313.76/291.47 WORST_CASE(Omega(n^2), ?) 313.76/291.48 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 313.76/291.48 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 313.76/291.48 313.76/291.48 313.76/291.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 313.76/291.48 313.76/291.48 (0) CpxTRS 313.76/291.48 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 313.76/291.48 (2) CpxTRS 313.76/291.48 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 313.76/291.48 (4) typed CpxTrs 313.76/291.48 (5) OrderProof [LOWER BOUND(ID), 0 ms] 313.76/291.48 (6) typed CpxTrs 313.76/291.48 (7) RewriteLemmaProof [LOWER BOUND(ID), 302 ms] 313.76/291.48 (8) BEST 313.76/291.48 (9) proven lower bound 313.76/291.48 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 313.76/291.48 (11) BOUNDS(n^1, INF) 313.76/291.48 (12) typed CpxTrs 313.76/291.48 (13) RewriteLemmaProof [LOWER BOUND(ID), 77 ms] 313.76/291.48 (14) typed CpxTrs 313.76/291.48 (15) RewriteLemmaProof [LOWER BOUND(ID), 45 ms] 313.76/291.48 (16) BEST 313.76/291.48 (17) proven lower bound 313.76/291.48 (18) LowerBoundPropagationProof [FINISHED, 0 ms] 313.76/291.48 (19) BOUNDS(n^2, INF) 313.76/291.48 (20) typed CpxTrs 313.76/291.48 313.76/291.48 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (0) 313.76/291.48 Obligation: 313.76/291.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 313.76/291.48 313.76/291.48 313.76/291.48 The TRS R consists of the following rules: 313.76/291.48 313.76/291.48 le(s(x), 0) -> false 313.76/291.48 le(0, y) -> true 313.76/291.48 le(s(x), s(y)) -> le(x, y) 313.76/291.48 plus(0, y) -> y 313.76/291.48 plus(s(x), y) -> s(plus(x, y)) 313.76/291.48 times(0, y) -> 0 313.76/291.48 times(s(x), y) -> plus(y, times(x, y)) 313.76/291.48 log(x, 0) -> baseError 313.76/291.48 log(x, s(0)) -> baseError 313.76/291.48 log(0, s(s(b))) -> logZeroError 313.76/291.48 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0) 313.76/291.48 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 313.76/291.48 if(true, x, b, y, z) -> z 313.76/291.48 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 313.76/291.48 313.76/291.48 S is empty. 313.76/291.48 Rewrite Strategy: FULL 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 313.76/291.48 Renamed function symbols to avoid clashes with predefined symbol. 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (2) 313.76/291.48 Obligation: 313.76/291.48 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 313.76/291.48 313.76/291.48 313.76/291.48 The TRS R consists of the following rules: 313.76/291.48 313.76/291.48 le(s(x), 0') -> false 313.76/291.48 le(0', y) -> true 313.76/291.48 le(s(x), s(y)) -> le(x, y) 313.76/291.48 plus(0', y) -> y 313.76/291.48 plus(s(x), y) -> s(plus(x, y)) 313.76/291.48 times(0', y) -> 0' 313.76/291.48 times(s(x), y) -> plus(y, times(x, y)) 313.76/291.48 log(x, 0') -> baseError 313.76/291.48 log(x, s(0')) -> baseError 313.76/291.48 log(0', s(s(b))) -> logZeroError 313.76/291.48 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') 313.76/291.48 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 313.76/291.48 if(true, x, b, y, z) -> z 313.76/291.48 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 313.76/291.48 313.76/291.48 S is empty. 313.76/291.48 Rewrite Strategy: FULL 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 313.76/291.48 Infered types. 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (4) 313.76/291.48 Obligation: 313.76/291.48 TRS: 313.76/291.48 Rules: 313.76/291.48 le(s(x), 0') -> false 313.76/291.48 le(0', y) -> true 313.76/291.48 le(s(x), s(y)) -> le(x, y) 313.76/291.48 plus(0', y) -> y 313.76/291.48 plus(s(x), y) -> s(plus(x, y)) 313.76/291.48 times(0', y) -> 0' 313.76/291.48 times(s(x), y) -> plus(y, times(x, y)) 313.76/291.48 log(x, 0') -> baseError 313.76/291.48 log(x, s(0')) -> baseError 313.76/291.48 log(0', s(s(b))) -> logZeroError 313.76/291.48 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') 313.76/291.48 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 313.76/291.48 if(true, x, b, y, z) -> z 313.76/291.48 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 313.76/291.48 313.76/291.48 Types: 313.76/291.48 le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true 313.76/291.48 s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 0' :: s:0':baseError:logZeroError 313.76/291.48 false :: false:true 313.76/291.48 true :: false:true 313.76/291.48 plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 baseError :: s:0':baseError:logZeroError 313.76/291.48 logZeroError :: s:0':baseError:logZeroError 313.76/291.48 loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 hole_false:true1_0 :: false:true 313.76/291.48 hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError 313.76/291.48 gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError 313.76/291.48 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (5) OrderProof (LOWER BOUND(ID)) 313.76/291.48 Heuristically decided to analyse the following defined symbols: 313.76/291.48 le, plus, times, loop 313.76/291.48 313.76/291.48 They will be analysed ascendingly in the following order: 313.76/291.48 le < loop 313.76/291.48 plus < times 313.76/291.48 times < loop 313.76/291.48 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (6) 313.76/291.48 Obligation: 313.76/291.48 TRS: 313.76/291.48 Rules: 313.76/291.48 le(s(x), 0') -> false 313.76/291.48 le(0', y) -> true 313.76/291.48 le(s(x), s(y)) -> le(x, y) 313.76/291.48 plus(0', y) -> y 313.76/291.48 plus(s(x), y) -> s(plus(x, y)) 313.76/291.48 times(0', y) -> 0' 313.76/291.48 times(s(x), y) -> plus(y, times(x, y)) 313.76/291.48 log(x, 0') -> baseError 313.76/291.48 log(x, s(0')) -> baseError 313.76/291.48 log(0', s(s(b))) -> logZeroError 313.76/291.48 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') 313.76/291.48 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 313.76/291.48 if(true, x, b, y, z) -> z 313.76/291.48 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 313.76/291.48 313.76/291.48 Types: 313.76/291.48 le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true 313.76/291.48 s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 0' :: s:0':baseError:logZeroError 313.76/291.48 false :: false:true 313.76/291.48 true :: false:true 313.76/291.48 plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 baseError :: s:0':baseError:logZeroError 313.76/291.48 logZeroError :: s:0':baseError:logZeroError 313.76/291.48 loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 hole_false:true1_0 :: false:true 313.76/291.48 hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError 313.76/291.48 gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError 313.76/291.48 313.76/291.48 313.76/291.48 Generator Equations: 313.76/291.48 gen_s:0':baseError:logZeroError3_0(0) <=> 0' 313.76/291.48 gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x)) 313.76/291.48 313.76/291.48 313.76/291.48 The following defined symbols remain to be analysed: 313.76/291.48 le, plus, times, loop 313.76/291.48 313.76/291.48 They will be analysed ascendingly in the following order: 313.76/291.48 le < loop 313.76/291.48 plus < times 313.76/291.48 times < loop 313.76/291.48 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (7) RewriteLemmaProof (LOWER BOUND(ID)) 313.76/291.48 Proved the following rewrite lemma: 313.76/291.48 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) 313.76/291.48 313.76/291.48 Induction Base: 313.76/291.48 le(gen_s:0':baseError:logZeroError3_0(+(1, 0)), gen_s:0':baseError:logZeroError3_0(0)) ->_R^Omega(1) 313.76/291.48 false 313.76/291.48 313.76/291.48 Induction Step: 313.76/291.48 le(gen_s:0':baseError:logZeroError3_0(+(1, +(n5_0, 1))), gen_s:0':baseError:logZeroError3_0(+(n5_0, 1))) ->_R^Omega(1) 313.76/291.48 le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) ->_IH 313.76/291.48 false 313.76/291.48 313.76/291.48 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (8) 313.76/291.48 Complex Obligation (BEST) 313.76/291.48 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (9) 313.76/291.48 Obligation: 313.76/291.48 Proved the lower bound n^1 for the following obligation: 313.76/291.48 313.76/291.48 TRS: 313.76/291.48 Rules: 313.76/291.48 le(s(x), 0') -> false 313.76/291.48 le(0', y) -> true 313.76/291.48 le(s(x), s(y)) -> le(x, y) 313.76/291.48 plus(0', y) -> y 313.76/291.48 plus(s(x), y) -> s(plus(x, y)) 313.76/291.48 times(0', y) -> 0' 313.76/291.48 times(s(x), y) -> plus(y, times(x, y)) 313.76/291.48 log(x, 0') -> baseError 313.76/291.48 log(x, s(0')) -> baseError 313.76/291.48 log(0', s(s(b))) -> logZeroError 313.76/291.48 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') 313.76/291.48 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 313.76/291.48 if(true, x, b, y, z) -> z 313.76/291.48 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 313.76/291.48 313.76/291.48 Types: 313.76/291.48 le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true 313.76/291.48 s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 0' :: s:0':baseError:logZeroError 313.76/291.48 false :: false:true 313.76/291.48 true :: false:true 313.76/291.48 plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 baseError :: s:0':baseError:logZeroError 313.76/291.48 logZeroError :: s:0':baseError:logZeroError 313.76/291.48 loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 hole_false:true1_0 :: false:true 313.76/291.48 hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError 313.76/291.48 gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError 313.76/291.48 313.76/291.48 313.76/291.48 Generator Equations: 313.76/291.48 gen_s:0':baseError:logZeroError3_0(0) <=> 0' 313.76/291.48 gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x)) 313.76/291.48 313.76/291.48 313.76/291.48 The following defined symbols remain to be analysed: 313.76/291.48 le, plus, times, loop 313.76/291.48 313.76/291.48 They will be analysed ascendingly in the following order: 313.76/291.48 le < loop 313.76/291.48 plus < times 313.76/291.48 times < loop 313.76/291.48 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (10) LowerBoundPropagationProof (FINISHED) 313.76/291.48 Propagated lower bound. 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (11) 313.76/291.48 BOUNDS(n^1, INF) 313.76/291.48 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (12) 313.76/291.48 Obligation: 313.76/291.48 TRS: 313.76/291.48 Rules: 313.76/291.48 le(s(x), 0') -> false 313.76/291.48 le(0', y) -> true 313.76/291.48 le(s(x), s(y)) -> le(x, y) 313.76/291.48 plus(0', y) -> y 313.76/291.48 plus(s(x), y) -> s(plus(x, y)) 313.76/291.48 times(0', y) -> 0' 313.76/291.48 times(s(x), y) -> plus(y, times(x, y)) 313.76/291.48 log(x, 0') -> baseError 313.76/291.48 log(x, s(0')) -> baseError 313.76/291.48 log(0', s(s(b))) -> logZeroError 313.76/291.48 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') 313.76/291.48 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 313.76/291.48 if(true, x, b, y, z) -> z 313.76/291.48 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 313.76/291.48 313.76/291.48 Types: 313.76/291.48 le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true 313.76/291.48 s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 0' :: s:0':baseError:logZeroError 313.76/291.48 false :: false:true 313.76/291.48 true :: false:true 313.76/291.48 plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 baseError :: s:0':baseError:logZeroError 313.76/291.48 logZeroError :: s:0':baseError:logZeroError 313.76/291.48 loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 hole_false:true1_0 :: false:true 313.76/291.48 hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError 313.76/291.48 gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError 313.76/291.48 313.76/291.48 313.76/291.48 Lemmas: 313.76/291.48 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) 313.76/291.48 313.76/291.48 313.76/291.48 Generator Equations: 313.76/291.48 gen_s:0':baseError:logZeroError3_0(0) <=> 0' 313.76/291.48 gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x)) 313.76/291.48 313.76/291.48 313.76/291.48 The following defined symbols remain to be analysed: 313.76/291.48 plus, times, loop 313.76/291.48 313.76/291.48 They will be analysed ascendingly in the following order: 313.76/291.48 plus < times 313.76/291.48 times < loop 313.76/291.48 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (13) RewriteLemmaProof (LOWER BOUND(ID)) 313.76/291.48 Proved the following rewrite lemma: 313.76/291.48 plus(gen_s:0':baseError:logZeroError3_0(n276_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(+(n276_0, b)), rt in Omega(1 + n276_0) 313.76/291.48 313.76/291.48 Induction Base: 313.76/291.48 plus(gen_s:0':baseError:logZeroError3_0(0), gen_s:0':baseError:logZeroError3_0(b)) ->_R^Omega(1) 313.76/291.48 gen_s:0':baseError:logZeroError3_0(b) 313.76/291.48 313.76/291.48 Induction Step: 313.76/291.48 plus(gen_s:0':baseError:logZeroError3_0(+(n276_0, 1)), gen_s:0':baseError:logZeroError3_0(b)) ->_R^Omega(1) 313.76/291.48 s(plus(gen_s:0':baseError:logZeroError3_0(n276_0), gen_s:0':baseError:logZeroError3_0(b))) ->_IH 313.76/291.48 s(gen_s:0':baseError:logZeroError3_0(+(b, c277_0))) 313.76/291.48 313.76/291.48 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (14) 313.76/291.48 Obligation: 313.76/291.48 TRS: 313.76/291.48 Rules: 313.76/291.48 le(s(x), 0') -> false 313.76/291.48 le(0', y) -> true 313.76/291.48 le(s(x), s(y)) -> le(x, y) 313.76/291.48 plus(0', y) -> y 313.76/291.48 plus(s(x), y) -> s(plus(x, y)) 313.76/291.48 times(0', y) -> 0' 313.76/291.48 times(s(x), y) -> plus(y, times(x, y)) 313.76/291.48 log(x, 0') -> baseError 313.76/291.48 log(x, s(0')) -> baseError 313.76/291.48 log(0', s(s(b))) -> logZeroError 313.76/291.48 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') 313.76/291.48 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 313.76/291.48 if(true, x, b, y, z) -> z 313.76/291.48 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 313.76/291.48 313.76/291.48 Types: 313.76/291.48 le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true 313.76/291.48 s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 0' :: s:0':baseError:logZeroError 313.76/291.48 false :: false:true 313.76/291.48 true :: false:true 313.76/291.48 plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 baseError :: s:0':baseError:logZeroError 313.76/291.48 logZeroError :: s:0':baseError:logZeroError 313.76/291.48 loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 hole_false:true1_0 :: false:true 313.76/291.48 hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError 313.76/291.48 gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError 313.76/291.48 313.76/291.48 313.76/291.48 Lemmas: 313.76/291.48 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) 313.76/291.48 plus(gen_s:0':baseError:logZeroError3_0(n276_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(+(n276_0, b)), rt in Omega(1 + n276_0) 313.76/291.48 313.76/291.48 313.76/291.48 Generator Equations: 313.76/291.48 gen_s:0':baseError:logZeroError3_0(0) <=> 0' 313.76/291.48 gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x)) 313.76/291.48 313.76/291.48 313.76/291.48 The following defined symbols remain to be analysed: 313.76/291.48 times, loop 313.76/291.48 313.76/291.48 They will be analysed ascendingly in the following order: 313.76/291.48 times < loop 313.76/291.48 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (15) RewriteLemmaProof (LOWER BOUND(ID)) 313.76/291.48 Proved the following rewrite lemma: 313.76/291.48 times(gen_s:0':baseError:logZeroError3_0(n873_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(*(n873_0, b)), rt in Omega(1 + b*n873_0 + n873_0) 313.76/291.48 313.76/291.48 Induction Base: 313.76/291.48 times(gen_s:0':baseError:logZeroError3_0(0), gen_s:0':baseError:logZeroError3_0(b)) ->_R^Omega(1) 313.76/291.48 0' 313.76/291.48 313.76/291.48 Induction Step: 313.76/291.48 times(gen_s:0':baseError:logZeroError3_0(+(n873_0, 1)), gen_s:0':baseError:logZeroError3_0(b)) ->_R^Omega(1) 313.76/291.48 plus(gen_s:0':baseError:logZeroError3_0(b), times(gen_s:0':baseError:logZeroError3_0(n873_0), gen_s:0':baseError:logZeroError3_0(b))) ->_IH 313.76/291.48 plus(gen_s:0':baseError:logZeroError3_0(b), gen_s:0':baseError:logZeroError3_0(*(c874_0, b))) ->_L^Omega(1 + b) 313.76/291.48 gen_s:0':baseError:logZeroError3_0(+(b, *(n873_0, b))) 313.76/291.48 313.76/291.48 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (16) 313.76/291.48 Complex Obligation (BEST) 313.76/291.48 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (17) 313.76/291.48 Obligation: 313.76/291.48 Proved the lower bound n^2 for the following obligation: 313.76/291.48 313.76/291.48 TRS: 313.76/291.48 Rules: 313.76/291.48 le(s(x), 0') -> false 313.76/291.48 le(0', y) -> true 313.76/291.48 le(s(x), s(y)) -> le(x, y) 313.76/291.48 plus(0', y) -> y 313.76/291.48 plus(s(x), y) -> s(plus(x, y)) 313.76/291.48 times(0', y) -> 0' 313.76/291.48 times(s(x), y) -> plus(y, times(x, y)) 313.76/291.48 log(x, 0') -> baseError 313.76/291.48 log(x, s(0')) -> baseError 313.76/291.48 log(0', s(s(b))) -> logZeroError 313.76/291.48 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') 313.76/291.48 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 313.76/291.48 if(true, x, b, y, z) -> z 313.76/291.48 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 313.76/291.48 313.76/291.48 Types: 313.76/291.48 le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true 313.76/291.48 s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 0' :: s:0':baseError:logZeroError 313.76/291.48 false :: false:true 313.76/291.48 true :: false:true 313.76/291.48 plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 baseError :: s:0':baseError:logZeroError 313.76/291.48 logZeroError :: s:0':baseError:logZeroError 313.76/291.48 loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 hole_false:true1_0 :: false:true 313.76/291.48 hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError 313.76/291.48 gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError 313.76/291.48 313.76/291.48 313.76/291.48 Lemmas: 313.76/291.48 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) 313.76/291.48 plus(gen_s:0':baseError:logZeroError3_0(n276_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(+(n276_0, b)), rt in Omega(1 + n276_0) 313.76/291.48 313.76/291.48 313.76/291.48 Generator Equations: 313.76/291.48 gen_s:0':baseError:logZeroError3_0(0) <=> 0' 313.76/291.48 gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x)) 313.76/291.48 313.76/291.48 313.76/291.48 The following defined symbols remain to be analysed: 313.76/291.48 times, loop 313.76/291.48 313.76/291.48 They will be analysed ascendingly in the following order: 313.76/291.48 times < loop 313.76/291.48 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (18) LowerBoundPropagationProof (FINISHED) 313.76/291.48 Propagated lower bound. 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (19) 313.76/291.48 BOUNDS(n^2, INF) 313.76/291.48 313.76/291.48 ---------------------------------------- 313.76/291.48 313.76/291.48 (20) 313.76/291.48 Obligation: 313.76/291.48 TRS: 313.76/291.48 Rules: 313.76/291.48 le(s(x), 0') -> false 313.76/291.48 le(0', y) -> true 313.76/291.48 le(s(x), s(y)) -> le(x, y) 313.76/291.48 plus(0', y) -> y 313.76/291.48 plus(s(x), y) -> s(plus(x, y)) 313.76/291.48 times(0', y) -> 0' 313.76/291.48 times(s(x), y) -> plus(y, times(x, y)) 313.76/291.48 log(x, 0') -> baseError 313.76/291.48 log(x, s(0')) -> baseError 313.76/291.48 log(0', s(s(b))) -> logZeroError 313.76/291.48 log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') 313.76/291.48 loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) 313.76/291.48 if(true, x, b, y, z) -> z 313.76/291.48 if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) 313.76/291.48 313.76/291.48 Types: 313.76/291.48 le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true 313.76/291.48 s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 0' :: s:0':baseError:logZeroError 313.76/291.48 false :: false:true 313.76/291.48 true :: false:true 313.76/291.48 plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 baseError :: s:0':baseError:logZeroError 313.76/291.48 logZeroError :: s:0':baseError:logZeroError 313.76/291.48 loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError 313.76/291.48 hole_false:true1_0 :: false:true 313.76/291.48 hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError 313.76/291.48 gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError 313.76/291.48 313.76/291.48 313.76/291.48 Lemmas: 313.76/291.48 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) 313.76/291.48 plus(gen_s:0':baseError:logZeroError3_0(n276_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(+(n276_0, b)), rt in Omega(1 + n276_0) 313.76/291.48 times(gen_s:0':baseError:logZeroError3_0(n873_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(*(n873_0, b)), rt in Omega(1 + b*n873_0 + n873_0) 313.76/291.48 313.76/291.48 313.76/291.48 Generator Equations: 313.76/291.48 gen_s:0':baseError:logZeroError3_0(0) <=> 0' 313.76/291.48 gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x)) 313.76/291.48 313.76/291.48 313.76/291.48 The following defined symbols remain to be analysed: 313.76/291.48 loop 313.79/291.52 EOF