306.53/291.53 WORST_CASE(Omega(n^1), ?) 306.53/291.54 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 306.53/291.54 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 306.53/291.54 306.53/291.54 306.53/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.53/291.54 306.53/291.54 (0) CpxTRS 306.53/291.54 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 306.53/291.54 (2) CpxTRS 306.53/291.54 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 306.53/291.54 (4) typed CpxTrs 306.53/291.54 (5) OrderProof [LOWER BOUND(ID), 0 ms] 306.53/291.54 (6) typed CpxTrs 306.53/291.54 (7) RewriteLemmaProof [LOWER BOUND(ID), 301 ms] 306.53/291.54 (8) BEST 306.53/291.54 (9) proven lower bound 306.53/291.54 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 306.53/291.54 (11) BOUNDS(n^1, INF) 306.53/291.54 (12) typed CpxTrs 306.53/291.54 306.53/291.54 306.53/291.54 ---------------------------------------- 306.53/291.54 306.53/291.54 (0) 306.53/291.54 Obligation: 306.53/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.53/291.54 306.53/291.54 306.53/291.54 The TRS R consists of the following rules: 306.53/291.54 306.53/291.54 le(0, y) -> true 306.53/291.54 le(s(x), 0) -> false 306.53/291.54 le(s(x), s(y)) -> le(x, y) 306.53/291.54 minus(0, y) -> 0 306.53/291.54 minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) 306.53/291.54 if_minus(true, s(x), y) -> 0 306.53/291.54 if_minus(false, s(x), y) -> s(minus(x, y)) 306.53/291.54 mod(0, y) -> 0 306.53/291.54 mod(s(x), 0) -> 0 306.53/291.54 mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) 306.53/291.54 if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) 306.53/291.54 if_mod(false, s(x), s(y)) -> s(x) 306.53/291.54 306.53/291.54 S is empty. 306.53/291.54 Rewrite Strategy: FULL 306.53/291.54 ---------------------------------------- 306.53/291.54 306.53/291.54 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 306.53/291.54 Renamed function symbols to avoid clashes with predefined symbol. 306.53/291.54 ---------------------------------------- 306.53/291.54 306.53/291.54 (2) 306.53/291.54 Obligation: 306.53/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 306.53/291.54 306.53/291.54 306.53/291.54 The TRS R consists of the following rules: 306.53/291.54 306.53/291.54 le(0', y) -> true 306.53/291.54 le(s(x), 0') -> false 306.53/291.54 le(s(x), s(y)) -> le(x, y) 306.53/291.54 minus(0', y) -> 0' 306.53/291.54 minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) 306.53/291.54 if_minus(true, s(x), y) -> 0' 306.53/291.54 if_minus(false, s(x), y) -> s(minus(x, y)) 306.53/291.54 mod(0', y) -> 0' 306.53/291.54 mod(s(x), 0') -> 0' 306.53/291.54 mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) 306.53/291.54 if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) 306.53/291.54 if_mod(false, s(x), s(y)) -> s(x) 306.53/291.54 306.53/291.54 S is empty. 306.53/291.54 Rewrite Strategy: FULL 306.53/291.54 ---------------------------------------- 306.53/291.54 306.53/291.54 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 306.53/291.54 Infered types. 306.53/291.54 ---------------------------------------- 306.53/291.54 306.53/291.54 (4) 306.53/291.54 Obligation: 306.53/291.54 TRS: 306.53/291.54 Rules: 306.53/291.54 le(0', y) -> true 306.53/291.54 le(s(x), 0') -> false 306.53/291.54 le(s(x), s(y)) -> le(x, y) 306.53/291.54 minus(0', y) -> 0' 306.53/291.54 minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) 306.53/291.54 if_minus(true, s(x), y) -> 0' 306.53/291.54 if_minus(false, s(x), y) -> s(minus(x, y)) 306.53/291.54 mod(0', y) -> 0' 306.53/291.54 mod(s(x), 0') -> 0' 306.53/291.54 mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) 306.53/291.54 if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) 306.53/291.54 if_mod(false, s(x), s(y)) -> s(x) 306.53/291.54 306.53/291.54 Types: 306.53/291.54 le :: 0':s -> 0':s -> true:false 306.53/291.54 0' :: 0':s 306.53/291.54 true :: true:false 306.53/291.54 s :: 0':s -> 0':s 306.53/291.54 false :: true:false 306.53/291.54 minus :: 0':s -> 0':s -> 0':s 306.53/291.54 if_minus :: true:false -> 0':s -> 0':s -> 0':s 306.53/291.54 mod :: 0':s -> 0':s -> 0':s 306.53/291.54 if_mod :: true:false -> 0':s -> 0':s -> 0':s 306.53/291.54 hole_true:false1_0 :: true:false 306.53/291.54 hole_0':s2_0 :: 0':s 306.53/291.54 gen_0':s3_0 :: Nat -> 0':s 306.53/291.54 306.53/291.54 ---------------------------------------- 306.53/291.54 306.53/291.54 (5) OrderProof (LOWER BOUND(ID)) 306.53/291.54 Heuristically decided to analyse the following defined symbols: 306.53/291.54 le, minus, mod 306.53/291.54 306.53/291.54 They will be analysed ascendingly in the following order: 306.53/291.54 le < minus 306.53/291.54 le < mod 306.53/291.54 minus < mod 306.53/291.54 306.53/291.54 ---------------------------------------- 306.53/291.54 306.53/291.54 (6) 306.53/291.54 Obligation: 306.53/291.54 TRS: 306.53/291.54 Rules: 306.53/291.54 le(0', y) -> true 306.53/291.54 le(s(x), 0') -> false 306.53/291.54 le(s(x), s(y)) -> le(x, y) 306.53/291.54 minus(0', y) -> 0' 306.53/291.54 minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) 306.53/291.54 if_minus(true, s(x), y) -> 0' 306.53/291.54 if_minus(false, s(x), y) -> s(minus(x, y)) 306.53/291.54 mod(0', y) -> 0' 306.53/291.54 mod(s(x), 0') -> 0' 306.53/291.54 mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) 306.53/291.54 if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) 306.53/291.54 if_mod(false, s(x), s(y)) -> s(x) 306.53/291.54 306.53/291.54 Types: 306.53/291.54 le :: 0':s -> 0':s -> true:false 306.53/291.54 0' :: 0':s 306.53/291.54 true :: true:false 306.53/291.54 s :: 0':s -> 0':s 306.53/291.54 false :: true:false 306.53/291.54 minus :: 0':s -> 0':s -> 0':s 306.53/291.54 if_minus :: true:false -> 0':s -> 0':s -> 0':s 306.53/291.54 mod :: 0':s -> 0':s -> 0':s 306.53/291.54 if_mod :: true:false -> 0':s -> 0':s -> 0':s 306.53/291.54 hole_true:false1_0 :: true:false 306.53/291.54 hole_0':s2_0 :: 0':s 306.53/291.54 gen_0':s3_0 :: Nat -> 0':s 306.53/291.54 306.53/291.54 306.53/291.54 Generator Equations: 306.53/291.54 gen_0':s3_0(0) <=> 0' 306.53/291.54 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 306.53/291.54 306.53/291.54 306.53/291.54 The following defined symbols remain to be analysed: 306.53/291.54 le, minus, mod 306.53/291.54 306.53/291.54 They will be analysed ascendingly in the following order: 306.53/291.54 le < minus 306.53/291.54 le < mod 306.53/291.54 minus < mod 306.53/291.54 306.53/291.54 ---------------------------------------- 306.53/291.54 306.53/291.54 (7) RewriteLemmaProof (LOWER BOUND(ID)) 306.53/291.54 Proved the following rewrite lemma: 306.53/291.54 le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 306.53/291.54 306.53/291.54 Induction Base: 306.53/291.54 le(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 306.53/291.54 true 306.53/291.54 306.53/291.54 Induction Step: 306.53/291.54 le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1) 306.53/291.54 le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH 306.53/291.54 true 306.53/291.54 306.53/291.54 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 306.53/291.54 ---------------------------------------- 306.53/291.54 306.53/291.54 (8) 306.53/291.54 Complex Obligation (BEST) 306.53/291.54 306.53/291.54 ---------------------------------------- 306.53/291.54 306.53/291.54 (9) 306.53/291.54 Obligation: 306.53/291.54 Proved the lower bound n^1 for the following obligation: 306.53/291.54 306.53/291.54 TRS: 306.53/291.54 Rules: 306.53/291.54 le(0', y) -> true 306.53/291.54 le(s(x), 0') -> false 306.53/291.54 le(s(x), s(y)) -> le(x, y) 306.53/291.54 minus(0', y) -> 0' 306.53/291.54 minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) 306.53/291.54 if_minus(true, s(x), y) -> 0' 306.53/291.54 if_minus(false, s(x), y) -> s(minus(x, y)) 306.53/291.54 mod(0', y) -> 0' 306.53/291.54 mod(s(x), 0') -> 0' 306.53/291.54 mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) 306.53/291.54 if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) 306.53/291.54 if_mod(false, s(x), s(y)) -> s(x) 306.53/291.54 306.53/291.54 Types: 306.53/291.54 le :: 0':s -> 0':s -> true:false 306.53/291.54 0' :: 0':s 306.53/291.54 true :: true:false 306.53/291.54 s :: 0':s -> 0':s 306.53/291.54 false :: true:false 306.53/291.54 minus :: 0':s -> 0':s -> 0':s 306.53/291.54 if_minus :: true:false -> 0':s -> 0':s -> 0':s 306.53/291.54 mod :: 0':s -> 0':s -> 0':s 306.53/291.54 if_mod :: true:false -> 0':s -> 0':s -> 0':s 306.53/291.54 hole_true:false1_0 :: true:false 306.53/291.54 hole_0':s2_0 :: 0':s 306.53/291.54 gen_0':s3_0 :: Nat -> 0':s 306.53/291.54 306.53/291.54 306.53/291.54 Generator Equations: 306.53/291.54 gen_0':s3_0(0) <=> 0' 306.53/291.54 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 306.53/291.54 306.53/291.54 306.53/291.54 The following defined symbols remain to be analysed: 306.53/291.54 le, minus, mod 306.53/291.54 306.53/291.54 They will be analysed ascendingly in the following order: 306.53/291.54 le < minus 306.53/291.54 le < mod 306.53/291.54 minus < mod 306.53/291.54 306.53/291.54 ---------------------------------------- 306.53/291.54 306.53/291.54 (10) LowerBoundPropagationProof (FINISHED) 306.53/291.54 Propagated lower bound. 306.53/291.54 ---------------------------------------- 306.53/291.54 306.53/291.54 (11) 306.53/291.54 BOUNDS(n^1, INF) 306.53/291.54 306.53/291.54 ---------------------------------------- 306.53/291.54 306.53/291.54 (12) 306.53/291.54 Obligation: 306.53/291.54 TRS: 306.53/291.54 Rules: 306.53/291.54 le(0', y) -> true 306.53/291.54 le(s(x), 0') -> false 306.53/291.54 le(s(x), s(y)) -> le(x, y) 306.53/291.54 minus(0', y) -> 0' 306.53/291.54 minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) 306.53/291.54 if_minus(true, s(x), y) -> 0' 306.53/291.54 if_minus(false, s(x), y) -> s(minus(x, y)) 306.53/291.54 mod(0', y) -> 0' 306.53/291.54 mod(s(x), 0') -> 0' 306.53/291.54 mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) 306.53/291.54 if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) 306.53/291.54 if_mod(false, s(x), s(y)) -> s(x) 306.53/291.54 306.53/291.54 Types: 306.53/291.54 le :: 0':s -> 0':s -> true:false 306.53/291.54 0' :: 0':s 306.53/291.54 true :: true:false 306.53/291.54 s :: 0':s -> 0':s 306.53/291.54 false :: true:false 306.53/291.54 minus :: 0':s -> 0':s -> 0':s 306.53/291.54 if_minus :: true:false -> 0':s -> 0':s -> 0':s 306.53/291.54 mod :: 0':s -> 0':s -> 0':s 306.53/291.54 if_mod :: true:false -> 0':s -> 0':s -> 0':s 306.53/291.54 hole_true:false1_0 :: true:false 306.53/291.54 hole_0':s2_0 :: 0':s 306.53/291.54 gen_0':s3_0 :: Nat -> 0':s 306.53/291.54 306.53/291.54 306.53/291.54 Lemmas: 306.53/291.54 le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0) 306.53/291.54 306.53/291.54 306.53/291.54 Generator Equations: 306.53/291.54 gen_0':s3_0(0) <=> 0' 306.53/291.54 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 306.53/291.54 306.53/291.54 306.53/291.54 The following defined symbols remain to be analysed: 306.53/291.54 minus, mod 306.53/291.54 306.53/291.54 They will be analysed ascendingly in the following order: 306.53/291.54 minus < mod 306.70/291.61 EOF