1131.74/291.54 WORST_CASE(Omega(n^3), ?) 1131.74/291.60 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1131.74/291.60 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1131.74/291.60 1131.74/291.60 1131.74/291.60 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF). 1131.74/291.60 1131.74/291.60 (0) CpxTRS 1131.74/291.60 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1131.74/291.60 (2) CpxTRS 1131.74/291.60 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1131.74/291.60 (4) typed CpxTrs 1131.74/291.60 (5) OrderProof [LOWER BOUND(ID), 0 ms] 1131.74/291.60 (6) typed CpxTrs 1131.74/291.60 (7) RewriteLemmaProof [LOWER BOUND(ID), 294 ms] 1131.74/291.60 (8) BEST 1131.74/291.60 (9) proven lower bound 1131.74/291.60 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 1131.74/291.60 (11) BOUNDS(n^1, INF) 1131.74/291.60 (12) typed CpxTrs 1131.74/291.60 (13) RewriteLemmaProof [LOWER BOUND(ID), 0 ms] 1131.74/291.60 (14) BEST 1131.74/291.60 (15) proven lower bound 1131.74/291.60 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 1131.74/291.60 (17) BOUNDS(n^2, INF) 1131.74/291.60 (18) typed CpxTrs 1131.74/291.60 (19) RewriteLemmaProof [LOWER BOUND(ID), 11 ms] 1131.74/291.60 (20) BEST 1131.74/291.60 (21) proven lower bound 1131.74/291.60 (22) LowerBoundPropagationProof [FINISHED, 0 ms] 1131.74/291.60 (23) BOUNDS(n^3, INF) 1131.74/291.60 (24) typed CpxTrs 1131.74/291.60 (25) RewriteLemmaProof [LOWER BOUND(ID), 585 ms] 1131.74/291.60 (26) typed CpxTrs 1131.74/291.60 1131.74/291.60 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (0) 1131.74/291.60 Obligation: 1131.74/291.60 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF). 1131.74/291.60 1131.74/291.60 1131.74/291.60 The TRS R consists of the following rules: 1131.74/291.60 1131.74/291.60 plus(0, x) -> x 1131.74/291.60 plus(s(x), y) -> s(plus(p(s(x)), y)) 1131.74/291.60 times(0, y) -> 0 1131.74/291.60 times(s(x), y) -> plus(y, times(p(s(x)), y)) 1131.74/291.60 exp(x, 0) -> s(0) 1131.74/291.60 exp(x, s(y)) -> times(x, exp(x, y)) 1131.74/291.60 p(s(0)) -> 0 1131.74/291.60 p(s(s(x))) -> s(p(s(x))) 1131.74/291.60 tower(x, y) -> towerIter(x, y, s(0)) 1131.74/291.60 towerIter(0, y, z) -> z 1131.74/291.60 towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) 1131.74/291.60 1131.74/291.60 S is empty. 1131.74/291.60 Rewrite Strategy: INNERMOST 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 1131.74/291.60 Renamed function symbols to avoid clashes with predefined symbol. 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (2) 1131.74/291.60 Obligation: 1131.74/291.60 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF). 1131.74/291.60 1131.74/291.60 1131.74/291.60 The TRS R consists of the following rules: 1131.74/291.60 1131.74/291.60 plus(0', x) -> x 1131.74/291.60 plus(s(x), y) -> s(plus(p(s(x)), y)) 1131.74/291.60 times(0', y) -> 0' 1131.74/291.60 times(s(x), y) -> plus(y, times(p(s(x)), y)) 1131.74/291.60 exp(x, 0') -> s(0') 1131.74/291.60 exp(x, s(y)) -> times(x, exp(x, y)) 1131.74/291.60 p(s(0')) -> 0' 1131.74/291.60 p(s(s(x))) -> s(p(s(x))) 1131.74/291.60 tower(x, y) -> towerIter(x, y, s(0')) 1131.74/291.60 towerIter(0', y, z) -> z 1131.74/291.60 towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) 1131.74/291.60 1131.74/291.60 S is empty. 1131.74/291.60 Rewrite Strategy: INNERMOST 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1131.74/291.60 Infered types. 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (4) 1131.74/291.60 Obligation: 1131.74/291.60 Innermost TRS: 1131.74/291.60 Rules: 1131.74/291.60 plus(0', x) -> x 1131.74/291.60 plus(s(x), y) -> s(plus(p(s(x)), y)) 1131.74/291.60 times(0', y) -> 0' 1131.74/291.60 times(s(x), y) -> plus(y, times(p(s(x)), y)) 1131.74/291.60 exp(x, 0') -> s(0') 1131.74/291.60 exp(x, s(y)) -> times(x, exp(x, y)) 1131.74/291.60 p(s(0')) -> 0' 1131.74/291.60 p(s(s(x))) -> s(p(s(x))) 1131.74/291.60 tower(x, y) -> towerIter(x, y, s(0')) 1131.74/291.60 towerIter(0', y, z) -> z 1131.74/291.60 towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) 1131.74/291.60 1131.74/291.60 Types: 1131.74/291.60 plus :: 0':s -> 0':s -> 0':s 1131.74/291.60 0' :: 0':s 1131.74/291.60 s :: 0':s -> 0':s 1131.74/291.60 p :: 0':s -> 0':s 1131.74/291.60 times :: 0':s -> 0':s -> 0':s 1131.74/291.60 exp :: 0':s -> 0':s -> 0':s 1131.74/291.60 tower :: 0':s -> 0':s -> 0':s 1131.74/291.60 towerIter :: 0':s -> 0':s -> 0':s -> 0':s 1131.74/291.60 hole_0':s1_0 :: 0':s 1131.74/291.60 gen_0':s2_0 :: Nat -> 0':s 1131.74/291.60 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (5) OrderProof (LOWER BOUND(ID)) 1131.74/291.60 Heuristically decided to analyse the following defined symbols: 1131.74/291.60 plus, p, times, exp, towerIter 1131.74/291.60 1131.74/291.60 They will be analysed ascendingly in the following order: 1131.74/291.60 p < plus 1131.74/291.60 plus < times 1131.74/291.60 p < times 1131.74/291.60 p < towerIter 1131.74/291.60 times < exp 1131.74/291.60 exp < towerIter 1131.74/291.60 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (6) 1131.74/291.60 Obligation: 1131.74/291.60 Innermost TRS: 1131.74/291.60 Rules: 1131.74/291.60 plus(0', x) -> x 1131.74/291.60 plus(s(x), y) -> s(plus(p(s(x)), y)) 1131.74/291.60 times(0', y) -> 0' 1131.74/291.60 times(s(x), y) -> plus(y, times(p(s(x)), y)) 1131.74/291.60 exp(x, 0') -> s(0') 1131.74/291.60 exp(x, s(y)) -> times(x, exp(x, y)) 1131.74/291.60 p(s(0')) -> 0' 1131.74/291.60 p(s(s(x))) -> s(p(s(x))) 1131.74/291.60 tower(x, y) -> towerIter(x, y, s(0')) 1131.74/291.60 towerIter(0', y, z) -> z 1131.74/291.60 towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) 1131.74/291.60 1131.74/291.60 Types: 1131.74/291.60 plus :: 0':s -> 0':s -> 0':s 1131.74/291.60 0' :: 0':s 1131.74/291.60 s :: 0':s -> 0':s 1131.74/291.60 p :: 0':s -> 0':s 1131.74/291.60 times :: 0':s -> 0':s -> 0':s 1131.74/291.60 exp :: 0':s -> 0':s -> 0':s 1131.74/291.60 tower :: 0':s -> 0':s -> 0':s 1131.74/291.60 towerIter :: 0':s -> 0':s -> 0':s -> 0':s 1131.74/291.60 hole_0':s1_0 :: 0':s 1131.74/291.60 gen_0':s2_0 :: Nat -> 0':s 1131.74/291.60 1131.74/291.60 1131.74/291.60 Generator Equations: 1131.74/291.60 gen_0':s2_0(0) <=> 0' 1131.74/291.60 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 1131.74/291.60 1131.74/291.60 1131.74/291.60 The following defined symbols remain to be analysed: 1131.74/291.60 p, plus, times, exp, towerIter 1131.74/291.60 1131.74/291.60 They will be analysed ascendingly in the following order: 1131.74/291.60 p < plus 1131.74/291.60 plus < times 1131.74/291.60 p < times 1131.74/291.60 p < towerIter 1131.74/291.60 times < exp 1131.74/291.60 exp < towerIter 1131.74/291.60 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (7) RewriteLemmaProof (LOWER BOUND(ID)) 1131.74/291.60 Proved the following rewrite lemma: 1131.74/291.60 p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0) 1131.74/291.60 1131.74/291.60 Induction Base: 1131.74/291.60 p(gen_0':s2_0(+(1, 0))) ->_R^Omega(1) 1131.74/291.60 0' 1131.74/291.60 1131.74/291.60 Induction Step: 1131.74/291.60 p(gen_0':s2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1) 1131.74/291.60 s(p(s(gen_0':s2_0(n4_0)))) ->_IH 1131.74/291.60 s(gen_0':s2_0(c5_0)) 1131.74/291.60 1131.74/291.60 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (8) 1131.74/291.60 Complex Obligation (BEST) 1131.74/291.60 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (9) 1131.74/291.60 Obligation: 1131.74/291.60 Proved the lower bound n^1 for the following obligation: 1131.74/291.60 1131.74/291.60 Innermost TRS: 1131.74/291.60 Rules: 1131.74/291.60 plus(0', x) -> x 1131.74/291.60 plus(s(x), y) -> s(plus(p(s(x)), y)) 1131.74/291.60 times(0', y) -> 0' 1131.74/291.60 times(s(x), y) -> plus(y, times(p(s(x)), y)) 1131.74/291.60 exp(x, 0') -> s(0') 1131.74/291.60 exp(x, s(y)) -> times(x, exp(x, y)) 1131.74/291.60 p(s(0')) -> 0' 1131.74/291.60 p(s(s(x))) -> s(p(s(x))) 1131.74/291.60 tower(x, y) -> towerIter(x, y, s(0')) 1131.74/291.60 towerIter(0', y, z) -> z 1131.74/291.60 towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) 1131.74/291.60 1131.74/291.60 Types: 1131.74/291.60 plus :: 0':s -> 0':s -> 0':s 1131.74/291.60 0' :: 0':s 1131.74/291.60 s :: 0':s -> 0':s 1131.74/291.60 p :: 0':s -> 0':s 1131.74/291.60 times :: 0':s -> 0':s -> 0':s 1131.74/291.60 exp :: 0':s -> 0':s -> 0':s 1131.74/291.60 tower :: 0':s -> 0':s -> 0':s 1131.74/291.60 towerIter :: 0':s -> 0':s -> 0':s -> 0':s 1131.74/291.60 hole_0':s1_0 :: 0':s 1131.74/291.60 gen_0':s2_0 :: Nat -> 0':s 1131.74/291.60 1131.74/291.60 1131.74/291.60 Generator Equations: 1131.74/291.60 gen_0':s2_0(0) <=> 0' 1131.74/291.60 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 1131.74/291.60 1131.74/291.60 1131.74/291.60 The following defined symbols remain to be analysed: 1131.74/291.60 p, plus, times, exp, towerIter 1131.74/291.60 1131.74/291.60 They will be analysed ascendingly in the following order: 1131.74/291.60 p < plus 1131.74/291.60 plus < times 1131.74/291.60 p < times 1131.74/291.60 p < towerIter 1131.74/291.60 times < exp 1131.74/291.60 exp < towerIter 1131.74/291.60 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (10) LowerBoundPropagationProof (FINISHED) 1131.74/291.60 Propagated lower bound. 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (11) 1131.74/291.60 BOUNDS(n^1, INF) 1131.74/291.60 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (12) 1131.74/291.60 Obligation: 1131.74/291.60 Innermost TRS: 1131.74/291.60 Rules: 1131.74/291.60 plus(0', x) -> x 1131.74/291.60 plus(s(x), y) -> s(plus(p(s(x)), y)) 1131.74/291.60 times(0', y) -> 0' 1131.74/291.60 times(s(x), y) -> plus(y, times(p(s(x)), y)) 1131.74/291.60 exp(x, 0') -> s(0') 1131.74/291.60 exp(x, s(y)) -> times(x, exp(x, y)) 1131.74/291.60 p(s(0')) -> 0' 1131.74/291.60 p(s(s(x))) -> s(p(s(x))) 1131.74/291.60 tower(x, y) -> towerIter(x, y, s(0')) 1131.74/291.60 towerIter(0', y, z) -> z 1131.74/291.60 towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) 1131.74/291.60 1131.74/291.60 Types: 1131.74/291.60 plus :: 0':s -> 0':s -> 0':s 1131.74/291.60 0' :: 0':s 1131.74/291.60 s :: 0':s -> 0':s 1131.74/291.60 p :: 0':s -> 0':s 1131.74/291.60 times :: 0':s -> 0':s -> 0':s 1131.74/291.60 exp :: 0':s -> 0':s -> 0':s 1131.74/291.60 tower :: 0':s -> 0':s -> 0':s 1131.74/291.60 towerIter :: 0':s -> 0':s -> 0':s -> 0':s 1131.74/291.60 hole_0':s1_0 :: 0':s 1131.74/291.60 gen_0':s2_0 :: Nat -> 0':s 1131.74/291.60 1131.74/291.60 1131.74/291.60 Lemmas: 1131.74/291.60 p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0) 1131.74/291.60 1131.74/291.60 1131.74/291.60 Generator Equations: 1131.74/291.60 gen_0':s2_0(0) <=> 0' 1131.74/291.60 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 1131.74/291.60 1131.74/291.60 1131.74/291.60 The following defined symbols remain to be analysed: 1131.74/291.60 plus, times, exp, towerIter 1131.74/291.60 1131.74/291.60 They will be analysed ascendingly in the following order: 1131.74/291.60 plus < times 1131.74/291.60 times < exp 1131.74/291.60 exp < towerIter 1131.74/291.60 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (13) RewriteLemmaProof (LOWER BOUND(ID)) 1131.74/291.60 Proved the following rewrite lemma: 1131.74/291.60 plus(gen_0':s2_0(n245_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n245_0, b)), rt in Omega(1 + n245_0 + n245_0^2) 1131.74/291.60 1131.74/291.60 Induction Base: 1131.74/291.60 plus(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1) 1131.74/291.60 gen_0':s2_0(b) 1131.74/291.60 1131.74/291.60 Induction Step: 1131.74/291.60 plus(gen_0':s2_0(+(n245_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1) 1131.74/291.60 s(plus(p(s(gen_0':s2_0(n245_0))), gen_0':s2_0(b))) ->_L^Omega(1 + n245_0) 1131.74/291.60 s(plus(gen_0':s2_0(n245_0), gen_0':s2_0(b))) ->_IH 1131.74/291.60 s(gen_0':s2_0(+(b, c246_0))) 1131.74/291.60 1131.74/291.60 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (14) 1131.74/291.60 Complex Obligation (BEST) 1131.74/291.60 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (15) 1131.74/291.60 Obligation: 1131.74/291.60 Proved the lower bound n^2 for the following obligation: 1131.74/291.60 1131.74/291.60 Innermost TRS: 1131.74/291.60 Rules: 1131.74/291.60 plus(0', x) -> x 1131.74/291.60 plus(s(x), y) -> s(plus(p(s(x)), y)) 1131.74/291.60 times(0', y) -> 0' 1131.74/291.60 times(s(x), y) -> plus(y, times(p(s(x)), y)) 1131.74/291.60 exp(x, 0') -> s(0') 1131.74/291.60 exp(x, s(y)) -> times(x, exp(x, y)) 1131.74/291.60 p(s(0')) -> 0' 1131.74/291.60 p(s(s(x))) -> s(p(s(x))) 1131.74/291.60 tower(x, y) -> towerIter(x, y, s(0')) 1131.74/291.60 towerIter(0', y, z) -> z 1131.74/291.60 towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) 1131.74/291.60 1131.74/291.60 Types: 1131.74/291.60 plus :: 0':s -> 0':s -> 0':s 1131.74/291.60 0' :: 0':s 1131.74/291.60 s :: 0':s -> 0':s 1131.74/291.60 p :: 0':s -> 0':s 1131.74/291.60 times :: 0':s -> 0':s -> 0':s 1131.74/291.60 exp :: 0':s -> 0':s -> 0':s 1131.74/291.60 tower :: 0':s -> 0':s -> 0':s 1131.74/291.60 towerIter :: 0':s -> 0':s -> 0':s -> 0':s 1131.74/291.60 hole_0':s1_0 :: 0':s 1131.74/291.60 gen_0':s2_0 :: Nat -> 0':s 1131.74/291.60 1131.74/291.60 1131.74/291.60 Lemmas: 1131.74/291.60 p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0) 1131.74/291.60 1131.74/291.60 1131.74/291.60 Generator Equations: 1131.74/291.60 gen_0':s2_0(0) <=> 0' 1131.74/291.60 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 1131.74/291.60 1131.74/291.60 1131.74/291.60 The following defined symbols remain to be analysed: 1131.74/291.60 plus, times, exp, towerIter 1131.74/291.60 1131.74/291.60 They will be analysed ascendingly in the following order: 1131.74/291.60 plus < times 1131.74/291.60 times < exp 1131.74/291.60 exp < towerIter 1131.74/291.60 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (16) LowerBoundPropagationProof (FINISHED) 1131.74/291.60 Propagated lower bound. 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (17) 1131.74/291.60 BOUNDS(n^2, INF) 1131.74/291.60 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (18) 1131.74/291.60 Obligation: 1131.74/291.60 Innermost TRS: 1131.74/291.60 Rules: 1131.74/291.60 plus(0', x) -> x 1131.74/291.60 plus(s(x), y) -> s(plus(p(s(x)), y)) 1131.74/291.60 times(0', y) -> 0' 1131.74/291.60 times(s(x), y) -> plus(y, times(p(s(x)), y)) 1131.74/291.60 exp(x, 0') -> s(0') 1131.74/291.60 exp(x, s(y)) -> times(x, exp(x, y)) 1131.74/291.60 p(s(0')) -> 0' 1131.74/291.60 p(s(s(x))) -> s(p(s(x))) 1131.74/291.60 tower(x, y) -> towerIter(x, y, s(0')) 1131.74/291.60 towerIter(0', y, z) -> z 1131.74/291.60 towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) 1131.74/291.60 1131.74/291.60 Types: 1131.74/291.60 plus :: 0':s -> 0':s -> 0':s 1131.74/291.60 0' :: 0':s 1131.74/291.60 s :: 0':s -> 0':s 1131.74/291.60 p :: 0':s -> 0':s 1131.74/291.60 times :: 0':s -> 0':s -> 0':s 1131.74/291.60 exp :: 0':s -> 0':s -> 0':s 1131.74/291.60 tower :: 0':s -> 0':s -> 0':s 1131.74/291.60 towerIter :: 0':s -> 0':s -> 0':s -> 0':s 1131.74/291.60 hole_0':s1_0 :: 0':s 1131.74/291.60 gen_0':s2_0 :: Nat -> 0':s 1131.74/291.60 1131.74/291.60 1131.74/291.60 Lemmas: 1131.74/291.60 p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0) 1131.74/291.60 plus(gen_0':s2_0(n245_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n245_0, b)), rt in Omega(1 + n245_0 + n245_0^2) 1131.74/291.60 1131.74/291.60 1131.74/291.60 Generator Equations: 1131.74/291.60 gen_0':s2_0(0) <=> 0' 1131.74/291.60 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 1131.74/291.60 1131.74/291.60 1131.74/291.60 The following defined symbols remain to be analysed: 1131.74/291.60 times, exp, towerIter 1131.74/291.60 1131.74/291.60 They will be analysed ascendingly in the following order: 1131.74/291.60 times < exp 1131.74/291.60 exp < towerIter 1131.74/291.60 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (19) RewriteLemmaProof (LOWER BOUND(ID)) 1131.74/291.60 Proved the following rewrite lemma: 1131.74/291.60 times(gen_0':s2_0(n767_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n767_0, b)), rt in Omega(1 + b*n767_0 + b^2*n767_0 + n767_0 + n767_0^2) 1131.74/291.60 1131.74/291.60 Induction Base: 1131.74/291.60 times(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1) 1131.74/291.60 0' 1131.74/291.60 1131.74/291.60 Induction Step: 1131.74/291.60 times(gen_0':s2_0(+(n767_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1) 1131.74/291.60 plus(gen_0':s2_0(b), times(p(s(gen_0':s2_0(n767_0))), gen_0':s2_0(b))) ->_L^Omega(1 + n767_0) 1131.74/291.60 plus(gen_0':s2_0(b), times(gen_0':s2_0(n767_0), gen_0':s2_0(b))) ->_IH 1131.74/291.60 plus(gen_0':s2_0(b), gen_0':s2_0(*(c768_0, b))) ->_L^Omega(1 + b + b^2) 1131.74/291.60 gen_0':s2_0(+(b, *(n767_0, b))) 1131.74/291.60 1131.74/291.60 We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3). 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (20) 1131.74/291.60 Complex Obligation (BEST) 1131.74/291.60 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (21) 1131.74/291.60 Obligation: 1131.74/291.60 Proved the lower bound n^3 for the following obligation: 1131.74/291.60 1131.74/291.60 Innermost TRS: 1131.74/291.60 Rules: 1131.74/291.60 plus(0', x) -> x 1131.74/291.60 plus(s(x), y) -> s(plus(p(s(x)), y)) 1131.74/291.60 times(0', y) -> 0' 1131.74/291.60 times(s(x), y) -> plus(y, times(p(s(x)), y)) 1131.74/291.60 exp(x, 0') -> s(0') 1131.74/291.60 exp(x, s(y)) -> times(x, exp(x, y)) 1131.74/291.60 p(s(0')) -> 0' 1131.74/291.60 p(s(s(x))) -> s(p(s(x))) 1131.74/291.60 tower(x, y) -> towerIter(x, y, s(0')) 1131.74/291.60 towerIter(0', y, z) -> z 1131.74/291.60 towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) 1131.74/291.60 1131.74/291.60 Types: 1131.74/291.60 plus :: 0':s -> 0':s -> 0':s 1131.74/291.60 0' :: 0':s 1131.74/291.60 s :: 0':s -> 0':s 1131.74/291.60 p :: 0':s -> 0':s 1131.74/291.60 times :: 0':s -> 0':s -> 0':s 1131.74/291.60 exp :: 0':s -> 0':s -> 0':s 1131.74/291.60 tower :: 0':s -> 0':s -> 0':s 1131.74/291.60 towerIter :: 0':s -> 0':s -> 0':s -> 0':s 1131.74/291.60 hole_0':s1_0 :: 0':s 1131.74/291.60 gen_0':s2_0 :: Nat -> 0':s 1131.74/291.60 1131.74/291.60 1131.74/291.60 Lemmas: 1131.74/291.60 p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0) 1131.74/291.60 plus(gen_0':s2_0(n245_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n245_0, b)), rt in Omega(1 + n245_0 + n245_0^2) 1131.74/291.60 1131.74/291.60 1131.74/291.60 Generator Equations: 1131.74/291.60 gen_0':s2_0(0) <=> 0' 1131.74/291.60 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 1131.74/291.60 1131.74/291.60 1131.74/291.60 The following defined symbols remain to be analysed: 1131.74/291.60 times, exp, towerIter 1131.74/291.60 1131.74/291.60 They will be analysed ascendingly in the following order: 1131.74/291.60 times < exp 1131.74/291.60 exp < towerIter 1131.74/291.60 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (22) LowerBoundPropagationProof (FINISHED) 1131.74/291.60 Propagated lower bound. 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (23) 1131.74/291.60 BOUNDS(n^3, INF) 1131.74/291.60 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (24) 1131.74/291.60 Obligation: 1131.74/291.60 Innermost TRS: 1131.74/291.60 Rules: 1131.74/291.60 plus(0', x) -> x 1131.74/291.60 plus(s(x), y) -> s(plus(p(s(x)), y)) 1131.74/291.60 times(0', y) -> 0' 1131.74/291.60 times(s(x), y) -> plus(y, times(p(s(x)), y)) 1131.74/291.60 exp(x, 0') -> s(0') 1131.74/291.60 exp(x, s(y)) -> times(x, exp(x, y)) 1131.74/291.60 p(s(0')) -> 0' 1131.74/291.60 p(s(s(x))) -> s(p(s(x))) 1131.74/291.60 tower(x, y) -> towerIter(x, y, s(0')) 1131.74/291.60 towerIter(0', y, z) -> z 1131.74/291.60 towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) 1131.74/291.60 1131.74/291.60 Types: 1131.74/291.60 plus :: 0':s -> 0':s -> 0':s 1131.74/291.60 0' :: 0':s 1131.74/291.60 s :: 0':s -> 0':s 1131.74/291.60 p :: 0':s -> 0':s 1131.74/291.60 times :: 0':s -> 0':s -> 0':s 1131.74/291.60 exp :: 0':s -> 0':s -> 0':s 1131.74/291.60 tower :: 0':s -> 0':s -> 0':s 1131.74/291.60 towerIter :: 0':s -> 0':s -> 0':s -> 0':s 1131.74/291.60 hole_0':s1_0 :: 0':s 1131.74/291.60 gen_0':s2_0 :: Nat -> 0':s 1131.74/291.60 1131.74/291.60 1131.74/291.60 Lemmas: 1131.74/291.60 p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0) 1131.74/291.60 plus(gen_0':s2_0(n245_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n245_0, b)), rt in Omega(1 + n245_0 + n245_0^2) 1131.74/291.60 times(gen_0':s2_0(n767_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n767_0, b)), rt in Omega(1 + b*n767_0 + b^2*n767_0 + n767_0 + n767_0^2) 1131.74/291.60 1131.74/291.60 1131.74/291.60 Generator Equations: 1131.74/291.60 gen_0':s2_0(0) <=> 0' 1131.74/291.60 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 1131.74/291.60 1131.74/291.60 1131.74/291.60 The following defined symbols remain to be analysed: 1131.74/291.60 exp, towerIter 1131.74/291.60 1131.74/291.60 They will be analysed ascendingly in the following order: 1131.74/291.60 exp < towerIter 1131.74/291.60 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (25) RewriteLemmaProof (LOWER BOUND(ID)) 1131.74/291.60 Proved the following rewrite lemma: 1131.74/291.60 exp(gen_0':s2_0(a), gen_0':s2_0(+(1, n1497_0))) -> *3_0, rt in Omega(n1497_0) 1131.74/291.60 1131.74/291.60 Induction Base: 1131.74/291.60 exp(gen_0':s2_0(a), gen_0':s2_0(+(1, 0))) 1131.74/291.60 1131.74/291.60 Induction Step: 1131.74/291.60 exp(gen_0':s2_0(a), gen_0':s2_0(+(1, +(n1497_0, 1)))) ->_R^Omega(1) 1131.74/291.60 times(gen_0':s2_0(a), exp(gen_0':s2_0(a), gen_0':s2_0(+(1, n1497_0)))) ->_IH 1131.74/291.60 times(gen_0':s2_0(a), *3_0) 1131.74/291.60 1131.74/291.60 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1131.74/291.60 ---------------------------------------- 1131.74/291.60 1131.74/291.60 (26) 1131.74/291.60 Obligation: 1131.74/291.60 Innermost TRS: 1131.74/291.60 Rules: 1131.74/291.60 plus(0', x) -> x 1131.74/291.60 plus(s(x), y) -> s(plus(p(s(x)), y)) 1131.74/291.60 times(0', y) -> 0' 1131.74/291.60 times(s(x), y) -> plus(y, times(p(s(x)), y)) 1131.74/291.60 exp(x, 0') -> s(0') 1131.74/291.60 exp(x, s(y)) -> times(x, exp(x, y)) 1131.74/291.60 p(s(0')) -> 0' 1131.74/291.60 p(s(s(x))) -> s(p(s(x))) 1131.74/291.60 tower(x, y) -> towerIter(x, y, s(0')) 1131.74/291.60 towerIter(0', y, z) -> z 1131.74/291.60 towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) 1131.74/291.60 1131.74/291.60 Types: 1131.74/291.60 plus :: 0':s -> 0':s -> 0':s 1131.74/291.60 0' :: 0':s 1131.74/291.60 s :: 0':s -> 0':s 1131.74/291.60 p :: 0':s -> 0':s 1131.74/291.60 times :: 0':s -> 0':s -> 0':s 1131.74/291.60 exp :: 0':s -> 0':s -> 0':s 1131.74/291.60 tower :: 0':s -> 0':s -> 0':s 1131.74/291.60 towerIter :: 0':s -> 0':s -> 0':s -> 0':s 1131.74/291.60 hole_0':s1_0 :: 0':s 1131.74/291.60 gen_0':s2_0 :: Nat -> 0':s 1131.74/291.60 1131.74/291.60 1131.74/291.60 Lemmas: 1131.74/291.60 p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0) 1131.74/291.60 plus(gen_0':s2_0(n245_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n245_0, b)), rt in Omega(1 + n245_0 + n245_0^2) 1131.74/291.60 times(gen_0':s2_0(n767_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n767_0, b)), rt in Omega(1 + b*n767_0 + b^2*n767_0 + n767_0 + n767_0^2) 1131.74/291.60 exp(gen_0':s2_0(a), gen_0':s2_0(+(1, n1497_0))) -> *3_0, rt in Omega(n1497_0) 1131.74/291.60 1131.74/291.60 1131.74/291.60 Generator Equations: 1131.74/291.60 gen_0':s2_0(0) <=> 0' 1131.74/291.60 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 1131.74/291.60 1131.74/291.60 1131.74/291.60 The following defined symbols remain to be analysed: 1131.74/291.60 towerIter 1132.25/291.72 EOF