313.79/291.52 WORST_CASE(Omega(n^1), ?) 313.79/291.52 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 313.79/291.52 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 313.79/291.52 313.79/291.52 313.79/291.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 313.79/291.52 313.79/291.52 (0) CpxTRS 313.79/291.52 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 313.79/291.52 (2) CpxTRS 313.79/291.52 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 313.79/291.52 (4) typed CpxTrs 313.79/291.52 (5) OrderProof [LOWER BOUND(ID), 0 ms] 313.79/291.52 (6) typed CpxTrs 313.79/291.52 (7) RewriteLemmaProof [LOWER BOUND(ID), 323 ms] 313.79/291.52 (8) BEST 313.79/291.52 (9) proven lower bound 313.79/291.52 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 313.79/291.52 (11) BOUNDS(n^1, INF) 313.79/291.52 (12) typed CpxTrs 313.79/291.52 (13) RewriteLemmaProof [LOWER BOUND(ID), 1252 ms] 313.79/291.52 (14) typed CpxTrs 313.79/291.52 (15) RewriteLemmaProof [LOWER BOUND(ID), 2034 ms] 313.79/291.52 (16) typed CpxTrs 313.79/291.52 (17) RewriteLemmaProof [LOWER BOUND(ID), 2530 ms] 313.79/291.52 (18) BOUNDS(1, INF) 313.79/291.52 313.79/291.52 313.79/291.52 ---------------------------------------- 313.79/291.52 313.79/291.52 (0) 313.79/291.52 Obligation: 313.79/291.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 313.79/291.53 313.79/291.53 313.79/291.53 The TRS R consists of the following rules: 313.79/291.53 313.79/291.53 -(x, 0) -> x 313.79/291.53 -(0, s(y)) -> 0 313.79/291.53 -(s(x), s(y)) -> -(x, y) 313.79/291.53 f(0) -> 0 313.79/291.53 f(s(x)) -> -(s(x), g(f(x))) 313.79/291.53 g(0) -> s(0) 313.79/291.53 g(s(x)) -> -(s(x), f(g(x))) 313.79/291.53 313.79/291.53 S is empty. 313.79/291.53 Rewrite Strategy: FULL 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 313.79/291.53 Renamed function symbols to avoid clashes with predefined symbol. 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (2) 313.79/291.53 Obligation: 313.79/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 313.79/291.53 313.79/291.53 313.79/291.53 The TRS R consists of the following rules: 313.79/291.53 313.79/291.53 -(x, 0') -> x 313.79/291.53 -(0', s(y)) -> 0' 313.79/291.53 -(s(x), s(y)) -> -(x, y) 313.79/291.53 f(0') -> 0' 313.79/291.53 f(s(x)) -> -(s(x), g(f(x))) 313.79/291.53 g(0') -> s(0') 313.79/291.53 g(s(x)) -> -(s(x), f(g(x))) 313.79/291.53 313.79/291.53 S is empty. 313.79/291.53 Rewrite Strategy: FULL 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 313.79/291.53 Infered types. 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (4) 313.79/291.53 Obligation: 313.79/291.53 TRS: 313.79/291.53 Rules: 313.79/291.53 -(x, 0') -> x 313.79/291.53 -(0', s(y)) -> 0' 313.79/291.53 -(s(x), s(y)) -> -(x, y) 313.79/291.53 f(0') -> 0' 313.79/291.53 f(s(x)) -> -(s(x), g(f(x))) 313.79/291.53 g(0') -> s(0') 313.79/291.53 g(s(x)) -> -(s(x), f(g(x))) 313.79/291.53 313.79/291.53 Types: 313.79/291.53 - :: 0':s -> 0':s -> 0':s 313.79/291.53 0' :: 0':s 313.79/291.53 s :: 0':s -> 0':s 313.79/291.53 f :: 0':s -> 0':s 313.79/291.53 g :: 0':s -> 0':s 313.79/291.53 hole_0':s1_0 :: 0':s 313.79/291.53 gen_0':s2_0 :: Nat -> 0':s 313.79/291.53 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (5) OrderProof (LOWER BOUND(ID)) 313.79/291.53 Heuristically decided to analyse the following defined symbols: 313.79/291.53 -, f, g 313.79/291.53 313.79/291.53 They will be analysed ascendingly in the following order: 313.79/291.53 - < f 313.79/291.53 - < g 313.79/291.53 f = g 313.79/291.53 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (6) 313.79/291.53 Obligation: 313.79/291.53 TRS: 313.79/291.53 Rules: 313.79/291.53 -(x, 0') -> x 313.79/291.53 -(0', s(y)) -> 0' 313.79/291.53 -(s(x), s(y)) -> -(x, y) 313.79/291.53 f(0') -> 0' 313.79/291.53 f(s(x)) -> -(s(x), g(f(x))) 313.79/291.53 g(0') -> s(0') 313.79/291.53 g(s(x)) -> -(s(x), f(g(x))) 313.79/291.53 313.79/291.53 Types: 313.79/291.53 - :: 0':s -> 0':s -> 0':s 313.79/291.53 0' :: 0':s 313.79/291.53 s :: 0':s -> 0':s 313.79/291.53 f :: 0':s -> 0':s 313.79/291.53 g :: 0':s -> 0':s 313.79/291.53 hole_0':s1_0 :: 0':s 313.79/291.53 gen_0':s2_0 :: Nat -> 0':s 313.79/291.53 313.79/291.53 313.79/291.53 Generator Equations: 313.79/291.53 gen_0':s2_0(0) <=> 0' 313.79/291.53 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 313.79/291.53 313.79/291.53 313.79/291.53 The following defined symbols remain to be analysed: 313.79/291.53 -, f, g 313.79/291.53 313.79/291.53 They will be analysed ascendingly in the following order: 313.79/291.53 - < f 313.79/291.53 - < g 313.79/291.53 f = g 313.79/291.53 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (7) RewriteLemmaProof (LOWER BOUND(ID)) 313.79/291.53 Proved the following rewrite lemma: 313.79/291.53 -(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0) 313.79/291.53 313.79/291.53 Induction Base: 313.79/291.53 -(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1) 313.79/291.53 gen_0':s2_0(0) 313.79/291.53 313.79/291.53 Induction Step: 313.79/291.53 -(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1) 313.79/291.53 -(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) ->_IH 313.79/291.53 gen_0':s2_0(0) 313.79/291.53 313.79/291.53 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (8) 313.79/291.53 Complex Obligation (BEST) 313.79/291.53 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (9) 313.79/291.53 Obligation: 313.79/291.53 Proved the lower bound n^1 for the following obligation: 313.79/291.53 313.79/291.53 TRS: 313.79/291.53 Rules: 313.79/291.53 -(x, 0') -> x 313.79/291.53 -(0', s(y)) -> 0' 313.79/291.53 -(s(x), s(y)) -> -(x, y) 313.79/291.53 f(0') -> 0' 313.79/291.53 f(s(x)) -> -(s(x), g(f(x))) 313.79/291.53 g(0') -> s(0') 313.79/291.53 g(s(x)) -> -(s(x), f(g(x))) 313.79/291.53 313.79/291.53 Types: 313.79/291.53 - :: 0':s -> 0':s -> 0':s 313.79/291.53 0' :: 0':s 313.79/291.53 s :: 0':s -> 0':s 313.79/291.53 f :: 0':s -> 0':s 313.79/291.53 g :: 0':s -> 0':s 313.79/291.53 hole_0':s1_0 :: 0':s 313.79/291.53 gen_0':s2_0 :: Nat -> 0':s 313.79/291.53 313.79/291.53 313.79/291.53 Generator Equations: 313.79/291.53 gen_0':s2_0(0) <=> 0' 313.79/291.53 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 313.79/291.53 313.79/291.53 313.79/291.53 The following defined symbols remain to be analysed: 313.79/291.53 -, f, g 313.79/291.53 313.79/291.53 They will be analysed ascendingly in the following order: 313.79/291.53 - < f 313.79/291.53 - < g 313.79/291.53 f = g 313.79/291.53 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (10) LowerBoundPropagationProof (FINISHED) 313.79/291.53 Propagated lower bound. 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (11) 313.79/291.53 BOUNDS(n^1, INF) 313.79/291.53 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (12) 313.79/291.53 Obligation: 313.79/291.53 TRS: 313.79/291.53 Rules: 313.79/291.53 -(x, 0') -> x 313.79/291.53 -(0', s(y)) -> 0' 313.79/291.53 -(s(x), s(y)) -> -(x, y) 313.79/291.53 f(0') -> 0' 313.79/291.53 f(s(x)) -> -(s(x), g(f(x))) 313.79/291.53 g(0') -> s(0') 313.79/291.53 g(s(x)) -> -(s(x), f(g(x))) 313.79/291.53 313.79/291.53 Types: 313.79/291.53 - :: 0':s -> 0':s -> 0':s 313.79/291.53 0' :: 0':s 313.79/291.53 s :: 0':s -> 0':s 313.79/291.53 f :: 0':s -> 0':s 313.79/291.53 g :: 0':s -> 0':s 313.79/291.53 hole_0':s1_0 :: 0':s 313.79/291.53 gen_0':s2_0 :: Nat -> 0':s 313.79/291.53 313.79/291.53 313.79/291.53 Lemmas: 313.79/291.53 -(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0) 313.79/291.53 313.79/291.53 313.79/291.53 Generator Equations: 313.79/291.53 gen_0':s2_0(0) <=> 0' 313.79/291.53 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 313.79/291.53 313.79/291.53 313.79/291.53 The following defined symbols remain to be analysed: 313.79/291.53 g, f 313.79/291.53 313.79/291.53 They will be analysed ascendingly in the following order: 313.79/291.53 f = g 313.79/291.53 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (13) RewriteLemmaProof (LOWER BOUND(ID)) 313.79/291.53 Proved the following rewrite lemma: 313.79/291.53 g(gen_0':s2_0(+(1, n335_0))) -> *3_0, rt in Omega(n335_0) 313.79/291.53 313.79/291.53 Induction Base: 313.79/291.53 g(gen_0':s2_0(+(1, 0))) 313.79/291.53 313.79/291.53 Induction Step: 313.79/291.53 g(gen_0':s2_0(+(1, +(n335_0, 1)))) ->_R^Omega(1) 313.79/291.53 -(s(gen_0':s2_0(+(1, n335_0))), f(g(gen_0':s2_0(+(1, n335_0))))) ->_IH 313.79/291.53 -(s(gen_0':s2_0(+(1, n335_0))), f(*3_0)) 313.79/291.53 313.79/291.53 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (14) 313.79/291.53 Obligation: 313.79/291.53 TRS: 313.79/291.53 Rules: 313.79/291.53 -(x, 0') -> x 313.79/291.53 -(0', s(y)) -> 0' 313.79/291.53 -(s(x), s(y)) -> -(x, y) 313.79/291.53 f(0') -> 0' 313.79/291.53 f(s(x)) -> -(s(x), g(f(x))) 313.79/291.53 g(0') -> s(0') 313.79/291.53 g(s(x)) -> -(s(x), f(g(x))) 313.79/291.53 313.79/291.53 Types: 313.79/291.53 - :: 0':s -> 0':s -> 0':s 313.79/291.53 0' :: 0':s 313.79/291.53 s :: 0':s -> 0':s 313.79/291.53 f :: 0':s -> 0':s 313.79/291.53 g :: 0':s -> 0':s 313.79/291.53 hole_0':s1_0 :: 0':s 313.79/291.53 gen_0':s2_0 :: Nat -> 0':s 313.79/291.53 313.79/291.53 313.79/291.53 Lemmas: 313.79/291.53 -(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0) 313.79/291.53 g(gen_0':s2_0(+(1, n335_0))) -> *3_0, rt in Omega(n335_0) 313.79/291.53 313.79/291.53 313.79/291.53 Generator Equations: 313.79/291.53 gen_0':s2_0(0) <=> 0' 313.79/291.53 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 313.79/291.53 313.79/291.53 313.79/291.53 The following defined symbols remain to be analysed: 313.79/291.53 f 313.79/291.53 313.79/291.53 They will be analysed ascendingly in the following order: 313.79/291.53 f = g 313.79/291.53 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (15) RewriteLemmaProof (LOWER BOUND(ID)) 313.79/291.53 Proved the following rewrite lemma: 313.79/291.53 f(gen_0':s2_0(+(1, n4115_0))) -> *3_0, rt in Omega(n4115_0) 313.79/291.53 313.79/291.53 Induction Base: 313.79/291.53 f(gen_0':s2_0(+(1, 0))) 313.79/291.53 313.79/291.53 Induction Step: 313.79/291.53 f(gen_0':s2_0(+(1, +(n4115_0, 1)))) ->_R^Omega(1) 313.79/291.53 -(s(gen_0':s2_0(+(1, n4115_0))), g(f(gen_0':s2_0(+(1, n4115_0))))) ->_IH 313.79/291.53 -(s(gen_0':s2_0(+(1, n4115_0))), g(*3_0)) 313.79/291.53 313.79/291.53 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (16) 313.79/291.53 Obligation: 313.79/291.53 TRS: 313.79/291.53 Rules: 313.79/291.53 -(x, 0') -> x 313.79/291.53 -(0', s(y)) -> 0' 313.79/291.53 -(s(x), s(y)) -> -(x, y) 313.79/291.53 f(0') -> 0' 313.79/291.53 f(s(x)) -> -(s(x), g(f(x))) 313.79/291.53 g(0') -> s(0') 313.79/291.53 g(s(x)) -> -(s(x), f(g(x))) 313.79/291.53 313.79/291.53 Types: 313.79/291.53 - :: 0':s -> 0':s -> 0':s 313.79/291.53 0' :: 0':s 313.79/291.53 s :: 0':s -> 0':s 313.79/291.53 f :: 0':s -> 0':s 313.79/291.53 g :: 0':s -> 0':s 313.79/291.53 hole_0':s1_0 :: 0':s 313.79/291.53 gen_0':s2_0 :: Nat -> 0':s 313.79/291.53 313.79/291.53 313.79/291.53 Lemmas: 313.79/291.53 -(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0) 313.79/291.53 g(gen_0':s2_0(+(1, n335_0))) -> *3_0, rt in Omega(n335_0) 313.79/291.53 f(gen_0':s2_0(+(1, n4115_0))) -> *3_0, rt in Omega(n4115_0) 313.79/291.53 313.79/291.53 313.79/291.53 Generator Equations: 313.79/291.53 gen_0':s2_0(0) <=> 0' 313.79/291.53 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 313.79/291.53 313.79/291.53 313.79/291.53 The following defined symbols remain to be analysed: 313.79/291.53 g 313.79/291.53 313.79/291.53 They will be analysed ascendingly in the following order: 313.79/291.53 f = g 313.79/291.53 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (17) RewriteLemmaProof (LOWER BOUND(ID)) 313.79/291.53 Proved the following rewrite lemma: 313.79/291.53 g(gen_0':s2_0(+(1, n62695_0))) -> *3_0, rt in Omega(n62695_0) 313.79/291.53 313.79/291.53 Induction Base: 313.79/291.53 g(gen_0':s2_0(+(1, 0))) 313.79/291.53 313.79/291.53 Induction Step: 313.79/291.53 g(gen_0':s2_0(+(1, +(n62695_0, 1)))) ->_R^Omega(1) 313.79/291.53 -(s(gen_0':s2_0(+(1, n62695_0))), f(g(gen_0':s2_0(+(1, n62695_0))))) ->_IH 313.79/291.53 -(s(gen_0':s2_0(+(1, n62695_0))), f(*3_0)) 313.79/291.53 313.79/291.53 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 313.79/291.53 ---------------------------------------- 313.79/291.53 313.79/291.53 (18) 313.79/291.53 BOUNDS(1, INF) 313.79/291.56 EOF