1134.79/291.55 WORST_CASE(Omega(n^2), ?) 1134.79/291.58 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1134.79/291.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1134.79/291.58 1134.79/291.58 1134.79/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1134.79/291.58 1134.79/291.58 (0) CpxTRS 1134.79/291.58 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1134.79/291.58 (2) CpxTRS 1134.79/291.58 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1134.79/291.58 (4) typed CpxTrs 1134.79/291.58 (5) OrderProof [LOWER BOUND(ID), 0 ms] 1134.79/291.58 (6) typed CpxTrs 1134.79/291.58 (7) RewriteLemmaProof [LOWER BOUND(ID), 277 ms] 1134.79/291.58 (8) BEST 1134.79/291.58 (9) proven lower bound 1134.79/291.58 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 1134.79/291.58 (11) BOUNDS(n^1, INF) 1134.79/291.58 (12) typed CpxTrs 1134.79/291.58 (13) RewriteLemmaProof [LOWER BOUND(ID), 65 ms] 1134.79/291.58 (14) typed CpxTrs 1134.79/291.58 (15) RewriteLemmaProof [LOWER BOUND(ID), 34 ms] 1134.79/291.58 (16) BEST 1134.79/291.58 (17) proven lower bound 1134.79/291.58 (18) LowerBoundPropagationProof [FINISHED, 0 ms] 1134.79/291.58 (19) BOUNDS(n^2, INF) 1134.79/291.58 (20) typed CpxTrs 1134.79/291.58 1134.79/291.58 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (0) 1134.79/291.58 Obligation: 1134.79/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1134.79/291.58 1134.79/291.58 1134.79/291.58 The TRS R consists of the following rules: 1134.79/291.58 1134.79/291.58 le(s(x), 0) -> false 1134.79/291.58 le(0, y) -> true 1134.79/291.58 le(s(x), s(y)) -> le(x, y) 1134.79/291.58 plus(0, y) -> y 1134.79/291.58 plus(s(x), y) -> s(plus(x, y)) 1134.79/291.58 times(0, y) -> 0 1134.79/291.58 times(s(x), y) -> plus(y, times(x, y)) 1134.79/291.58 log(x, 0) -> baseError 1134.79/291.58 log(x, s(0)) -> baseError 1134.79/291.58 log(0, s(s(b))) -> logZeroError 1134.79/291.58 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0) 1134.79/291.58 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 1134.79/291.58 if(true, x, b, y, z) -> z 1134.79/291.58 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 1134.79/291.58 1134.79/291.58 S is empty. 1134.79/291.58 Rewrite Strategy: INNERMOST 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 1134.79/291.58 Renamed function symbols to avoid clashes with predefined symbol. 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (2) 1134.79/291.58 Obligation: 1134.79/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1134.79/291.58 1134.79/291.58 1134.79/291.58 The TRS R consists of the following rules: 1134.79/291.58 1134.79/291.58 le(s(x), 0') -> false 1134.79/291.58 le(0', y) -> true 1134.79/291.58 le(s(x), s(y)) -> le(x, y) 1134.79/291.58 plus(0', y) -> y 1134.79/291.58 plus(s(x), y) -> s(plus(x, y)) 1134.79/291.58 times(0', y) -> 0' 1134.79/291.58 times(s(x), y) -> plus(y, times(x, y)) 1134.79/291.58 log(x, 0') -> baseError 1134.79/291.58 log(x, s(0')) -> baseError 1134.79/291.58 log(0', s(s(b))) -> logZeroError 1134.79/291.58 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') 1134.79/291.58 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 1134.79/291.58 if(true, x, b, y, z) -> z 1134.79/291.58 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 1134.79/291.58 1134.79/291.58 S is empty. 1134.79/291.58 Rewrite Strategy: INNERMOST 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1134.79/291.58 Infered types. 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (4) 1134.79/291.58 Obligation: 1134.79/291.58 Innermost TRS: 1134.79/291.58 Rules: 1134.79/291.58 le(s(x), 0') -> false 1134.79/291.58 le(0', y) -> true 1134.79/291.58 le(s(x), s(y)) -> le(x, y) 1134.79/291.58 plus(0', y) -> y 1134.79/291.58 plus(s(x), y) -> s(plus(x, y)) 1134.79/291.58 times(0', y) -> 0' 1134.79/291.58 times(s(x), y) -> plus(y, times(x, y)) 1134.79/291.58 log(x, 0') -> baseError 1134.79/291.58 log(x, s(0')) -> baseError 1134.79/291.58 log(0', s(s(b))) -> logZeroError 1134.79/291.58 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') 1134.79/291.58 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 1134.79/291.58 if(true, x, b, y, z) -> z 1134.79/291.58 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 1134.79/291.58 1134.79/291.58 Types: 1134.79/291.58 le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true 1134.79/291.58 s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 0' :: s:0':baseError:logZeroError 1134.79/291.58 false :: false:true 1134.79/291.58 true :: false:true 1134.79/291.58 plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 baseError :: s:0':baseError:logZeroError 1134.79/291.58 logZeroError :: s:0':baseError:logZeroError 1134.79/291.58 loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 hole_false:true1_0 :: false:true 1134.79/291.58 hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError 1134.79/291.58 gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError 1134.79/291.58 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (5) OrderProof (LOWER BOUND(ID)) 1134.79/291.58 Heuristically decided to analyse the following defined symbols: 1134.79/291.58 le, plus, times, loop 1134.79/291.58 1134.79/291.58 They will be analysed ascendingly in the following order: 1134.79/291.58 le < loop 1134.79/291.58 plus < times 1134.79/291.58 times < loop 1134.79/291.58 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (6) 1134.79/291.58 Obligation: 1134.79/291.58 Innermost TRS: 1134.79/291.58 Rules: 1134.79/291.58 le(s(x), 0') -> false 1134.79/291.58 le(0', y) -> true 1134.79/291.58 le(s(x), s(y)) -> le(x, y) 1134.79/291.58 plus(0', y) -> y 1134.79/291.58 plus(s(x), y) -> s(plus(x, y)) 1134.79/291.58 times(0', y) -> 0' 1134.79/291.58 times(s(x), y) -> plus(y, times(x, y)) 1134.79/291.58 log(x, 0') -> baseError 1134.79/291.58 log(x, s(0')) -> baseError 1134.79/291.58 log(0', s(s(b))) -> logZeroError 1134.79/291.58 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') 1134.79/291.58 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 1134.79/291.58 if(true, x, b, y, z) -> z 1134.79/291.58 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 1134.79/291.58 1134.79/291.58 Types: 1134.79/291.58 le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true 1134.79/291.58 s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 0' :: s:0':baseError:logZeroError 1134.79/291.58 false :: false:true 1134.79/291.58 true :: false:true 1134.79/291.58 plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 baseError :: s:0':baseError:logZeroError 1134.79/291.58 logZeroError :: s:0':baseError:logZeroError 1134.79/291.58 loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 hole_false:true1_0 :: false:true 1134.79/291.58 hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError 1134.79/291.58 gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError 1134.79/291.58 1134.79/291.58 1134.79/291.58 Generator Equations: 1134.79/291.58 gen_s:0':baseError:logZeroError3_0(0) <=> 0' 1134.79/291.58 gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x)) 1134.79/291.58 1134.79/291.58 1134.79/291.58 The following defined symbols remain to be analysed: 1134.79/291.58 le, plus, times, loop 1134.79/291.58 1134.79/291.58 They will be analysed ascendingly in the following order: 1134.79/291.58 le < loop 1134.79/291.58 plus < times 1134.79/291.58 times < loop 1134.79/291.58 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (7) RewriteLemmaProof (LOWER BOUND(ID)) 1134.79/291.58 Proved the following rewrite lemma: 1134.79/291.58 le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) -> false, rt in Omega(1 + n5_0) 1134.79/291.58 1134.79/291.58 Induction Base: 1134.79/291.58 le(gen_s:0':baseError:logZeroError3_0(+(1, 0)), gen_s:0':baseError:logZeroError3_0(0)) ->_R^Omega(1) 1134.79/291.58 false 1134.79/291.58 1134.79/291.58 Induction Step: 1134.79/291.58 le(gen_s:0':baseError:logZeroError3_0(+(1, +(n5_0, 1))), gen_s:0':baseError:logZeroError3_0(+(n5_0, 1))) ->_R^Omega(1) 1134.79/291.58 le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) ->_IH 1134.79/291.58 false 1134.79/291.58 1134.79/291.58 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (8) 1134.79/291.58 Complex Obligation (BEST) 1134.79/291.58 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (9) 1134.79/291.58 Obligation: 1134.79/291.58 Proved the lower bound n^1 for the following obligation: 1134.79/291.58 1134.79/291.58 Innermost TRS: 1134.79/291.58 Rules: 1134.79/291.58 le(s(x), 0') -> false 1134.79/291.58 le(0', y) -> true 1134.79/291.58 le(s(x), s(y)) -> le(x, y) 1134.79/291.58 plus(0', y) -> y 1134.79/291.58 plus(s(x), y) -> s(plus(x, y)) 1134.79/291.58 times(0', y) -> 0' 1134.79/291.58 times(s(x), y) -> plus(y, times(x, y)) 1134.79/291.58 log(x, 0') -> baseError 1134.79/291.58 log(x, s(0')) -> baseError 1134.79/291.58 log(0', s(s(b))) -> logZeroError 1134.79/291.58 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') 1134.79/291.58 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 1134.79/291.58 if(true, x, b, y, z) -> z 1134.79/291.58 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 1134.79/291.58 1134.79/291.58 Types: 1134.79/291.58 le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true 1134.79/291.58 s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 0' :: s:0':baseError:logZeroError 1134.79/291.58 false :: false:true 1134.79/291.58 true :: false:true 1134.79/291.58 plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 baseError :: s:0':baseError:logZeroError 1134.79/291.58 logZeroError :: s:0':baseError:logZeroError 1134.79/291.58 loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 hole_false:true1_0 :: false:true 1134.79/291.58 hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError 1134.79/291.58 gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError 1134.79/291.58 1134.79/291.58 1134.79/291.58 Generator Equations: 1134.79/291.58 gen_s:0':baseError:logZeroError3_0(0) <=> 0' 1134.79/291.58 gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x)) 1134.79/291.58 1134.79/291.58 1134.79/291.58 The following defined symbols remain to be analysed: 1134.79/291.58 le, plus, times, loop 1134.79/291.58 1134.79/291.58 They will be analysed ascendingly in the following order: 1134.79/291.58 le < loop 1134.79/291.58 plus < times 1134.79/291.58 times < loop 1134.79/291.58 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (10) LowerBoundPropagationProof (FINISHED) 1134.79/291.58 Propagated lower bound. 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (11) 1134.79/291.58 BOUNDS(n^1, INF) 1134.79/291.58 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (12) 1134.79/291.58 Obligation: 1134.79/291.58 Innermost TRS: 1134.79/291.58 Rules: 1134.79/291.58 le(s(x), 0') -> false 1134.79/291.58 le(0', y) -> true 1134.79/291.58 le(s(x), s(y)) -> le(x, y) 1134.79/291.58 plus(0', y) -> y 1134.79/291.58 plus(s(x), y) -> s(plus(x, y)) 1134.79/291.58 times(0', y) -> 0' 1134.79/291.58 times(s(x), y) -> plus(y, times(x, y)) 1134.79/291.58 log(x, 0') -> baseError 1134.79/291.58 log(x, s(0')) -> baseError 1134.79/291.58 log(0', s(s(b))) -> logZeroError 1134.79/291.58 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') 1134.79/291.58 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 1134.79/291.58 if(true, x, b, y, z) -> z 1134.79/291.58 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 1134.79/291.58 1134.79/291.58 Types: 1134.79/291.58 le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true 1134.79/291.58 s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 0' :: s:0':baseError:logZeroError 1134.79/291.58 false :: false:true 1134.79/291.58 true :: false:true 1134.79/291.58 plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 baseError :: s:0':baseError:logZeroError 1134.79/291.58 logZeroError :: s:0':baseError:logZeroError 1134.79/291.58 loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 hole_false:true1_0 :: false:true 1134.79/291.58 hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError 1134.79/291.58 gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError 1134.79/291.58 1134.79/291.58 1134.79/291.58 Lemmas: 1134.79/291.58 le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) -> false, rt in Omega(1 + n5_0) 1134.79/291.58 1134.79/291.58 1134.79/291.58 Generator Equations: 1134.79/291.58 gen_s:0':baseError:logZeroError3_0(0) <=> 0' 1134.79/291.58 gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x)) 1134.79/291.58 1134.79/291.58 1134.79/291.58 The following defined symbols remain to be analysed: 1134.79/291.58 plus, times, loop 1134.79/291.58 1134.79/291.58 They will be analysed ascendingly in the following order: 1134.79/291.58 plus < times 1134.79/291.58 times < loop 1134.79/291.58 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (13) RewriteLemmaProof (LOWER BOUND(ID)) 1134.79/291.58 Proved the following rewrite lemma: 1134.79/291.58 plus(gen_s:0':baseError:logZeroError3_0(n342_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(+(n342_0, b)), rt in Omega(1 + n342_0) 1134.79/291.58 1134.79/291.58 Induction Base: 1134.79/291.58 plus(gen_s:0':baseError:logZeroError3_0(0), gen_s:0':baseError:logZeroError3_0(b)) ->_R^Omega(1) 1134.79/291.58 gen_s:0':baseError:logZeroError3_0(b) 1134.79/291.58 1134.79/291.58 Induction Step: 1134.79/291.58 plus(gen_s:0':baseError:logZeroError3_0(+(n342_0, 1)), gen_s:0':baseError:logZeroError3_0(b)) ->_R^Omega(1) 1134.79/291.58 s(plus(gen_s:0':baseError:logZeroError3_0(n342_0), gen_s:0':baseError:logZeroError3_0(b))) ->_IH 1134.79/291.58 s(gen_s:0':baseError:logZeroError3_0(+(b, c343_0))) 1134.79/291.58 1134.79/291.58 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (14) 1134.79/291.58 Obligation: 1134.79/291.58 Innermost TRS: 1134.79/291.58 Rules: 1134.79/291.58 le(s(x), 0') -> false 1134.79/291.58 le(0', y) -> true 1134.79/291.58 le(s(x), s(y)) -> le(x, y) 1134.79/291.58 plus(0', y) -> y 1134.79/291.58 plus(s(x), y) -> s(plus(x, y)) 1134.79/291.58 times(0', y) -> 0' 1134.79/291.58 times(s(x), y) -> plus(y, times(x, y)) 1134.79/291.58 log(x, 0') -> baseError 1134.79/291.58 log(x, s(0')) -> baseError 1134.79/291.58 log(0', s(s(b))) -> logZeroError 1134.79/291.58 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') 1134.79/291.58 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 1134.79/291.58 if(true, x, b, y, z) -> z 1134.79/291.58 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 1134.79/291.58 1134.79/291.58 Types: 1134.79/291.58 le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true 1134.79/291.58 s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 0' :: s:0':baseError:logZeroError 1134.79/291.58 false :: false:true 1134.79/291.58 true :: false:true 1134.79/291.58 plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 baseError :: s:0':baseError:logZeroError 1134.79/291.58 logZeroError :: s:0':baseError:logZeroError 1134.79/291.58 loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 hole_false:true1_0 :: false:true 1134.79/291.58 hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError 1134.79/291.58 gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError 1134.79/291.58 1134.79/291.58 1134.79/291.58 Lemmas: 1134.79/291.58 le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) -> false, rt in Omega(1 + n5_0) 1134.79/291.58 plus(gen_s:0':baseError:logZeroError3_0(n342_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(+(n342_0, b)), rt in Omega(1 + n342_0) 1134.79/291.58 1134.79/291.58 1134.79/291.58 Generator Equations: 1134.79/291.58 gen_s:0':baseError:logZeroError3_0(0) <=> 0' 1134.79/291.58 gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x)) 1134.79/291.58 1134.79/291.58 1134.79/291.58 The following defined symbols remain to be analysed: 1134.79/291.58 times, loop 1134.79/291.58 1134.79/291.58 They will be analysed ascendingly in the following order: 1134.79/291.58 times < loop 1134.79/291.58 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (15) RewriteLemmaProof (LOWER BOUND(ID)) 1134.79/291.58 Proved the following rewrite lemma: 1134.79/291.58 times(gen_s:0':baseError:logZeroError3_0(n1027_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(*(n1027_0, b)), rt in Omega(1 + b*n1027_0 + n1027_0) 1134.79/291.58 1134.79/291.58 Induction Base: 1134.79/291.58 times(gen_s:0':baseError:logZeroError3_0(0), gen_s:0':baseError:logZeroError3_0(b)) ->_R^Omega(1) 1134.79/291.58 0' 1134.79/291.58 1134.79/291.58 Induction Step: 1134.79/291.58 times(gen_s:0':baseError:logZeroError3_0(+(n1027_0, 1)), gen_s:0':baseError:logZeroError3_0(b)) ->_R^Omega(1) 1134.79/291.58 plus(gen_s:0':baseError:logZeroError3_0(b), times(gen_s:0':baseError:logZeroError3_0(n1027_0), gen_s:0':baseError:logZeroError3_0(b))) ->_IH 1134.79/291.58 plus(gen_s:0':baseError:logZeroError3_0(b), gen_s:0':baseError:logZeroError3_0(*(c1028_0, b))) ->_L^Omega(1 + b) 1134.79/291.58 gen_s:0':baseError:logZeroError3_0(+(b, *(n1027_0, b))) 1134.79/291.58 1134.79/291.58 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (16) 1134.79/291.58 Complex Obligation (BEST) 1134.79/291.58 1134.79/291.58 ---------------------------------------- 1134.79/291.58 1134.79/291.58 (17) 1134.79/291.58 Obligation: 1134.79/291.58 Proved the lower bound n^2 for the following obligation: 1134.79/291.58 1134.79/291.58 Innermost TRS: 1134.79/291.58 Rules: 1134.79/291.58 le(s(x), 0') -> false 1134.79/291.58 le(0', y) -> true 1134.79/291.58 le(s(x), s(y)) -> le(x, y) 1134.79/291.58 plus(0', y) -> y 1134.79/291.58 plus(s(x), y) -> s(plus(x, y)) 1134.79/291.58 times(0', y) -> 0' 1134.79/291.58 times(s(x), y) -> plus(y, times(x, y)) 1134.79/291.58 log(x, 0') -> baseError 1134.79/291.58 log(x, s(0')) -> baseError 1134.79/291.58 log(0', s(s(b))) -> logZeroError 1134.79/291.58 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') 1134.79/291.58 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 1134.79/291.58 if(true, x, b, y, z) -> z 1134.79/291.58 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 1134.79/291.58 1134.79/291.58 Types: 1134.79/291.58 le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true 1134.79/291.58 s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 0' :: s:0':baseError:logZeroError 1134.79/291.58 false :: false:true 1134.79/291.58 true :: false:true 1134.79/291.58 plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 baseError :: s:0':baseError:logZeroError 1134.79/291.58 logZeroError :: s:0':baseError:logZeroError 1134.79/291.58 loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.58 hole_false:true1_0 :: false:true 1134.79/291.58 hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError 1134.79/291.58 gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError 1134.79/291.58 1134.79/291.58 1134.79/291.58 Lemmas: 1134.79/291.58 le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) -> false, rt in Omega(1 + n5_0) 1134.79/291.59 plus(gen_s:0':baseError:logZeroError3_0(n342_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(+(n342_0, b)), rt in Omega(1 + n342_0) 1134.79/291.59 1134.79/291.59 1134.79/291.59 Generator Equations: 1134.79/291.59 gen_s:0':baseError:logZeroError3_0(0) <=> 0' 1134.79/291.59 gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x)) 1134.79/291.59 1134.79/291.59 1134.79/291.59 The following defined symbols remain to be analysed: 1134.79/291.59 times, loop 1134.79/291.59 1134.79/291.59 They will be analysed ascendingly in the following order: 1134.79/291.59 times < loop 1134.79/291.59 1134.79/291.59 ---------------------------------------- 1134.79/291.59 1134.79/291.59 (18) LowerBoundPropagationProof (FINISHED) 1134.79/291.59 Propagated lower bound. 1134.79/291.59 ---------------------------------------- 1134.79/291.59 1134.79/291.59 (19) 1134.79/291.59 BOUNDS(n^2, INF) 1134.79/291.59 1134.79/291.59 ---------------------------------------- 1134.79/291.59 1134.79/291.59 (20) 1134.79/291.59 Obligation: 1134.79/291.59 Innermost TRS: 1134.79/291.59 Rules: 1134.79/291.59 le(s(x), 0') -> false 1134.79/291.59 le(0', y) -> true 1134.79/291.59 le(s(x), s(y)) -> le(x, y) 1134.79/291.59 plus(0', y) -> y 1134.79/291.59 plus(s(x), y) -> s(plus(x, y)) 1134.79/291.59 times(0', y) -> 0' 1134.79/291.59 times(s(x), y) -> plus(y, times(x, y)) 1134.79/291.59 log(x, 0') -> baseError 1134.79/291.59 log(x, s(0')) -> baseError 1134.79/291.59 log(0', s(s(b))) -> logZeroError 1134.79/291.59 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') 1134.79/291.59 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 1134.79/291.59 if(true, x, b, y, z) -> z 1134.79/291.59 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 1134.79/291.59 1134.79/291.59 Types: 1134.79/291.59 le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true 1134.79/291.59 s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.59 0' :: s:0':baseError:logZeroError 1134.79/291.59 false :: false:true 1134.79/291.59 true :: false:true 1134.79/291.59 plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.59 times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.59 log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.59 baseError :: s:0':baseError:logZeroError 1134.79/291.59 logZeroError :: s:0':baseError:logZeroError 1134.79/291.59 loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.59 if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 1134.79/291.59 hole_false:true1_0 :: false:true 1134.79/291.59 hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError 1134.79/291.59 gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError 1134.79/291.59 1134.79/291.59 1134.79/291.59 Lemmas: 1134.79/291.59 le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) -> false, rt in Omega(1 + n5_0) 1134.79/291.59 plus(gen_s:0':baseError:logZeroError3_0(n342_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(+(n342_0, b)), rt in Omega(1 + n342_0) 1134.79/291.59 times(gen_s:0':baseError:logZeroError3_0(n1027_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(*(n1027_0, b)), rt in Omega(1 + b*n1027_0 + n1027_0) 1134.79/291.59 1134.79/291.59 1134.79/291.59 Generator Equations: 1134.79/291.59 gen_s:0':baseError:logZeroError3_0(0) <=> 0' 1134.79/291.59 gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x)) 1134.79/291.59 1134.79/291.59 1134.79/291.59 The following defined symbols remain to be analysed: 1134.79/291.59 loop 1135.08/291.66 EOF