1099.67/291.59 WORST_CASE(Omega(n^1), ?) 1100.04/291.67 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1100.04/291.67 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1100.04/291.67 1100.04/291.67 1100.04/291.67 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1100.04/291.67 1100.04/291.67 (0) CpxTRS 1100.04/291.67 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1100.04/291.67 (2) CpxTRS 1100.04/291.67 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1100.04/291.67 (4) typed CpxTrs 1100.04/291.67 (5) OrderProof [LOWER BOUND(ID), 0 ms] 1100.04/291.67 (6) typed CpxTrs 1100.04/291.67 (7) RewriteLemmaProof [LOWER BOUND(ID), 293 ms] 1100.04/291.67 (8) BEST 1100.04/291.67 (9) proven lower bound 1100.04/291.67 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 1100.04/291.67 (11) BOUNDS(n^1, INF) 1100.04/291.67 (12) typed CpxTrs 1100.04/291.67 (13) RewriteLemmaProof [LOWER BOUND(ID), 76 ms] 1100.04/291.67 (14) typed CpxTrs 1100.04/291.67 1100.04/291.67 1100.04/291.67 ---------------------------------------- 1100.04/291.67 1100.04/291.67 (0) 1100.04/291.67 Obligation: 1100.04/291.67 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1100.04/291.67 1100.04/291.67 1100.04/291.67 The TRS R consists of the following rules: 1100.04/291.67 1100.04/291.67 lt(x, 0) -> false 1100.04/291.67 lt(0, s(y)) -> true 1100.04/291.67 lt(s(x), s(y)) -> lt(x, y) 1100.04/291.67 plus(x, 0) -> x 1100.04/291.67 plus(x, s(y)) -> s(plus(x, y)) 1100.04/291.67 quot(x, s(y)) -> help(x, s(y), 0) 1100.04/291.67 help(x, s(y), c) -> if(lt(c, x), x, s(y), c) 1100.04/291.67 if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) 1100.04/291.67 if(false, x, s(y), c) -> 0 1100.04/291.67 1100.04/291.67 S is empty. 1100.04/291.67 Rewrite Strategy: INNERMOST 1100.04/291.67 ---------------------------------------- 1100.04/291.67 1100.04/291.67 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 1100.04/291.67 Renamed function symbols to avoid clashes with predefined symbol. 1100.04/291.67 ---------------------------------------- 1100.04/291.67 1100.04/291.67 (2) 1100.04/291.67 Obligation: 1100.04/291.67 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1100.04/291.67 1100.04/291.67 1100.04/291.67 The TRS R consists of the following rules: 1100.04/291.67 1100.04/291.67 lt(x, 0') -> false 1100.04/291.67 lt(0', s(y)) -> true 1100.04/291.67 lt(s(x), s(y)) -> lt(x, y) 1100.04/291.67 plus(x, 0') -> x 1100.04/291.67 plus(x, s(y)) -> s(plus(x, y)) 1100.04/291.67 quot(x, s(y)) -> help(x, s(y), 0') 1100.04/291.67 help(x, s(y), c) -> if(lt(c, x), x, s(y), c) 1100.04/291.67 if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) 1100.04/291.67 if(false, x, s(y), c) -> 0' 1100.04/291.67 1100.04/291.67 S is empty. 1100.04/291.67 Rewrite Strategy: INNERMOST 1100.04/291.67 ---------------------------------------- 1100.04/291.67 1100.04/291.67 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1100.04/291.67 Infered types. 1100.04/291.67 ---------------------------------------- 1100.04/291.67 1100.04/291.67 (4) 1100.04/291.67 Obligation: 1100.04/291.67 Innermost TRS: 1100.04/291.67 Rules: 1100.04/291.67 lt(x, 0') -> false 1100.04/291.67 lt(0', s(y)) -> true 1100.04/291.67 lt(s(x), s(y)) -> lt(x, y) 1100.04/291.67 plus(x, 0') -> x 1100.04/291.67 plus(x, s(y)) -> s(plus(x, y)) 1100.04/291.67 quot(x, s(y)) -> help(x, s(y), 0') 1100.04/291.67 help(x, s(y), c) -> if(lt(c, x), x, s(y), c) 1100.04/291.67 if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) 1100.04/291.67 if(false, x, s(y), c) -> 0' 1100.04/291.67 1100.04/291.67 Types: 1100.04/291.67 lt :: 0':s -> 0':s -> false:true 1100.04/291.67 0' :: 0':s 1100.04/291.67 false :: false:true 1100.04/291.67 s :: 0':s -> 0':s 1100.04/291.67 true :: false:true 1100.04/291.67 plus :: 0':s -> 0':s -> 0':s 1100.04/291.67 quot :: 0':s -> 0':s -> 0':s 1100.04/291.67 help :: 0':s -> 0':s -> 0':s -> 0':s 1100.04/291.67 if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s 1100.04/291.67 hole_false:true1_0 :: false:true 1100.04/291.67 hole_0':s2_0 :: 0':s 1100.04/291.67 gen_0':s3_0 :: Nat -> 0':s 1100.04/291.67 1100.04/291.67 ---------------------------------------- 1100.04/291.67 1100.04/291.67 (5) OrderProof (LOWER BOUND(ID)) 1100.04/291.67 Heuristically decided to analyse the following defined symbols: 1100.04/291.67 lt, plus, help 1100.04/291.67 1100.04/291.67 They will be analysed ascendingly in the following order: 1100.04/291.67 lt < help 1100.04/291.67 plus < help 1100.04/291.67 1100.04/291.67 ---------------------------------------- 1100.04/291.67 1100.04/291.67 (6) 1100.04/291.67 Obligation: 1100.04/291.67 Innermost TRS: 1100.04/291.67 Rules: 1100.04/291.67 lt(x, 0') -> false 1100.04/291.67 lt(0', s(y)) -> true 1100.04/291.67 lt(s(x), s(y)) -> lt(x, y) 1100.04/291.67 plus(x, 0') -> x 1100.04/291.67 plus(x, s(y)) -> s(plus(x, y)) 1100.04/291.67 quot(x, s(y)) -> help(x, s(y), 0') 1100.04/291.67 help(x, s(y), c) -> if(lt(c, x), x, s(y), c) 1100.04/291.67 if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) 1100.04/291.67 if(false, x, s(y), c) -> 0' 1100.04/291.67 1100.04/291.67 Types: 1100.04/291.67 lt :: 0':s -> 0':s -> false:true 1100.04/291.67 0' :: 0':s 1100.04/291.67 false :: false:true 1100.04/291.67 s :: 0':s -> 0':s 1100.04/291.67 true :: false:true 1100.04/291.67 plus :: 0':s -> 0':s -> 0':s 1100.04/291.67 quot :: 0':s -> 0':s -> 0':s 1100.04/291.67 help :: 0':s -> 0':s -> 0':s -> 0':s 1100.04/291.67 if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s 1100.04/291.67 hole_false:true1_0 :: false:true 1100.04/291.67 hole_0':s2_0 :: 0':s 1100.04/291.67 gen_0':s3_0 :: Nat -> 0':s 1100.04/291.67 1100.04/291.67 1100.04/291.67 Generator Equations: 1100.04/291.67 gen_0':s3_0(0) <=> 0' 1100.04/291.67 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1100.04/291.67 1100.04/291.67 1100.04/291.67 The following defined symbols remain to be analysed: 1100.04/291.67 lt, plus, help 1100.04/291.67 1100.04/291.67 They will be analysed ascendingly in the following order: 1100.04/291.67 lt < help 1100.04/291.67 plus < help 1100.04/291.67 1100.04/291.67 ---------------------------------------- 1100.04/291.67 1100.04/291.67 (7) RewriteLemmaProof (LOWER BOUND(ID)) 1100.04/291.67 Proved the following rewrite lemma: 1100.04/291.67 lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, rt in Omega(1 + n5_0) 1100.04/291.67 1100.04/291.67 Induction Base: 1100.04/291.67 lt(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 1100.04/291.67 false 1100.04/291.67 1100.04/291.67 Induction Step: 1100.04/291.67 lt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1) 1100.04/291.67 lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH 1100.04/291.67 false 1100.04/291.67 1100.04/291.67 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1100.04/291.67 ---------------------------------------- 1100.04/291.67 1100.04/291.67 (8) 1100.04/291.67 Complex Obligation (BEST) 1100.04/291.67 1100.04/291.67 ---------------------------------------- 1100.04/291.67 1100.04/291.67 (9) 1100.04/291.67 Obligation: 1100.04/291.67 Proved the lower bound n^1 for the following obligation: 1100.04/291.67 1100.04/291.67 Innermost TRS: 1100.04/291.67 Rules: 1100.04/291.67 lt(x, 0') -> false 1100.04/291.67 lt(0', s(y)) -> true 1100.04/291.67 lt(s(x), s(y)) -> lt(x, y) 1100.04/291.67 plus(x, 0') -> x 1100.04/291.67 plus(x, s(y)) -> s(plus(x, y)) 1100.04/291.67 quot(x, s(y)) -> help(x, s(y), 0') 1100.04/291.67 help(x, s(y), c) -> if(lt(c, x), x, s(y), c) 1100.04/291.67 if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) 1100.04/291.67 if(false, x, s(y), c) -> 0' 1100.04/291.67 1100.04/291.67 Types: 1100.04/291.67 lt :: 0':s -> 0':s -> false:true 1100.04/291.67 0' :: 0':s 1100.04/291.67 false :: false:true 1100.04/291.67 s :: 0':s -> 0':s 1100.04/291.67 true :: false:true 1100.04/291.67 plus :: 0':s -> 0':s -> 0':s 1100.04/291.67 quot :: 0':s -> 0':s -> 0':s 1100.04/291.67 help :: 0':s -> 0':s -> 0':s -> 0':s 1100.04/291.67 if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s 1100.04/291.67 hole_false:true1_0 :: false:true 1100.04/291.67 hole_0':s2_0 :: 0':s 1100.04/291.67 gen_0':s3_0 :: Nat -> 0':s 1100.04/291.67 1100.04/291.67 1100.04/291.67 Generator Equations: 1100.04/291.67 gen_0':s3_0(0) <=> 0' 1100.04/291.67 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1100.04/291.67 1100.04/291.67 1100.04/291.67 The following defined symbols remain to be analysed: 1100.04/291.67 lt, plus, help 1100.04/291.67 1100.04/291.67 They will be analysed ascendingly in the following order: 1100.04/291.67 lt < help 1100.04/291.67 plus < help 1100.04/291.67 1100.04/291.67 ---------------------------------------- 1100.04/291.67 1100.04/291.67 (10) LowerBoundPropagationProof (FINISHED) 1100.04/291.67 Propagated lower bound. 1100.04/291.67 ---------------------------------------- 1100.04/291.67 1100.04/291.67 (11) 1100.04/291.67 BOUNDS(n^1, INF) 1100.04/291.67 1100.04/291.67 ---------------------------------------- 1100.04/291.67 1100.04/291.67 (12) 1100.04/291.67 Obligation: 1100.04/291.67 Innermost TRS: 1100.04/291.67 Rules: 1100.04/291.67 lt(x, 0') -> false 1100.04/291.67 lt(0', s(y)) -> true 1100.04/291.67 lt(s(x), s(y)) -> lt(x, y) 1100.04/291.67 plus(x, 0') -> x 1100.04/291.67 plus(x, s(y)) -> s(plus(x, y)) 1100.04/291.67 quot(x, s(y)) -> help(x, s(y), 0') 1100.04/291.67 help(x, s(y), c) -> if(lt(c, x), x, s(y), c) 1100.04/291.67 if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) 1100.04/291.67 if(false, x, s(y), c) -> 0' 1100.04/291.67 1100.04/291.67 Types: 1100.04/291.67 lt :: 0':s -> 0':s -> false:true 1100.04/291.67 0' :: 0':s 1100.04/291.67 false :: false:true 1100.04/291.67 s :: 0':s -> 0':s 1100.04/291.67 true :: false:true 1100.04/291.67 plus :: 0':s -> 0':s -> 0':s 1100.04/291.67 quot :: 0':s -> 0':s -> 0':s 1100.04/291.67 help :: 0':s -> 0':s -> 0':s -> 0':s 1100.04/291.67 if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s 1100.04/291.67 hole_false:true1_0 :: false:true 1100.04/291.67 hole_0':s2_0 :: 0':s 1100.04/291.67 gen_0':s3_0 :: Nat -> 0':s 1100.04/291.67 1100.04/291.67 1100.04/291.67 Lemmas: 1100.04/291.67 lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, rt in Omega(1 + n5_0) 1100.04/291.67 1100.04/291.67 1100.04/291.67 Generator Equations: 1100.04/291.67 gen_0':s3_0(0) <=> 0' 1100.04/291.67 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1100.04/291.67 1100.04/291.67 1100.04/291.67 The following defined symbols remain to be analysed: 1100.04/291.67 plus, help 1100.04/291.67 1100.04/291.67 They will be analysed ascendingly in the following order: 1100.04/291.67 plus < help 1100.04/291.67 1100.04/291.67 ---------------------------------------- 1100.04/291.67 1100.04/291.67 (13) RewriteLemmaProof (LOWER BOUND(ID)) 1100.04/291.67 Proved the following rewrite lemma: 1100.04/291.67 plus(gen_0':s3_0(a), gen_0':s3_0(n281_0)) -> gen_0':s3_0(+(n281_0, a)), rt in Omega(1 + n281_0) 1100.04/291.67 1100.04/291.67 Induction Base: 1100.04/291.67 plus(gen_0':s3_0(a), gen_0':s3_0(0)) ->_R^Omega(1) 1100.04/291.67 gen_0':s3_0(a) 1100.04/291.67 1100.04/291.67 Induction Step: 1100.04/291.67 plus(gen_0':s3_0(a), gen_0':s3_0(+(n281_0, 1))) ->_R^Omega(1) 1100.04/291.67 s(plus(gen_0':s3_0(a), gen_0':s3_0(n281_0))) ->_IH 1100.04/291.67 s(gen_0':s3_0(+(a, c282_0))) 1100.04/291.67 1100.04/291.67 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1100.04/291.67 ---------------------------------------- 1100.04/291.67 1100.04/291.67 (14) 1100.04/291.67 Obligation: 1100.04/291.67 Innermost TRS: 1100.04/291.67 Rules: 1100.04/291.67 lt(x, 0') -> false 1100.04/291.67 lt(0', s(y)) -> true 1100.04/291.67 lt(s(x), s(y)) -> lt(x, y) 1100.04/291.67 plus(x, 0') -> x 1100.04/291.67 plus(x, s(y)) -> s(plus(x, y)) 1100.04/291.67 quot(x, s(y)) -> help(x, s(y), 0') 1100.04/291.67 help(x, s(y), c) -> if(lt(c, x), x, s(y), c) 1100.04/291.67 if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) 1100.04/291.67 if(false, x, s(y), c) -> 0' 1100.04/291.67 1100.04/291.67 Types: 1100.04/291.67 lt :: 0':s -> 0':s -> false:true 1100.04/291.67 0' :: 0':s 1100.04/291.67 false :: false:true 1100.04/291.67 s :: 0':s -> 0':s 1100.04/291.67 true :: false:true 1100.04/291.67 plus :: 0':s -> 0':s -> 0':s 1100.04/291.67 quot :: 0':s -> 0':s -> 0':s 1100.04/291.67 help :: 0':s -> 0':s -> 0':s -> 0':s 1100.04/291.67 if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s 1100.04/291.67 hole_false:true1_0 :: false:true 1100.04/291.67 hole_0':s2_0 :: 0':s 1100.04/291.67 gen_0':s3_0 :: Nat -> 0':s 1100.04/291.67 1100.04/291.67 1100.04/291.67 Lemmas: 1100.04/291.67 lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, rt in Omega(1 + n5_0) 1100.04/291.67 plus(gen_0':s3_0(a), gen_0':s3_0(n281_0)) -> gen_0':s3_0(+(n281_0, a)), rt in Omega(1 + n281_0) 1100.04/291.67 1100.04/291.67 1100.04/291.67 Generator Equations: 1100.04/291.67 gen_0':s3_0(0) <=> 0' 1100.04/291.67 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1100.04/291.67 1100.04/291.67 1100.04/291.67 The following defined symbols remain to be analysed: 1100.04/291.67 help 1100.39/291.72 EOF