly in the following order: 1110.26/291.60 *' < f 1110.26/291.60 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (18) LowerBoundPropagationProof (FINISHED) 1110.26/291.60 Propagated lower bound. 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (19) 1110.26/291.60 BOUNDS(n^2, INF) 1110.26/291.60 1110.26/291.60 ---------------------------------------- 1110.26/291.60 1110.26/291.60 (20) 1110.26/291.60 Obligation: 1110.26/291.60 Innermost TRS: 1110.26/291.60 Rules: 1110.26/291.60 -(x, 0') -> x 1110.26/291.60 -(s(x), s(y)) -> -(x, y) 1110.26/291.60 +'(0', y) -> y 1110.26/291.60 +'(s(x), y) -> s(+'(x, y)) 1110.26/291.60 *'(x, 0') -> 0' 1110.26/291.60 *'(x, s(y)) -> +'(x, *'(x, y)) 1110.26/291.60 p(s(x)) -> x 1110.26/291.60 f(s(x)) -> f(-(p(*'(s(x), s(x))), *'(s(x), s(x)))) 1110.26/291.60 1110.26/291.60 Types: 1110.26/291.60 - :: 0':s -> 0':s -> 0':s 1110.26/291.60 0' :: 0':s 1110.26/291.60 s :: 0':s -> 0':s 1110.26/291.60 +' :: 0':s -> 0':s -> 0':s 1110.26/291.60 *' :: 0':s -> 0':s -> 0':s 1110.26/291.60 p :: 0':s -> 0':s 1110.26/291.60 f :: 0':s -> f 1110.26/291.60 hole_0':s1_0 :: 0':s 1110.26/291.60 hole_f2_0 :: f 1110.26/291.60 gen_0':s3_0 :: Nat -> 0':s 1110.26/291.60 1110.26/291.60 1110.26/291.60 Lemmas: 1110.26/291.60 -(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0) 1110.26/291.60 +'(gen_0':s3_0(n289_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n289_0, b)), rt in Omega(1 + n289_0) 1110.26/291.60 *'(gen_0':s3_0(a), gen_0':s3_0(n834_0)) -> gen_0':s3_0(*(n834_0, a)), rt in Omega(1 + a*n834_0 + n834_0) 1110.26/291.60 1110.26/291.60 1110.26/291.60 Generator Equations: 1110.26/291.60 gen_0':s3_0(0) <=> 0' 1110.26/291.60 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1110.26/291.60 1110.26/291.60 1110.26/291.60 The following defined symbols remain to be analysed: 1110.26/291.60 f 1110.53/291.72 EOF