312.77/291.55 WORST_CASE(Omega(n^1), ?) 312.77/291.56 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 312.77/291.56 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 312.77/291.56 312.77/291.56 312.77/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 312.77/291.56 312.77/291.56 (0) CpxTRS 312.77/291.56 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 312.77/291.56 (2) CpxTRS 312.77/291.56 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 312.77/291.56 (4) typed CpxTrs 312.77/291.56 (5) OrderProof [LOWER BOUND(ID), 0 ms] 312.77/291.56 (6) typed CpxTrs 312.77/291.56 (7) RewriteLemmaProof [LOWER BOUND(ID), 308 ms] 312.77/291.56 (8) BEST 312.77/291.56 (9) proven lower bound 312.77/291.56 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 312.77/291.56 (11) BOUNDS(n^1, INF) 312.77/291.56 (12) typed CpxTrs 312.77/291.56 (13) RewriteLemmaProof [LOWER BOUND(ID), 70 ms] 312.77/291.56 (14) typed CpxTrs 312.77/291.56 312.77/291.56 312.77/291.56 ---------------------------------------- 312.77/291.56 312.77/291.56 (0) 312.77/291.56 Obligation: 312.77/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 312.77/291.56 312.77/291.56 312.77/291.56 The TRS R consists of the following rules: 312.77/291.56 312.77/291.56 le(s(x), 0) -> false 312.77/291.56 le(0, y) -> true 312.77/291.56 le(s(x), s(y)) -> le(x, y) 312.77/291.56 double(0) -> 0 312.77/291.56 double(s(x)) -> s(s(double(x))) 312.77/291.56 log(0) -> logError 312.77/291.56 log(s(x)) -> loop(s(x), s(0), 0) 312.77/291.56 loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) 312.77/291.56 if(true, x, y, z) -> z 312.77/291.56 if(false, x, y, z) -> loop(x, double(y), s(z)) 312.77/291.56 312.77/291.56 S is empty. 312.77/291.56 Rewrite Strategy: FULL 312.77/291.56 ---------------------------------------- 312.77/291.56 312.77/291.56 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 312.77/291.56 Renamed function symbols to avoid clashes with predefined symbol. 312.77/291.56 ---------------------------------------- 312.77/291.56 312.77/291.56 (2) 312.77/291.56 Obligation: 312.77/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 312.77/291.56 312.77/291.56 312.77/291.56 The TRS R consists of the following rules: 312.77/291.56 312.77/291.56 le(s(x), 0') -> false 312.77/291.56 le(0', y) -> true 312.77/291.56 le(s(x), s(y)) -> le(x, y) 312.77/291.56 double(0') -> 0' 312.77/291.56 double(s(x)) -> s(s(double(x))) 312.77/291.56 log(0') -> logError 312.77/291.56 log(s(x)) -> loop(s(x), s(0'), 0') 312.77/291.56 loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) 312.77/291.56 if(true, x, y, z) -> z 312.77/291.56 if(false, x, y, z) -> loop(x, double(y), s(z)) 312.77/291.56 312.77/291.56 S is empty. 312.77/291.56 Rewrite Strategy: FULL 312.77/291.56 ---------------------------------------- 312.77/291.56 312.77/291.56 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 312.77/291.56 Infered types. 312.77/291.56 ---------------------------------------- 312.77/291.56 312.77/291.56 (4) 312.77/291.56 Obligation: 312.77/291.56 TRS: 312.77/291.56 Rules: 312.77/291.56 le(s(x), 0') -> false 312.77/291.56 le(0', y) -> true 312.77/291.56 le(s(x), s(y)) -> le(x, y) 312.77/291.56 double(0') -> 0' 312.77/291.56 double(s(x)) -> s(s(double(x))) 312.77/291.56 log(0') -> logError 312.77/291.56 log(s(x)) -> loop(s(x), s(0'), 0') 312.77/291.56 loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) 312.77/291.56 if(true, x, y, z) -> z 312.77/291.56 if(false, x, y, z) -> loop(x, double(y), s(z)) 312.77/291.56 312.77/291.56 Types: 312.77/291.56 le :: s:0':logError -> s:0':logError -> false:true 312.77/291.56 s :: s:0':logError -> s:0':logError 312.77/291.56 0' :: s:0':logError 312.77/291.56 false :: false:true 312.77/291.56 true :: false:true 312.77/291.56 double :: s:0':logError -> s:0':logError 312.77/291.56 log :: s:0':logError -> s:0':logError 312.77/291.56 logError :: s:0':logError 312.77/291.56 loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError 312.77/291.56 if :: false:true -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError 312.77/291.56 hole_false:true1_0 :: false:true 312.77/291.56 hole_s:0':logError2_0 :: s:0':logError 312.77/291.56 gen_s:0':logError3_0 :: Nat -> s:0':logError 312.77/291.56 312.77/291.56 ---------------------------------------- 312.77/291.56 312.77/291.56 (5) OrderProof (LOWER BOUND(ID)) 312.77/291.56 Heuristically decided to analyse the following defined symbols: 312.77/291.56 le, double, loop 312.77/291.56 312.77/291.56 They will be analysed ascendingly in the following order: 312.77/291.56 le < loop 312.77/291.56 double < loop 312.77/291.56 312.77/291.56 ---------------------------------------- 312.77/291.56 312.77/291.56 (6) 312.77/291.56 Obligation: 312.77/291.56 TRS: 312.77/291.56 Rules: 312.77/291.56 le(s(x), 0') -> false 312.77/291.56 le(0', y) -> true 312.77/291.56 le(s(x), s(y)) -> le(x, y) 312.77/291.56 double(0') -> 0' 312.77/291.56 double(s(x)) -> s(s(double(x))) 312.77/291.56 log(0') -> logError 312.77/291.56 log(s(x)) -> loop(s(x), s(0'), 0') 312.77/291.56 loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) 312.77/291.56 if(true, x, y, z) -> z 312.77/291.56 if(false, x, y, z) -> loop(x, double(y), s(z)) 312.77/291.56 312.77/291.56 Types: 312.77/291.56 le :: s:0':logError -> s:0':logError -> false:true 312.77/291.56 s :: s:0':logError -> s:0':logError 312.77/291.56 0' :: s:0':logError 312.77/291.56 false :: false:true 312.77/291.56 true :: false:true 312.77/291.56 double :: s:0':logError -> s:0':logError 312.77/291.56 log :: s:0':logError -> s:0':logError 312.77/291.56 logError :: s:0':logError 312.77/291.56 loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError 312.77/291.56 if :: false:true -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError 312.77/291.56 hole_false:true1_0 :: false:true 312.77/291.56 hole_s:0':logError2_0 :: s:0':logError 312.77/291.56 gen_s:0':logError3_0 :: Nat -> s:0':logError 312.77/291.56 312.77/291.56 312.77/291.56 Generator Equations: 312.77/291.56 gen_s:0':logError3_0(0) <=> 0' 312.77/291.56 gen_s:0':logError3_0(+(x, 1)) <=> s(gen_s:0':logError3_0(x)) 312.77/291.56 312.77/291.56 312.77/291.56 The following defined symbols remain to be analysed: 312.77/291.56 le, double, loop 312.77/291.56 312.77/291.56 They will be analysed ascendingly in the following order: 312.77/291.56 le < loop 312.77/291.56 double < loop 312.77/291.56 312.77/291.56 ---------------------------------------- 312.77/291.56 312.77/291.56 (7) RewriteLemmaProof (LOWER BOUND(ID)) 312.77/291.56 Proved the following rewrite lemma: 312.77/291.56 le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) -> false, rt in Omega(1 + n5_0) 312.77/291.56 312.77/291.56 Induction Base: 312.77/291.56 le(gen_s:0':logError3_0(+(1, 0)), gen_s:0':logError3_0(0)) ->_R^Omega(1) 312.77/291.56 false 312.77/291.56 312.77/291.56 Induction Step: 312.77/291.56 le(gen_s:0':logError3_0(+(1, +(n5_0, 1))), gen_s:0':logError3_0(+(n5_0, 1))) ->_R^Omega(1) 312.77/291.56 le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) ->_IH 312.77/291.56 false 312.77/291.56 312.77/291.56 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 312.77/291.56 ---------------------------------------- 312.77/291.56 312.77/291.56 (8) 312.77/291.56 Complex Obligation (BEST) 312.77/291.56 312.77/291.56 ---------------------------------------- 312.77/291.56 312.77/291.56 (9) 312.77/291.56 Obligation: 312.77/291.56 Proved the lower bound n^1 for the following obligation: 312.77/291.56 312.77/291.56 TRS: 312.77/291.56 Rules: 312.77/291.56 le(s(x), 0') -> false 312.77/291.56 le(0', y) -> true 312.77/291.56 le(s(x), s(y)) -> le(x, y) 312.77/291.56 double(0') -> 0' 312.77/291.56 double(s(x)) -> s(s(double(x))) 312.77/291.56 log(0') -> logError 312.77/291.56 log(s(x)) -> loop(s(x), s(0'), 0') 312.77/291.56 loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) 312.77/291.56 if(true, x, y, z) -> z 312.77/291.56 if(false, x, y, z) -> loop(x, double(y), s(z)) 312.77/291.56 312.77/291.56 Types: 312.77/291.56 le :: s:0':logError -> s:0':logError -> false:true 312.77/291.56 s :: s:0':logError -> s:0':logError 312.77/291.56 0' :: s:0':logError 312.77/291.56 false :: false:true 312.77/291.56 true :: false:true 312.77/291.56 double :: s:0':logError -> s:0':logError 312.77/291.56 log :: s:0':logError -> s:0':logError 312.77/291.56 logError :: s:0':logError 312.77/291.56 loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError 312.77/291.56 if :: false:true -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError 312.77/291.56 hole_false:true1_0 :: false:true 312.77/291.56 hole_s:0':logError2_0 :: s:0':logError 312.77/291.56 gen_s:0':logError3_0 :: Nat -> s:0':logError 312.77/291.56 312.77/291.56 312.77/291.56 Generator Equations: 312.77/291.56 gen_s:0':logError3_0(0) <=> 0' 312.77/291.56 gen_s:0':logError3_0(+(x, 1)) <=> s(gen_s:0':logError3_0(x)) 312.77/291.56 312.77/291.56 312.77/291.56 The following defined symbols remain to be analysed: 312.77/291.56 le, double, loop 312.77/291.56 312.77/291.56 They will be analysed ascendingly in the following order: 312.77/291.56 le < loop 312.77/291.56 double < loop 312.77/291.56 312.77/291.56 ---------------------------------------- 312.77/291.56 312.77/291.56 (10) LowerBoundPropagationProof (FINISHED) 312.77/291.56 Propagated lower bound. 312.77/291.56 ---------------------------------------- 312.77/291.56 312.77/291.56 (11) 312.77/291.56 BOUNDS(n^1, INF) 312.77/291.56 312.77/291.56 ---------------------------------------- 312.77/291.56 312.77/291.56 (12) 312.77/291.56 Obligation: 312.77/291.56 TRS: 312.77/291.56 Rules: 312.77/291.56 le(s(x), 0') -> false 312.77/291.56 le(0', y) -> true 312.77/291.56 le(s(x), s(y)) -> le(x, y) 312.77/291.56 double(0') -> 0' 312.77/291.56 double(s(x)) -> s(s(double(x))) 312.77/291.56 log(0') -> logError 312.77/291.56 log(s(x)) -> loop(s(x), s(0'), 0') 312.77/291.56 loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) 312.77/291.56 if(true, x, y, z) -> z 312.77/291.56 if(false, x, y, z) -> loop(x, double(y), s(z)) 312.77/291.56 312.77/291.56 Types: 312.77/291.56 le :: s:0':logError -> s:0':logError -> false:true 312.77/291.56 s :: s:0':logError -> s:0':logError 312.77/291.56 0' :: s:0':logError 312.77/291.56 false :: false:true 312.77/291.56 true :: false:true 312.77/291.56 double :: s:0':logError -> s:0':logError 312.77/291.56 log :: s:0':logError -> s:0':logError 312.77/291.56 logError :: s:0':logError 312.77/291.56 loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError 312.77/291.56 if :: false:true -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError 312.77/291.56 hole_false:true1_0 :: false:true 312.77/291.56 hole_s:0':logError2_0 :: s:0':logError 312.77/291.56 gen_s:0':logError3_0 :: Nat -> s:0':logError 312.77/291.56 312.77/291.56 312.77/291.56 Lemmas: 312.77/291.56 le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) -> false, rt in Omega(1 + n5_0) 312.77/291.56 312.77/291.56 312.77/291.56 Generator Equations: 312.77/291.56 gen_s:0':logError3_0(0) <=> 0' 312.77/291.56 gen_s:0':logError3_0(+(x, 1)) <=> s(gen_s:0':logError3_0(x)) 312.77/291.56 312.77/291.56 312.77/291.56 The following defined symbols remain to be analysed: 312.77/291.56 double, loop 312.77/291.56 312.77/291.56 They will be analysed ascendingly in the following order: 312.77/291.56 double < loop 312.77/291.56 312.77/291.56 ---------------------------------------- 312.77/291.56 312.77/291.56 (13) RewriteLemmaProof (LOWER BOUND(ID)) 312.77/291.56 Proved the following rewrite lemma: 312.77/291.56 double(gen_s:0':logError3_0(n252_0)) -> gen_s:0':logError3_0(*(2, n252_0)), rt in Omega(1 + n252_0) 312.77/291.56 312.77/291.56 Induction Base: 312.77/291.56 double(gen_s:0':logError3_0(0)) ->_R^Omega(1) 312.77/291.56 0' 312.77/291.56 312.77/291.56 Induction Step: 312.77/291.56 double(gen_s:0':logError3_0(+(n252_0, 1))) ->_R^Omega(1) 312.77/291.56 s(s(double(gen_s:0':logError3_0(n252_0)))) ->_IH 312.77/291.56 s(s(gen_s:0':logError3_0(*(2, c253_0)))) 312.77/291.56 312.77/291.56 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 312.77/291.56 ---------------------------------------- 312.77/291.56 312.77/291.56 (14) 312.77/291.56 Obligation: 312.77/291.56 TRS: 312.77/291.56 Rules: 312.77/291.56 le(s(x), 0') -> false 312.77/291.56 le(0', y) -> true 312.77/291.56 le(s(x), s(y)) -> le(x, y) 312.77/291.56 double(0') -> 0' 312.77/291.56 double(s(x)) -> s(s(double(x))) 312.77/291.56 log(0') -> logError 312.77/291.56 log(s(x)) -> loop(s(x), s(0'), 0') 312.77/291.56 loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) 312.77/291.56 if(true, x, y, z) -> z 312.77/291.56 if(false, x, y, z) -> loop(x, double(y), s(z)) 312.77/291.56 312.77/291.56 Types: 312.77/291.56 le :: s:0':logError -> s:0':logError -> false:true 312.77/291.56 s :: s:0':logError -> s:0':logError 312.77/291.56 0' :: s:0':logError 312.77/291.56 false :: false:true 312.77/291.56 true :: false:true 312.77/291.56 double :: s:0':logError -> s:0':logError 312.77/291.56 log :: s:0':logError -> s:0':logError 312.77/291.56 logError :: s:0':logError 312.77/291.56 loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError 312.77/291.56 if :: false:true -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError 312.77/291.56 hole_false:true1_0 :: false:true 312.77/291.56 hole_s:0':logError2_0 :: s:0':logError 312.77/291.56 gen_s:0':logError3_0 :: Nat -> s:0':logError 312.77/291.56 312.77/291.56 312.77/291.56 Lemmas: 312.77/291.56 le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) -> false, rt in Omega(1 + n5_0) 312.77/291.56 double(gen_s:0':logError3_0(n252_0)) -> gen_s:0':logError3_0(*(2, n252_0)), rt in Omega(1 + n252_0) 312.77/291.56 312.77/291.56 312.77/291.56 Generator Equations: 312.77/291.56 gen_s:0':logError3_0(0) <=> 0' 312.77/291.56 gen_s:0':logError3_0(+(x, 1)) <=> s(gen_s:0':logError3_0(x)) 312.77/291.56 312.77/291.56 312.77/291.56 The following defined symbols remain to be analysed: 312.77/291.56 loop 312.77/291.59 EOF