308.21/291.48 WORST_CASE(Omega(n^1), ?) 308.21/291.49 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 308.21/291.49 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 308.21/291.49 308.21/291.49 308.21/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 308.21/291.49 308.21/291.49 (0) CpxTRS 308.21/291.49 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 308.21/291.49 (2) CpxTRS 308.21/291.49 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 308.21/291.49 (4) typed CpxTrs 308.21/291.49 (5) OrderProof [LOWER BOUND(ID), 0 ms] 308.21/291.49 (6) typed CpxTrs 308.21/291.49 (7) RewriteLemmaProof [LOWER BOUND(ID), 269 ms] 308.21/291.49 (8) BEST 308.21/291.49 (9) proven lower bound 308.21/291.49 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 308.21/291.49 (11) BOUNDS(n^1, INF) 308.21/291.49 (12) typed CpxTrs 308.21/291.49 (13) RewriteLemmaProof [LOWER BOUND(ID), 47 ms] 308.21/291.49 (14) typed CpxTrs 308.21/291.49 308.21/291.49 308.21/291.49 ---------------------------------------- 308.21/291.49 308.21/291.49 (0) 308.21/291.49 Obligation: 308.21/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 308.21/291.49 308.21/291.49 308.21/291.49 The TRS R consists of the following rules: 308.21/291.49 308.21/291.49 minus(x, 0) -> x 308.21/291.49 minus(s(x), s(y)) -> minus(x, y) 308.21/291.49 le(0, y) -> true 308.21/291.49 le(s(x), 0) -> false 308.21/291.49 le(s(x), s(y)) -> le(x, y) 308.21/291.49 quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y)) 308.21/291.49 if_quot(true, x, y) -> s(quot(minus(x, y), y)) 308.21/291.49 if_quot(false, x, y) -> 0 308.21/291.49 308.21/291.49 S is empty. 308.21/291.49 Rewrite Strategy: FULL 308.21/291.49 ---------------------------------------- 308.21/291.49 308.21/291.49 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 308.21/291.49 Renamed function symbols to avoid clashes with predefined symbol. 308.21/291.49 ---------------------------------------- 308.21/291.49 308.21/291.49 (2) 308.21/291.49 Obligation: 308.21/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 308.21/291.49 308.21/291.49 308.21/291.49 The TRS R consists of the following rules: 308.21/291.49 308.21/291.49 minus(x, 0') -> x 308.21/291.49 minus(s(x), s(y)) -> minus(x, y) 308.21/291.49 le(0', y) -> true 308.21/291.49 le(s(x), 0') -> false 308.21/291.49 le(s(x), s(y)) -> le(x, y) 308.21/291.49 quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y)) 308.21/291.49 if_quot(true, x, y) -> s(quot(minus(x, y), y)) 308.21/291.49 if_quot(false, x, y) -> 0' 308.21/291.49 308.21/291.49 S is empty. 308.21/291.49 Rewrite Strategy: FULL 308.21/291.49 ---------------------------------------- 308.21/291.49 308.21/291.49 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 308.21/291.49 Infered types. 308.21/291.49 ---------------------------------------- 308.21/291.49 308.21/291.49 (4) 308.21/291.49 Obligation: 308.21/291.49 TRS: 308.21/291.49 Rules: 308.21/291.49 minus(x, 0') -> x 308.21/291.49 minus(s(x), s(y)) -> minus(x, y) 308.21/291.49 le(0', y) -> true 308.21/291.49 le(s(x), 0') -> false 308.21/291.49 le(s(x), s(y)) -> le(x, y) 308.21/291.49 quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y)) 308.21/291.49 if_quot(true, x, y) -> s(quot(minus(x, y), y)) 308.21/291.49 if_quot(false, x, y) -> 0' 308.21/291.49 308.21/291.49 Types: 308.21/291.49 minus :: 0':s -> 0':s -> 0':s 308.21/291.49 0' :: 0':s 308.21/291.49 s :: 0':s -> 0':s 308.21/291.49 le :: 0':s -> 0':s -> true:false 308.21/291.49 true :: true:false 308.21/291.49 false :: true:false 308.21/291.49 quot :: 0':s -> 0':s -> 0':s 308.21/291.49 if_quot :: true:false -> 0':s -> 0':s -> 0':s 308.21/291.49 hole_0':s1_0 :: 0':s 308.21/291.49 hole_true:false2_0 :: true:false 308.21/291.49 gen_0':s3_0 :: Nat -> 0':s 308.21/291.49 308.21/291.49 ---------------------------------------- 308.21/291.49 308.21/291.49 (5) OrderProof (LOWER BOUND(ID)) 308.21/291.49 Heuristically decided to analyse the following defined symbols: 308.21/291.49 minus, le, quot 308.21/291.49 308.21/291.49 They will be analysed ascendingly in the following order: 308.21/291.49 minus < quot 308.21/291.49 le < quot 308.21/291.49 308.21/291.49 ---------------------------------------- 308.21/291.49 308.21/291.49 (6) 308.21/291.49 Obligation: 308.21/291.49 TRS: 308.21/291.49 Rules: 308.21/291.49 minus(x, 0') -> x 308.21/291.49 minus(s(x), s(y)) -> minus(x, y) 308.21/291.49 le(0', y) -> true 308.21/291.49 le(s(x), 0') -> false 308.21/291.49 le(s(x), s(y)) -> le(x, y) 308.21/291.49 quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y)) 308.21/291.49 if_quot(true, x, y) -> s(quot(minus(x, y), y)) 308.21/291.49 if_quot(false, x, y) -> 0' 308.21/291.49 308.21/291.49 Types: 308.21/291.49 minus :: 0':s -> 0':s -> 0':s 308.21/291.49 0' :: 0':s 308.21/291.49 s :: 0':s -> 0':s 308.21/291.49 le :: 0':s -> 0':s -> true:false 308.21/291.49 true :: true:false 308.21/291.49 false :: true:false 308.21/291.49 quot :: 0':s -> 0':s -> 0':s 308.21/291.49 if_quot :: true:false -> 0':s -> 0':s -> 0':s 308.21/291.49 hole_0':s1_0 :: 0':s 308.21/291.49 hole_true:false2_0 :: true:false 308.21/291.49 gen_0':s3_0 :: Nat -> 0':s 308.21/291.49 308.21/291.49 308.21/291.49 Generator Equations: 308.21/291.49 gen_0':s3_0(0) <=> 0' 308.21/291.49 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 308.21/291.49 308.21/291.49 308.21/291.49 The following defined symbols remain to be analysed: 308.21/291.49 minus, le, quot 308.21/291.49 308.21/291.49 They will be analysed ascendingly in the following order: 308.21/291.49 minus < quot 308.21/291.49 le < quot 308.21/291.49 308.21/291.49 ---------------------------------------- 308.21/291.49 308.21/291.49 (7) RewriteLemmaProof (LOWER BOUND(ID)) 308.21/291.49 Proved the following rewrite lemma: 308.21/291.49 minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0) 308.21/291.49 308.21/291.49 Induction Base: 308.21/291.49 minus(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 308.21/291.49 gen_0':s3_0(0) 308.21/291.49 308.21/291.49 Induction Step: 308.21/291.49 minus(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1) 308.21/291.49 minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH 308.21/291.49 gen_0':s3_0(0) 308.21/291.49 308.21/291.49 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 308.21/291.49 ---------------------------------------- 308.21/291.49 308.21/291.49 (8) 308.21/291.49 Complex Obligation (BEST) 308.21/291.49 308.21/291.49 ---------------------------------------- 308.21/291.49 308.21/291.49 (9) 308.21/291.49 Obligation: 308.21/291.49 Proved the lower bound n^1 for the following obligation: 308.21/291.49 308.21/291.49 TRS: 308.21/291.49 Rules: 308.21/291.49 minus(x, 0') -> x 308.21/291.49 minus(s(x), s(y)) -> minus(x, y) 308.21/291.49 le(0', y) -> true 308.21/291.49 le(s(x), 0') -> false 308.21/291.49 le(s(x), s(y)) -> le(x, y) 308.21/291.49 quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y)) 308.21/291.49 if_quot(true, x, y) -> s(quot(minus(x, y), y)) 308.21/291.49 if_quot(false, x, y) -> 0' 308.21/291.49 308.21/291.49 Types: 308.21/291.49 minus :: 0':s -> 0':s -> 0':s 308.21/291.49 0' :: 0':s 308.21/291.49 s :: 0':s -> 0':s 308.21/291.49 le :: 0':s -> 0':s -> true:false 308.21/291.49 true :: true:false 308.21/291.49 false :: true:false 308.21/291.49 quot :: 0':s -> 0':s -> 0':s 308.21/291.49 if_quot :: true:false -> 0':s -> 0':s -> 0':s 308.21/291.49 hole_0':s1_0 :: 0':s 308.21/291.49 hole_true:false2_0 :: true:false 308.21/291.49 gen_0':s3_0 :: Nat -> 0':s 308.21/291.49 308.21/291.49 308.21/291.49 Generator Equations: 308.21/291.49 gen_0':s3_0(0) <=> 0' 308.21/291.49 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 308.21/291.49 308.21/291.49 308.21/291.49 The following defined symbols remain to be analysed: 308.21/291.49 minus, le, quot 308.21/291.49 308.21/291.49 They will be analysed ascendingly in the following order: 308.21/291.49 minus < quot 308.21/291.49 le < quot 308.21/291.49 308.21/291.49 ---------------------------------------- 308.21/291.49 308.21/291.49 (10) LowerBoundPropagationProof (FINISHED) 308.21/291.49 Propagated lower bound. 308.21/291.49 ---------------------------------------- 308.21/291.49 308.21/291.49 (11) 308.21/291.49 BOUNDS(n^1, INF) 308.21/291.49 308.21/291.49 ---------------------------------------- 308.21/291.49 308.21/291.49 (12) 308.21/291.49 Obligation: 308.21/291.49 TRS: 308.21/291.49 Rules: 308.21/291.49 minus(x, 0') -> x 308.21/291.49 minus(s(x), s(y)) -> minus(x, y) 308.21/291.49 le(0', y) -> true 308.21/291.49 le(s(x), 0') -> false 308.21/291.49 le(s(x), s(y)) -> le(x, y) 308.21/291.49 quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y)) 308.21/291.49 if_quot(true, x, y) -> s(quot(minus(x, y), y)) 308.21/291.49 if_quot(false, x, y) -> 0' 308.21/291.49 308.21/291.49 Types: 308.21/291.49 minus :: 0':s -> 0':s -> 0':s 308.21/291.49 0' :: 0':s 308.21/291.49 s :: 0':s -> 0':s 308.21/291.49 le :: 0':s -> 0':s -> true:false 308.21/291.49 true :: true:false 308.21/291.49 false :: true:false 308.21/291.49 quot :: 0':s -> 0':s -> 0':s 308.21/291.49 if_quot :: true:false -> 0':s -> 0':s -> 0':s 308.21/291.49 hole_0':s1_0 :: 0':s 308.21/291.49 hole_true:false2_0 :: true:false 308.21/291.49 gen_0':s3_0 :: Nat -> 0':s 308.21/291.49 308.21/291.49 308.21/291.49 Lemmas: 308.21/291.49 minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0) 308.21/291.49 308.21/291.49 308.21/291.49 Generator Equations: 308.21/291.49 gen_0':s3_0(0) <=> 0' 308.21/291.49 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 308.21/291.49 308.21/291.49 308.21/291.49 The following defined symbols remain to be analysed: 308.21/291.49 le, quot 308.21/291.49 308.21/291.49 They will be analysed ascendingly in the following order: 308.21/291.49 le < quot 308.21/291.49 308.21/291.49 ---------------------------------------- 308.21/291.49 308.21/291.49 (13) RewriteLemmaProof (LOWER BOUND(ID)) 308.21/291.49 Proved the following rewrite lemma: 308.21/291.49 le(gen_0':s3_0(n233_0), gen_0':s3_0(n233_0)) -> true, rt in Omega(1 + n233_0) 308.21/291.49 308.21/291.49 Induction Base: 308.21/291.49 le(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1) 308.21/291.49 true 308.21/291.49 308.21/291.49 Induction Step: 308.21/291.49 le(gen_0':s3_0(+(n233_0, 1)), gen_0':s3_0(+(n233_0, 1))) ->_R^Omega(1) 308.21/291.49 le(gen_0':s3_0(n233_0), gen_0':s3_0(n233_0)) ->_IH 308.21/291.49 true 308.21/291.49 308.21/291.49 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 308.21/291.49 ---------------------------------------- 308.21/291.49 308.21/291.49 (14) 308.21/291.49 Obligation: 308.21/291.49 TRS: 308.21/291.49 Rules: 308.21/291.49 minus(x, 0') -> x 308.21/291.49 minus(s(x), s(y)) -> minus(x, y) 308.21/291.49 le(0', y) -> true 308.21/291.49 le(s(x), 0') -> false 308.21/291.49 le(s(x), s(y)) -> le(x, y) 308.21/291.49 quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y)) 308.21/291.49 if_quot(true, x, y) -> s(quot(minus(x, y), y)) 308.21/291.49 if_quot(false, x, y) -> 0' 308.21/291.49 308.21/291.49 Types: 308.21/291.49 minus :: 0':s -> 0':s -> 0':s 308.21/291.49 0' :: 0':s 308.21/291.49 s :: 0':s -> 0':s 308.21/291.49 le :: 0':s -> 0':s -> true:false 308.21/291.49 true :: true:false 308.21/291.49 false :: true:false 308.21/291.49 quot :: 0':s -> 0':s -> 0':s 308.21/291.49 if_quot :: true:false -> 0':s -> 0':s -> 0':s 308.21/291.49 hole_0':s1_0 :: 0':s 308.21/291.49 hole_true:false2_0 :: true:false 308.21/291.49 gen_0':s3_0 :: Nat -> 0':s 308.21/291.49 308.21/291.49 308.21/291.49 Lemmas: 308.21/291.49 minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0) 308.21/291.49 le(gen_0':s3_0(n233_0), gen_0':s3_0(n233_0)) -> true, rt in Omega(1 + n233_0) 308.21/291.49 308.21/291.49 308.21/291.49 Generator Equations: 308.21/291.49 gen_0':s3_0(0) <=> 0' 308.21/291.49 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 308.21/291.49 308.21/291.49 308.21/291.49 The following defined symbols remain to be analysed: 308.21/291.49 quot 308.21/291.53 EOF