335.41/291.50 WORST_CASE(Omega(n^4), ?) 335.41/291.51 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 335.41/291.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 335.41/291.51 335.41/291.51 335.41/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^4, INF). 335.41/291.51 335.41/291.51 (0) CpxTRS 335.41/291.51 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 335.41/291.51 (2) CpxTRS 335.41/291.51 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 335.41/291.51 (4) typed CpxTrs 335.41/291.51 (5) OrderProof [LOWER BOUND(ID), 0 ms] 335.41/291.51 (6) typed CpxTrs 335.41/291.51 (7) RewriteLemmaProof [LOWER BOUND(ID), 257 ms] 335.41/291.51 (8) BEST 335.41/291.51 (9) proven lower bound 335.41/291.51 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 335.41/291.51 (11) BOUNDS(n^1, INF) 335.41/291.51 (12) typed CpxTrs 335.41/291.51 (13) RewriteLemmaProof [LOWER BOUND(ID), 21 ms] 335.41/291.51 (14) BEST 335.41/291.51 (15) proven lower bound 335.41/291.51 (16) LowerBoundPropagationProof [FINISHED, 0 ms] 335.41/291.51 (17) BOUNDS(n^2, INF) 335.41/291.51 (18) typed CpxTrs 335.41/291.51 (19) RewriteLemmaProof [LOWER BOUND(ID), 26 ms] 335.41/291.51 (20) BEST 335.41/291.51 (21) proven lower bound 335.41/291.51 (22) LowerBoundPropagationProof [FINISHED, 0 ms] 335.41/291.51 (23) BOUNDS(n^3, INF) 335.41/291.51 (24) typed CpxTrs 335.41/291.51 (25) RewriteLemmaProof [LOWER BOUND(ID), 1123 ms] 335.41/291.51 (26) proven lower bound 335.41/291.51 (27) LowerBoundPropagationProof [FINISHED, 0 ms] 335.41/291.51 (28) BOUNDS(n^4, INF) 335.41/291.51 335.41/291.51 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (0) 335.41/291.51 Obligation: 335.41/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^4, INF). 335.41/291.51 335.41/291.51 335.41/291.51 The TRS R consists of the following rules: 335.41/291.51 335.41/291.51 plus(0, x) -> x 335.41/291.51 plus(s(x), y) -> s(plus(p(s(x)), y)) 335.41/291.51 times(0, y) -> 0 335.41/291.51 times(s(x), y) -> plus(y, times(p(s(x)), y)) 335.41/291.51 p(s(0)) -> 0 335.41/291.51 p(s(s(x))) -> s(p(s(x))) 335.41/291.51 fac(0, x) -> x 335.41/291.51 fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) 335.41/291.51 factorial(x) -> fac(x, s(0)) 335.41/291.51 335.41/291.51 S is empty. 335.41/291.51 Rewrite Strategy: FULL 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 335.41/291.51 Renamed function symbols to avoid clashes with predefined symbol. 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (2) 335.41/291.51 Obligation: 335.41/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^4, INF). 335.41/291.51 335.41/291.51 335.41/291.51 The TRS R consists of the following rules: 335.41/291.51 335.41/291.51 plus(0', x) -> x 335.41/291.51 plus(s(x), y) -> s(plus(p(s(x)), y)) 335.41/291.51 times(0', y) -> 0' 335.41/291.51 times(s(x), y) -> plus(y, times(p(s(x)), y)) 335.41/291.51 p(s(0')) -> 0' 335.41/291.51 p(s(s(x))) -> s(p(s(x))) 335.41/291.51 fac(0', x) -> x 335.41/291.51 fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) 335.41/291.51 factorial(x) -> fac(x, s(0')) 335.41/291.51 335.41/291.51 S is empty. 335.41/291.51 Rewrite Strategy: FULL 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 335.41/291.51 Infered types. 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (4) 335.41/291.51 Obligation: 335.41/291.51 TRS: 335.41/291.51 Rules: 335.41/291.51 plus(0', x) -> x 335.41/291.51 plus(s(x), y) -> s(plus(p(s(x)), y)) 335.41/291.51 times(0', y) -> 0' 335.41/291.51 times(s(x), y) -> plus(y, times(p(s(x)), y)) 335.41/291.51 p(s(0')) -> 0' 335.41/291.51 p(s(s(x))) -> s(p(s(x))) 335.41/291.51 fac(0', x) -> x 335.41/291.51 fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) 335.41/291.51 factorial(x) -> fac(x, s(0')) 335.41/291.51 335.41/291.51 Types: 335.41/291.51 plus :: 0':s -> 0':s -> 0':s 335.41/291.51 0' :: 0':s 335.41/291.51 s :: 0':s -> 0':s 335.41/291.51 p :: 0':s -> 0':s 335.41/291.51 times :: 0':s -> 0':s -> 0':s 335.41/291.51 fac :: 0':s -> 0':s -> 0':s 335.41/291.51 factorial :: 0':s -> 0':s 335.41/291.51 hole_0':s1_0 :: 0':s 335.41/291.51 gen_0':s2_0 :: Nat -> 0':s 335.41/291.51 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (5) OrderProof (LOWER BOUND(ID)) 335.41/291.51 Heuristically decided to analyse the following defined symbols: 335.41/291.51 plus, p, times, fac 335.41/291.51 335.41/291.51 They will be analysed ascendingly in the following order: 335.41/291.51 p < plus 335.41/291.51 plus < times 335.41/291.51 p < times 335.41/291.51 p < fac 335.41/291.51 times < fac 335.41/291.51 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (6) 335.41/291.51 Obligation: 335.41/291.51 TRS: 335.41/291.51 Rules: 335.41/291.51 plus(0', x) -> x 335.41/291.51 plus(s(x), y) -> s(plus(p(s(x)), y)) 335.41/291.51 times(0', y) -> 0' 335.41/291.51 times(s(x), y) -> plus(y, times(p(s(x)), y)) 335.41/291.51 p(s(0')) -> 0' 335.41/291.51 p(s(s(x))) -> s(p(s(x))) 335.41/291.51 fac(0', x) -> x 335.41/291.51 fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) 335.41/291.51 factorial(x) -> fac(x, s(0')) 335.41/291.51 335.41/291.51 Types: 335.41/291.51 plus :: 0':s -> 0':s -> 0':s 335.41/291.51 0' :: 0':s 335.41/291.51 s :: 0':s -> 0':s 335.41/291.51 p :: 0':s -> 0':s 335.41/291.51 times :: 0':s -> 0':s -> 0':s 335.41/291.51 fac :: 0':s -> 0':s -> 0':s 335.41/291.51 factorial :: 0':s -> 0':s 335.41/291.51 hole_0':s1_0 :: 0':s 335.41/291.51 gen_0':s2_0 :: Nat -> 0':s 335.41/291.51 335.41/291.51 335.41/291.51 Generator Equations: 335.41/291.51 gen_0':s2_0(0) <=> 0' 335.41/291.51 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 335.41/291.51 335.41/291.51 335.41/291.51 The following defined symbols remain to be analysed: 335.41/291.51 p, plus, times, fac 335.41/291.51 335.41/291.51 They will be analysed ascendingly in the following order: 335.41/291.51 p < plus 335.41/291.51 plus < times 335.41/291.51 p < times 335.41/291.51 p < fac 335.41/291.51 times < fac 335.41/291.51 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (7) RewriteLemmaProof (LOWER BOUND(ID)) 335.41/291.51 Proved the following rewrite lemma: 335.41/291.51 p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0) 335.41/291.51 335.41/291.51 Induction Base: 335.41/291.51 p(gen_0':s2_0(+(1, 0))) ->_R^Omega(1) 335.41/291.51 0' 335.41/291.51 335.41/291.51 Induction Step: 335.41/291.51 p(gen_0':s2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1) 335.41/291.51 s(p(s(gen_0':s2_0(n4_0)))) ->_IH 335.41/291.51 s(gen_0':s2_0(c5_0)) 335.41/291.51 335.41/291.51 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (8) 335.41/291.51 Complex Obligation (BEST) 335.41/291.51 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (9) 335.41/291.51 Obligation: 335.41/291.51 Proved the lower bound n^1 for the following obligation: 335.41/291.51 335.41/291.51 TRS: 335.41/291.51 Rules: 335.41/291.51 plus(0', x) -> x 335.41/291.51 plus(s(x), y) -> s(plus(p(s(x)), y)) 335.41/291.51 times(0', y) -> 0' 335.41/291.51 times(s(x), y) -> plus(y, times(p(s(x)), y)) 335.41/291.51 p(s(0')) -> 0' 335.41/291.51 p(s(s(x))) -> s(p(s(x))) 335.41/291.51 fac(0', x) -> x 335.41/291.51 fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) 335.41/291.51 factorial(x) -> fac(x, s(0')) 335.41/291.51 335.41/291.51 Types: 335.41/291.51 plus :: 0':s -> 0':s -> 0':s 335.41/291.51 0' :: 0':s 335.41/291.51 s :: 0':s -> 0':s 335.41/291.51 p :: 0':s -> 0':s 335.41/291.51 times :: 0':s -> 0':s -> 0':s 335.41/291.51 fac :: 0':s -> 0':s -> 0':s 335.41/291.51 factorial :: 0':s -> 0':s 335.41/291.51 hole_0':s1_0 :: 0':s 335.41/291.51 gen_0':s2_0 :: Nat -> 0':s 335.41/291.51 335.41/291.51 335.41/291.51 Generator Equations: 335.41/291.51 gen_0':s2_0(0) <=> 0' 335.41/291.51 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 335.41/291.51 335.41/291.51 335.41/291.51 The following defined symbols remain to be analysed: 335.41/291.51 p, plus, times, fac 335.41/291.51 335.41/291.51 They will be analysed ascendingly in the following order: 335.41/291.51 p < plus 335.41/291.51 plus < times 335.41/291.51 p < times 335.41/291.51 p < fac 335.41/291.51 times < fac 335.41/291.51 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (10) LowerBoundPropagationProof (FINISHED) 335.41/291.51 Propagated lower bound. 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (11) 335.41/291.51 BOUNDS(n^1, INF) 335.41/291.51 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (12) 335.41/291.51 Obligation: 335.41/291.51 TRS: 335.41/291.51 Rules: 335.41/291.51 plus(0', x) -> x 335.41/291.51 plus(s(x), y) -> s(plus(p(s(x)), y)) 335.41/291.51 times(0', y) -> 0' 335.41/291.51 times(s(x), y) -> plus(y, times(p(s(x)), y)) 335.41/291.51 p(s(0')) -> 0' 335.41/291.51 p(s(s(x))) -> s(p(s(x))) 335.41/291.51 fac(0', x) -> x 335.41/291.51 fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) 335.41/291.51 factorial(x) -> fac(x, s(0')) 335.41/291.51 335.41/291.51 Types: 335.41/291.51 plus :: 0':s -> 0':s -> 0':s 335.41/291.51 0' :: 0':s 335.41/291.51 s :: 0':s -> 0':s 335.41/291.51 p :: 0':s -> 0':s 335.41/291.51 times :: 0':s -> 0':s -> 0':s 335.41/291.51 fac :: 0':s -> 0':s -> 0':s 335.41/291.51 factorial :: 0':s -> 0':s 335.41/291.51 hole_0':s1_0 :: 0':s 335.41/291.51 gen_0':s2_0 :: Nat -> 0':s 335.41/291.51 335.41/291.51 335.41/291.51 Lemmas: 335.41/291.51 p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0) 335.41/291.51 335.41/291.51 335.41/291.51 Generator Equations: 335.41/291.51 gen_0':s2_0(0) <=> 0' 335.41/291.51 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 335.41/291.51 335.41/291.51 335.41/291.51 The following defined symbols remain to be analysed: 335.41/291.51 plus, times, fac 335.41/291.51 335.41/291.51 They will be analysed ascendingly in the following order: 335.41/291.51 plus < times 335.41/291.51 times < fac 335.41/291.51 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (13) RewriteLemmaProof (LOWER BOUND(ID)) 335.41/291.51 Proved the following rewrite lemma: 335.41/291.51 plus(gen_0':s2_0(n193_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n193_0, b)), rt in Omega(1 + n193_0 + n193_0^2) 335.41/291.51 335.41/291.51 Induction Base: 335.41/291.51 plus(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1) 335.41/291.51 gen_0':s2_0(b) 335.41/291.51 335.41/291.51 Induction Step: 335.41/291.51 plus(gen_0':s2_0(+(n193_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1) 335.41/291.51 s(plus(p(s(gen_0':s2_0(n193_0))), gen_0':s2_0(b))) ->_L^Omega(1 + n193_0) 335.41/291.51 s(plus(gen_0':s2_0(n193_0), gen_0':s2_0(b))) ->_IH 335.41/291.51 s(gen_0':s2_0(+(b, c194_0))) 335.41/291.51 335.41/291.51 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (14) 335.41/291.51 Complex Obligation (BEST) 335.41/291.51 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (15) 335.41/291.51 Obligation: 335.41/291.51 Proved the lower bound n^2 for the following obligation: 335.41/291.51 335.41/291.51 TRS: 335.41/291.51 Rules: 335.41/291.51 plus(0', x) -> x 335.41/291.51 plus(s(x), y) -> s(plus(p(s(x)), y)) 335.41/291.51 times(0', y) -> 0' 335.41/291.51 times(s(x), y) -> plus(y, times(p(s(x)), y)) 335.41/291.51 p(s(0')) -> 0' 335.41/291.51 p(s(s(x))) -> s(p(s(x))) 335.41/291.51 fac(0', x) -> x 335.41/291.51 fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) 335.41/291.51 factorial(x) -> fac(x, s(0')) 335.41/291.51 335.41/291.51 Types: 335.41/291.51 plus :: 0':s -> 0':s -> 0':s 335.41/291.51 0' :: 0':s 335.41/291.51 s :: 0':s -> 0':s 335.41/291.51 p :: 0':s -> 0':s 335.41/291.51 times :: 0':s -> 0':s -> 0':s 335.41/291.51 fac :: 0':s -> 0':s -> 0':s 335.41/291.51 factorial :: 0':s -> 0':s 335.41/291.51 hole_0':s1_0 :: 0':s 335.41/291.51 gen_0':s2_0 :: Nat -> 0':s 335.41/291.51 335.41/291.51 335.41/291.51 Lemmas: 335.41/291.51 p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0) 335.41/291.51 335.41/291.51 335.41/291.51 Generator Equations: 335.41/291.51 gen_0':s2_0(0) <=> 0' 335.41/291.51 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 335.41/291.51 335.41/291.51 335.41/291.51 The following defined symbols remain to be analysed: 335.41/291.51 plus, times, fac 335.41/291.51 335.41/291.51 They will be analysed ascendingly in the following order: 335.41/291.51 plus < times 335.41/291.51 times < fac 335.41/291.51 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (16) LowerBoundPropagationProof (FINISHED) 335.41/291.51 Propagated lower bound. 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (17) 335.41/291.51 BOUNDS(n^2, INF) 335.41/291.51 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (18) 335.41/291.51 Obligation: 335.41/291.51 TRS: 335.41/291.51 Rules: 335.41/291.51 plus(0', x) -> x 335.41/291.51 plus(s(x), y) -> s(plus(p(s(x)), y)) 335.41/291.51 times(0', y) -> 0' 335.41/291.51 times(s(x), y) -> plus(y, times(p(s(x)), y)) 335.41/291.51 p(s(0')) -> 0' 335.41/291.51 p(s(s(x))) -> s(p(s(x))) 335.41/291.51 fac(0', x) -> x 335.41/291.51 fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) 335.41/291.51 factorial(x) -> fac(x, s(0')) 335.41/291.51 335.41/291.51 Types: 335.41/291.51 plus :: 0':s -> 0':s -> 0':s 335.41/291.51 0' :: 0':s 335.41/291.51 s :: 0':s -> 0':s 335.41/291.51 p :: 0':s -> 0':s 335.41/291.51 times :: 0':s -> 0':s -> 0':s 335.41/291.51 fac :: 0':s -> 0':s -> 0':s 335.41/291.51 factorial :: 0':s -> 0':s 335.41/291.51 hole_0':s1_0 :: 0':s 335.41/291.51 gen_0':s2_0 :: Nat -> 0':s 335.41/291.51 335.41/291.51 335.41/291.51 Lemmas: 335.41/291.51 p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0) 335.41/291.51 plus(gen_0':s2_0(n193_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n193_0, b)), rt in Omega(1 + n193_0 + n193_0^2) 335.41/291.51 335.41/291.51 335.41/291.51 Generator Equations: 335.41/291.51 gen_0':s2_0(0) <=> 0' 335.41/291.51 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 335.41/291.51 335.41/291.51 335.41/291.51 The following defined symbols remain to be analysed: 335.41/291.51 times, fac 335.41/291.51 335.41/291.51 They will be analysed ascendingly in the following order: 335.41/291.51 times < fac 335.41/291.51 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (19) RewriteLemmaProof (LOWER BOUND(ID)) 335.41/291.51 Proved the following rewrite lemma: 335.41/291.51 times(gen_0':s2_0(n578_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n578_0, b)), rt in Omega(1 + b*n578_0 + b^2*n578_0 + n578_0 + n578_0^2) 335.41/291.51 335.41/291.51 Induction Base: 335.41/291.51 times(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1) 335.41/291.51 0' 335.41/291.51 335.41/291.51 Induction Step: 335.41/291.51 times(gen_0':s2_0(+(n578_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1) 335.41/291.51 plus(gen_0':s2_0(b), times(p(s(gen_0':s2_0(n578_0))), gen_0':s2_0(b))) ->_L^Omega(1 + n578_0) 335.41/291.51 plus(gen_0':s2_0(b), times(gen_0':s2_0(n578_0), gen_0':s2_0(b))) ->_IH 335.41/291.51 plus(gen_0':s2_0(b), gen_0':s2_0(*(c579_0, b))) ->_L^Omega(1 + b + b^2) 335.41/291.51 gen_0':s2_0(+(b, *(n578_0, b))) 335.41/291.51 335.41/291.51 We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3). 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (20) 335.41/291.51 Complex Obligation (BEST) 335.41/291.51 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (21) 335.41/291.51 Obligation: 335.41/291.51 Proved the lower bound n^3 for the following obligation: 335.41/291.51 335.41/291.51 TRS: 335.41/291.51 Rules: 335.41/291.51 plus(0', x) -> x 335.41/291.51 plus(s(x), y) -> s(plus(p(s(x)), y)) 335.41/291.51 times(0', y) -> 0' 335.41/291.51 times(s(x), y) -> plus(y, times(p(s(x)), y)) 335.41/291.51 p(s(0')) -> 0' 335.41/291.51 p(s(s(x))) -> s(p(s(x))) 335.41/291.51 fac(0', x) -> x 335.41/291.51 fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) 335.41/291.51 factorial(x) -> fac(x, s(0')) 335.41/291.51 335.41/291.51 Types: 335.41/291.51 plus :: 0':s -> 0':s -> 0':s 335.41/291.51 0' :: 0':s 335.41/291.51 s :: 0':s -> 0':s 335.41/291.51 p :: 0':s -> 0':s 335.41/291.51 times :: 0':s -> 0':s -> 0':s 335.41/291.51 fac :: 0':s -> 0':s -> 0':s 335.41/291.51 factorial :: 0':s -> 0':s 335.41/291.51 hole_0':s1_0 :: 0':s 335.41/291.51 gen_0':s2_0 :: Nat -> 0':s 335.41/291.51 335.41/291.51 335.41/291.51 Lemmas: 335.41/291.51 p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0) 335.41/291.51 plus(gen_0':s2_0(n193_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n193_0, b)), rt in Omega(1 + n193_0 + n193_0^2) 335.41/291.51 335.41/291.51 335.41/291.51 Generator Equations: 335.41/291.51 gen_0':s2_0(0) <=> 0' 335.41/291.51 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 335.41/291.51 335.41/291.51 335.41/291.51 The following defined symbols remain to be analysed: 335.41/291.51 times, fac 335.41/291.51 335.41/291.51 They will be analysed ascendingly in the following order: 335.41/291.51 times < fac 335.41/291.51 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (22) LowerBoundPropagationProof (FINISHED) 335.41/291.51 Propagated lower bound. 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (23) 335.41/291.51 BOUNDS(n^3, INF) 335.41/291.51 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (24) 335.41/291.51 Obligation: 335.41/291.51 TRS: 335.41/291.51 Rules: 335.41/291.51 plus(0', x) -> x 335.41/291.51 plus(s(x), y) -> s(plus(p(s(x)), y)) 335.41/291.51 times(0', y) -> 0' 335.41/291.51 times(s(x), y) -> plus(y, times(p(s(x)), y)) 335.41/291.51 p(s(0')) -> 0' 335.41/291.51 p(s(s(x))) -> s(p(s(x))) 335.41/291.51 fac(0', x) -> x 335.41/291.51 fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) 335.41/291.51 factorial(x) -> fac(x, s(0')) 335.41/291.51 335.41/291.51 Types: 335.41/291.51 plus :: 0':s -> 0':s -> 0':s 335.41/291.51 0' :: 0':s 335.41/291.51 s :: 0':s -> 0':s 335.41/291.51 p :: 0':s -> 0':s 335.41/291.51 times :: 0':s -> 0':s -> 0':s 335.41/291.51 fac :: 0':s -> 0':s -> 0':s 335.41/291.51 factorial :: 0':s -> 0':s 335.41/291.51 hole_0':s1_0 :: 0':s 335.41/291.51 gen_0':s2_0 :: Nat -> 0':s 335.41/291.51 335.41/291.51 335.41/291.51 Lemmas: 335.41/291.51 p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0) 335.41/291.51 plus(gen_0':s2_0(n193_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n193_0, b)), rt in Omega(1 + n193_0 + n193_0^2) 335.41/291.51 times(gen_0':s2_0(n578_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n578_0, b)), rt in Omega(1 + b*n578_0 + b^2*n578_0 + n578_0 + n578_0^2) 335.41/291.51 335.41/291.51 335.41/291.51 Generator Equations: 335.41/291.51 gen_0':s2_0(0) <=> 0' 335.41/291.51 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 335.41/291.51 335.41/291.51 335.41/291.51 The following defined symbols remain to be analysed: 335.41/291.51 fac 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (25) RewriteLemmaProof (LOWER BOUND(ID)) 335.41/291.51 Proved the following rewrite lemma: 335.41/291.51 fac(gen_0':s2_0(n1122_0), gen_0':s2_0(b)) -> *3_0, rt in Omega(b*n1122_0 + b*n1122_0^2 + b^2*n1122_0 + b^2*n1122_0^2 + n1122_0 + n1122_0^2 + n1122_0^3) 335.41/291.51 335.41/291.51 Induction Base: 335.41/291.51 fac(gen_0':s2_0(0), gen_0':s2_0(b)) 335.41/291.51 335.41/291.51 Induction Step: 335.41/291.51 fac(gen_0':s2_0(+(n1122_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1) 335.41/291.51 fac(p(s(gen_0':s2_0(n1122_0))), times(s(gen_0':s2_0(n1122_0)), gen_0':s2_0(b))) ->_L^Omega(1 + n1122_0) 335.41/291.51 fac(gen_0':s2_0(n1122_0), times(s(gen_0':s2_0(n1122_0)), gen_0':s2_0(b))) ->_L^Omega(3 + b + b*n1122_0 + b^2 + b^2*n1122_0 + 3*n1122_0 + n1122_0^2) 335.41/291.51 fac(gen_0':s2_0(n1122_0), gen_0':s2_0(*(+(n1122_0, 1), b))) ->_IH 335.41/291.51 *3_0 335.41/291.51 335.41/291.51 We have rt in Omega(n^4) and sz in O(n). Thus, we have irc_R in Omega(n^4). 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (26) 335.41/291.51 Obligation: 335.41/291.51 Proved the lower bound n^4 for the following obligation: 335.41/291.51 335.41/291.51 TRS: 335.41/291.51 Rules: 335.41/291.51 plus(0', x) -> x 335.41/291.51 plus(s(x), y) -> s(plus(p(s(x)), y)) 335.41/291.51 times(0', y) -> 0' 335.41/291.51 times(s(x), y) -> plus(y, times(p(s(x)), y)) 335.41/291.51 p(s(0')) -> 0' 335.41/291.51 p(s(s(x))) -> s(p(s(x))) 335.41/291.51 fac(0', x) -> x 335.41/291.51 fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) 335.41/291.51 factorial(x) -> fac(x, s(0')) 335.41/291.51 335.41/291.51 Types: 335.41/291.51 plus :: 0':s -> 0':s -> 0':s 335.41/291.51 0' :: 0':s 335.41/291.51 s :: 0':s -> 0':s 335.41/291.51 p :: 0':s -> 0':s 335.41/291.51 times :: 0':s -> 0':s -> 0':s 335.41/291.51 fac :: 0':s -> 0':s -> 0':s 335.41/291.51 factorial :: 0':s -> 0':s 335.41/291.51 hole_0':s1_0 :: 0':s 335.41/291.51 gen_0':s2_0 :: Nat -> 0':s 335.41/291.51 335.41/291.51 335.41/291.51 Lemmas: 335.41/291.51 p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0) 335.41/291.51 plus(gen_0':s2_0(n193_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n193_0, b)), rt in Omega(1 + n193_0 + n193_0^2) 335.41/291.51 times(gen_0':s2_0(n578_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n578_0, b)), rt in Omega(1 + b*n578_0 + b^2*n578_0 + n578_0 + n578_0^2) 335.41/291.51 335.41/291.51 335.41/291.51 Generator Equations: 335.41/291.51 gen_0':s2_0(0) <=> 0' 335.41/291.51 gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x)) 335.41/291.51 335.41/291.51 335.41/291.51 The following defined symbols remain to be analysed: 335.41/291.51 fac 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (27) LowerBoundPropagationProof (FINISHED) 335.41/291.51 Propagated lower bound. 335.41/291.51 ---------------------------------------- 335.41/291.51 335.41/291.51 (28) 335.41/291.51 BOUNDS(n^4, INF) 335.41/291.55 EOF