1104.00/291.50 WORST_CASE(Omega(n^1), ?) 1104.00/291.51 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1104.00/291.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1104.00/291.51 1104.00/291.51 1104.00/291.51 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1104.00/291.51 1104.00/291.51 (0) CpxTRS 1104.00/291.51 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1104.00/291.51 (2) CpxTRS 1104.00/291.51 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1104.00/291.51 (4) typed CpxTrs 1104.00/291.51 (5) OrderProof [LOWER BOUND(ID), 0 ms] 1104.00/291.51 (6) typed CpxTrs 1104.00/291.51 (7) RewriteLemmaProof [LOWER BOUND(ID), 325 ms] 1104.00/291.51 (8) BEST 1104.00/291.51 (9) proven lower bound 1104.00/291.51 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 1104.00/291.51 (11) BOUNDS(n^1, INF) 1104.00/291.51 (12) typed CpxTrs 1104.00/291.51 (13) RewriteLemmaProof [LOWER BOUND(ID), 65 ms] 1104.00/291.51 (14) typed CpxTrs 1104.00/291.51 (15) RewriteLemmaProof [LOWER BOUND(ID), 584 ms] 1104.00/291.51 (16) typed CpxTrs 1104.00/291.51 1104.00/291.51 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (0) 1104.00/291.51 Obligation: 1104.00/291.51 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1104.00/291.51 1104.00/291.51 1104.00/291.51 The TRS R consists of the following rules: 1104.00/291.51 1104.00/291.51 tower(x) -> f(a, x, s(0)) 1104.00/291.51 f(a, 0, y) -> y 1104.00/291.51 f(a, s(x), y) -> f(b, y, s(x)) 1104.00/291.51 f(b, y, x) -> f(a, half(x), exp(y)) 1104.00/291.51 exp(0) -> s(0) 1104.00/291.51 exp(s(x)) -> double(exp(x)) 1104.00/291.51 double(0) -> 0 1104.00/291.51 double(s(x)) -> s(s(double(x))) 1104.00/291.51 half(0) -> double(0) 1104.00/291.51 half(s(0)) -> half(0) 1104.00/291.51 half(s(s(x))) -> s(half(x)) 1104.00/291.51 1104.00/291.51 S is empty. 1104.00/291.51 Rewrite Strategy: INNERMOST 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 1104.00/291.51 Renamed function symbols to avoid clashes with predefined symbol. 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (2) 1104.00/291.51 Obligation: 1104.00/291.51 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1104.00/291.51 1104.00/291.51 1104.00/291.51 The TRS R consists of the following rules: 1104.00/291.51 1104.00/291.51 tower(x) -> f(a, x, s(0')) 1104.00/291.51 f(a, 0', y) -> y 1104.00/291.51 f(a, s(x), y) -> f(b, y, s(x)) 1104.00/291.51 f(b, y, x) -> f(a, half(x), exp(y)) 1104.00/291.51 exp(0') -> s(0') 1104.00/291.51 exp(s(x)) -> double(exp(x)) 1104.00/291.51 double(0') -> 0' 1104.00/291.51 double(s(x)) -> s(s(double(x))) 1104.00/291.51 half(0') -> double(0') 1104.00/291.51 half(s(0')) -> half(0') 1104.00/291.51 half(s(s(x))) -> s(half(x)) 1104.00/291.51 1104.00/291.51 S is empty. 1104.00/291.51 Rewrite Strategy: INNERMOST 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1104.00/291.51 Infered types. 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (4) 1104.00/291.51 Obligation: 1104.00/291.51 Innermost TRS: 1104.00/291.51 Rules: 1104.00/291.51 tower(x) -> f(a, x, s(0')) 1104.00/291.51 f(a, 0', y) -> y 1104.00/291.51 f(a, s(x), y) -> f(b, y, s(x)) 1104.00/291.51 f(b, y, x) -> f(a, half(x), exp(y)) 1104.00/291.51 exp(0') -> s(0') 1104.00/291.51 exp(s(x)) -> double(exp(x)) 1104.00/291.51 double(0') -> 0' 1104.00/291.51 double(s(x)) -> s(s(double(x))) 1104.00/291.51 half(0') -> double(0') 1104.00/291.51 half(s(0')) -> half(0') 1104.00/291.51 half(s(s(x))) -> s(half(x)) 1104.00/291.51 1104.00/291.51 Types: 1104.00/291.51 tower :: 0':s -> 0':s 1104.00/291.51 f :: a:b -> 0':s -> 0':s -> 0':s 1104.00/291.51 a :: a:b 1104.00/291.51 s :: 0':s -> 0':s 1104.00/291.51 0' :: 0':s 1104.00/291.51 b :: a:b 1104.00/291.51 half :: 0':s -> 0':s 1104.00/291.51 exp :: 0':s -> 0':s 1104.00/291.51 double :: 0':s -> 0':s 1104.00/291.51 hole_0':s1_0 :: 0':s 1104.00/291.51 hole_a:b2_0 :: a:b 1104.00/291.51 gen_0':s3_0 :: Nat -> 0':s 1104.00/291.51 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (5) OrderProof (LOWER BOUND(ID)) 1104.00/291.51 Heuristically decided to analyse the following defined symbols: 1104.00/291.51 f, half, exp, double 1104.00/291.51 1104.00/291.51 They will be analysed ascendingly in the following order: 1104.00/291.51 half < f 1104.00/291.51 exp < f 1104.00/291.51 double < half 1104.00/291.51 double < exp 1104.00/291.51 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (6) 1104.00/291.51 Obligation: 1104.00/291.51 Innermost TRS: 1104.00/291.51 Rules: 1104.00/291.51 tower(x) -> f(a, x, s(0')) 1104.00/291.51 f(a, 0', y) -> y 1104.00/291.51 f(a, s(x), y) -> f(b, y, s(x)) 1104.00/291.51 f(b, y, x) -> f(a, half(x), exp(y)) 1104.00/291.51 exp(0') -> s(0') 1104.00/291.51 exp(s(x)) -> double(exp(x)) 1104.00/291.51 double(0') -> 0' 1104.00/291.51 double(s(x)) -> s(s(double(x))) 1104.00/291.51 half(0') -> double(0') 1104.00/291.51 half(s(0')) -> half(0') 1104.00/291.51 half(s(s(x))) -> s(half(x)) 1104.00/291.51 1104.00/291.51 Types: 1104.00/291.51 tower :: 0':s -> 0':s 1104.00/291.51 f :: a:b -> 0':s -> 0':s -> 0':s 1104.00/291.51 a :: a:b 1104.00/291.51 s :: 0':s -> 0':s 1104.00/291.51 0' :: 0':s 1104.00/291.51 b :: a:b 1104.00/291.51 half :: 0':s -> 0':s 1104.00/291.51 exp :: 0':s -> 0':s 1104.00/291.51 double :: 0':s -> 0':s 1104.00/291.51 hole_0':s1_0 :: 0':s 1104.00/291.51 hole_a:b2_0 :: a:b 1104.00/291.51 gen_0':s3_0 :: Nat -> 0':s 1104.00/291.51 1104.00/291.51 1104.00/291.51 Generator Equations: 1104.00/291.51 gen_0':s3_0(0) <=> 0' 1104.00/291.51 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1104.00/291.51 1104.00/291.51 1104.00/291.51 The following defined symbols remain to be analysed: 1104.00/291.51 double, f, half, exp 1104.00/291.51 1104.00/291.51 They will be analysed ascendingly in the following order: 1104.00/291.51 half < f 1104.00/291.51 exp < f 1104.00/291.51 double < half 1104.00/291.51 double < exp 1104.00/291.51 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (7) RewriteLemmaProof (LOWER BOUND(ID)) 1104.00/291.51 Proved the following rewrite lemma: 1104.00/291.51 double(gen_0':s3_0(n5_0)) -> gen_0':s3_0(*(2, n5_0)), rt in Omega(1 + n5_0) 1104.00/291.51 1104.00/291.51 Induction Base: 1104.00/291.51 double(gen_0':s3_0(0)) ->_R^Omega(1) 1104.00/291.51 0' 1104.00/291.51 1104.00/291.51 Induction Step: 1104.00/291.51 double(gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1) 1104.00/291.51 s(s(double(gen_0':s3_0(n5_0)))) ->_IH 1104.00/291.51 s(s(gen_0':s3_0(*(2, c6_0)))) 1104.00/291.51 1104.00/291.51 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (8) 1104.00/291.51 Complex Obligation (BEST) 1104.00/291.51 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (9) 1104.00/291.51 Obligation: 1104.00/291.51 Proved the lower bound n^1 for the following obligation: 1104.00/291.51 1104.00/291.51 Innermost TRS: 1104.00/291.51 Rules: 1104.00/291.51 tower(x) -> f(a, x, s(0')) 1104.00/291.51 f(a, 0', y) -> y 1104.00/291.51 f(a, s(x), y) -> f(b, y, s(x)) 1104.00/291.51 f(b, y, x) -> f(a, half(x), exp(y)) 1104.00/291.51 exp(0') -> s(0') 1104.00/291.51 exp(s(x)) -> double(exp(x)) 1104.00/291.51 double(0') -> 0' 1104.00/291.51 double(s(x)) -> s(s(double(x))) 1104.00/291.51 half(0') -> double(0') 1104.00/291.51 half(s(0')) -> half(0') 1104.00/291.51 half(s(s(x))) -> s(half(x)) 1104.00/291.51 1104.00/291.51 Types: 1104.00/291.51 tower :: 0':s -> 0':s 1104.00/291.51 f :: a:b -> 0':s -> 0':s -> 0':s 1104.00/291.51 a :: a:b 1104.00/291.51 s :: 0':s -> 0':s 1104.00/291.51 0' :: 0':s 1104.00/291.51 b :: a:b 1104.00/291.51 half :: 0':s -> 0':s 1104.00/291.51 exp :: 0':s -> 0':s 1104.00/291.51 double :: 0':s -> 0':s 1104.00/291.51 hole_0':s1_0 :: 0':s 1104.00/291.51 hole_a:b2_0 :: a:b 1104.00/291.51 gen_0':s3_0 :: Nat -> 0':s 1104.00/291.51 1104.00/291.51 1104.00/291.51 Generator Equations: 1104.00/291.51 gen_0':s3_0(0) <=> 0' 1104.00/291.51 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1104.00/291.51 1104.00/291.51 1104.00/291.51 The following defined symbols remain to be analysed: 1104.00/291.51 double, f, half, exp 1104.00/291.51 1104.00/291.51 They will be analysed ascendingly in the following order: 1104.00/291.51 half < f 1104.00/291.51 exp < f 1104.00/291.51 double < half 1104.00/291.51 double < exp 1104.00/291.51 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (10) LowerBoundPropagationProof (FINISHED) 1104.00/291.51 Propagated lower bound. 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (11) 1104.00/291.51 BOUNDS(n^1, INF) 1104.00/291.51 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (12) 1104.00/291.51 Obligation: 1104.00/291.51 Innermost TRS: 1104.00/291.51 Rules: 1104.00/291.51 tower(x) -> f(a, x, s(0')) 1104.00/291.51 f(a, 0', y) -> y 1104.00/291.51 f(a, s(x), y) -> f(b, y, s(x)) 1104.00/291.51 f(b, y, x) -> f(a, half(x), exp(y)) 1104.00/291.51 exp(0') -> s(0') 1104.00/291.51 exp(s(x)) -> double(exp(x)) 1104.00/291.51 double(0') -> 0' 1104.00/291.51 double(s(x)) -> s(s(double(x))) 1104.00/291.51 half(0') -> double(0') 1104.00/291.51 half(s(0')) -> half(0') 1104.00/291.51 half(s(s(x))) -> s(half(x)) 1104.00/291.51 1104.00/291.51 Types: 1104.00/291.51 tower :: 0':s -> 0':s 1104.00/291.51 f :: a:b -> 0':s -> 0':s -> 0':s 1104.00/291.51 a :: a:b 1104.00/291.51 s :: 0':s -> 0':s 1104.00/291.51 0' :: 0':s 1104.00/291.51 b :: a:b 1104.00/291.51 half :: 0':s -> 0':s 1104.00/291.51 exp :: 0':s -> 0':s 1104.00/291.51 double :: 0':s -> 0':s 1104.00/291.51 hole_0':s1_0 :: 0':s 1104.00/291.51 hole_a:b2_0 :: a:b 1104.00/291.51 gen_0':s3_0 :: Nat -> 0':s 1104.00/291.51 1104.00/291.51 1104.00/291.51 Lemmas: 1104.00/291.51 double(gen_0':s3_0(n5_0)) -> gen_0':s3_0(*(2, n5_0)), rt in Omega(1 + n5_0) 1104.00/291.51 1104.00/291.51 1104.00/291.51 Generator Equations: 1104.00/291.51 gen_0':s3_0(0) <=> 0' 1104.00/291.51 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1104.00/291.51 1104.00/291.51 1104.00/291.51 The following defined symbols remain to be analysed: 1104.00/291.51 half, f, exp 1104.00/291.51 1104.00/291.51 They will be analysed ascendingly in the following order: 1104.00/291.51 half < f 1104.00/291.51 exp < f 1104.00/291.51 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (13) RewriteLemmaProof (LOWER BOUND(ID)) 1104.00/291.51 Proved the following rewrite lemma: 1104.00/291.51 half(gen_0':s3_0(*(2, n297_0))) -> gen_0':s3_0(n297_0), rt in Omega(1 + n297_0) 1104.00/291.51 1104.00/291.51 Induction Base: 1104.00/291.51 half(gen_0':s3_0(*(2, 0))) ->_R^Omega(1) 1104.00/291.51 double(0') ->_L^Omega(1) 1104.00/291.51 gen_0':s3_0(*(2, 0)) 1104.00/291.51 1104.00/291.51 Induction Step: 1104.00/291.51 half(gen_0':s3_0(*(2, +(n297_0, 1)))) ->_R^Omega(1) 1104.00/291.51 s(half(gen_0':s3_0(*(2, n297_0)))) ->_IH 1104.00/291.51 s(gen_0':s3_0(c298_0)) 1104.00/291.51 1104.00/291.51 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (14) 1104.00/291.51 Obligation: 1104.00/291.51 Innermost TRS: 1104.00/291.51 Rules: 1104.00/291.51 tower(x) -> f(a, x, s(0')) 1104.00/291.51 f(a, 0', y) -> y 1104.00/291.51 f(a, s(x), y) -> f(b, y, s(x)) 1104.00/291.51 f(b, y, x) -> f(a, half(x), exp(y)) 1104.00/291.51 exp(0') -> s(0') 1104.00/291.51 exp(s(x)) -> double(exp(x)) 1104.00/291.51 double(0') -> 0' 1104.00/291.51 double(s(x)) -> s(s(double(x))) 1104.00/291.51 half(0') -> double(0') 1104.00/291.51 half(s(0')) -> half(0') 1104.00/291.51 half(s(s(x))) -> s(half(x)) 1104.00/291.51 1104.00/291.51 Types: 1104.00/291.51 tower :: 0':s -> 0':s 1104.00/291.51 f :: a:b -> 0':s -> 0':s -> 0':s 1104.00/291.51 a :: a:b 1104.00/291.51 s :: 0':s -> 0':s 1104.00/291.51 0' :: 0':s 1104.00/291.51 b :: a:b 1104.00/291.51 half :: 0':s -> 0':s 1104.00/291.51 exp :: 0':s -> 0':s 1104.00/291.51 double :: 0':s -> 0':s 1104.00/291.51 hole_0':s1_0 :: 0':s 1104.00/291.51 hole_a:b2_0 :: a:b 1104.00/291.51 gen_0':s3_0 :: Nat -> 0':s 1104.00/291.51 1104.00/291.51 1104.00/291.51 Lemmas: 1104.00/291.51 double(gen_0':s3_0(n5_0)) -> gen_0':s3_0(*(2, n5_0)), rt in Omega(1 + n5_0) 1104.00/291.51 half(gen_0':s3_0(*(2, n297_0))) -> gen_0':s3_0(n297_0), rt in Omega(1 + n297_0) 1104.00/291.51 1104.00/291.51 1104.00/291.51 Generator Equations: 1104.00/291.51 gen_0':s3_0(0) <=> 0' 1104.00/291.51 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1104.00/291.51 1104.00/291.51 1104.00/291.51 The following defined symbols remain to be analysed: 1104.00/291.51 exp, f 1104.00/291.51 1104.00/291.51 They will be analysed ascendingly in the following order: 1104.00/291.51 exp < f 1104.00/291.51 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (15) RewriteLemmaProof (LOWER BOUND(ID)) 1104.00/291.51 Proved the following rewrite lemma: 1104.00/291.51 exp(gen_0':s3_0(+(1, n685_0))) -> *4_0, rt in Omega(n685_0) 1104.00/291.51 1104.00/291.51 Induction Base: 1104.00/291.51 exp(gen_0':s3_0(+(1, 0))) 1104.00/291.51 1104.00/291.51 Induction Step: 1104.00/291.51 exp(gen_0':s3_0(+(1, +(n685_0, 1)))) ->_R^Omega(1) 1104.00/291.51 double(exp(gen_0':s3_0(+(1, n685_0)))) ->_IH 1104.00/291.51 double(*4_0) 1104.00/291.51 1104.00/291.51 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1104.00/291.51 ---------------------------------------- 1104.00/291.51 1104.00/291.51 (16) 1104.00/291.51 Obligation: 1104.00/291.51 Innermost TRS: 1104.00/291.51 Rules: 1104.00/291.51 tower(x) -> f(a, x, s(0')) 1104.00/291.51 f(a, 0', y) -> y 1104.00/291.51 f(a, s(x), y) -> f(b, y, s(x)) 1104.00/291.51 f(b, y, x) -> f(a, half(x), exp(y)) 1104.00/291.51 exp(0') -> s(0') 1104.00/291.51 exp(s(x)) -> double(exp(x)) 1104.00/291.51 double(0') -> 0' 1104.00/291.51 double(s(x)) -> s(s(double(x))) 1104.00/291.51 half(0') -> double(0') 1104.00/291.51 half(s(0')) -> half(0') 1104.00/291.51 half(s(s(x))) -> s(half(x)) 1104.00/291.51 1104.00/291.51 Types: 1104.00/291.51 tower :: 0':s -> 0':s 1104.00/291.51 f :: a:b -> 0':s -> 0':s -> 0':s 1104.00/291.51 a :: a:b 1104.00/291.51 s :: 0':s -> 0':s 1104.00/291.51 0' :: 0':s 1104.00/291.51 b :: a:b 1104.00/291.51 half :: 0':s -> 0':s 1104.00/291.51 exp :: 0':s -> 0':s 1104.00/291.51 double :: 0':s -> 0':s 1104.00/291.51 hole_0':s1_0 :: 0':s 1104.00/291.51 hole_a:b2_0 :: a:b 1104.00/291.51 gen_0':s3_0 :: Nat -> 0':s 1104.00/291.51 1104.00/291.51 1104.00/291.51 Lemmas: 1104.00/291.51 double(gen_0':s3_0(n5_0)) -> gen_0':s3_0(*(2, n5_0)), rt in Omega(1 + n5_0) 1104.00/291.51 half(gen_0':s3_0(*(2, n297_0))) -> gen_0':s3_0(n297_0), rt in Omega(1 + n297_0) 1104.00/291.51 exp(gen_0':s3_0(+(1, n685_0))) -> *4_0, rt in Omega(n685_0) 1104.00/291.51 1104.00/291.51 1104.00/291.51 Generator Equations: 1104.00/291.51 gen_0':s3_0(0) <=> 0' 1104.00/291.51 gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x)) 1104.00/291.51 1104.00/291.51 1104.00/291.51 The following defined symbols remain to be analysed: 1104.00/291.51 f 1104.21/291.56 EOF