1113.68/291.56 WORST_CASE(Omega(n^2), ?) 1113.68/291.57 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1113.68/291.57 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1113.68/291.57 1113.68/291.57 1113.68/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1113.68/291.57 1113.68/291.57 (0) CpxTRS 1113.68/291.57 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1113.68/291.57 (2) CpxTRS 1113.68/291.57 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1113.68/291.57 (4) typed CpxTrs 1113.68/291.57 (5) OrderProof [LOWER BOUND(ID), 0 ms] 1113.68/291.57 (6) typed CpxTrs 1113.68/291.57 (7) RewriteLemmaProof [LOWER BOUND(ID), 300 ms] 1113.68/291.57 (8) BEST 1113.68/291.57 (9) proven lower bound 1113.68/291.57 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 1113.68/291.57 (11) BOUNDS(n^1, INF) 1113.68/291.57 (12) typed CpxTrs 1113.68/291.57 (13) RewriteLemmaProof [LOWER BOUND(ID), 255 ms] 1113.68/291.57 (14) proven lower bound 1113.68/291.57 (15) LowerBoundPropagationProof [FINISHED, 0 ms] 1113.68/291.57 (16) BOUNDS(n^2, INF) 1113.68/291.57 1113.68/291.57 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (0) 1113.68/291.57 Obligation: 1113.68/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1113.68/291.57 1113.68/291.57 1113.68/291.57 The TRS R consists of the following rules: 1113.68/291.57 1113.68/291.57 isEmpty(empty) -> true 1113.68/291.57 isEmpty(node(l, r)) -> false 1113.68/291.57 left(empty) -> empty 1113.68/291.57 left(node(l, r)) -> l 1113.68/291.57 right(empty) -> empty 1113.68/291.57 right(node(l, r)) -> r 1113.68/291.57 inc(0) -> s(0) 1113.68/291.57 inc(s(x)) -> s(inc(x)) 1113.68/291.57 count(n, x) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), node(right(left(n)), right(n))), x, inc(x)) 1113.68/291.57 if(true, b, n, m, x, y) -> x 1113.68/291.57 if(false, false, n, m, x, y) -> count(m, x) 1113.68/291.57 if(false, true, n, m, x, y) -> count(n, y) 1113.68/291.57 nrOfNodes(n) -> count(n, 0) 1113.68/291.57 1113.68/291.57 S is empty. 1113.68/291.57 Rewrite Strategy: INNERMOST 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 1113.68/291.57 Renamed function symbols to avoid clashes with predefined symbol. 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (2) 1113.68/291.57 Obligation: 1113.68/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF). 1113.68/291.57 1113.68/291.57 1113.68/291.57 The TRS R consists of the following rules: 1113.68/291.57 1113.68/291.57 isEmpty(empty) -> true 1113.68/291.57 isEmpty(node(l, r)) -> false 1113.68/291.57 left(empty) -> empty 1113.68/291.57 left(node(l, r)) -> l 1113.68/291.57 right(empty) -> empty 1113.68/291.57 right(node(l, r)) -> r 1113.68/291.57 inc(0') -> s(0') 1113.68/291.57 inc(s(x)) -> s(inc(x)) 1113.68/291.57 count(n, x) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), node(right(left(n)), right(n))), x, inc(x)) 1113.68/291.57 if(true, b, n, m, x, y) -> x 1113.68/291.57 if(false, false, n, m, x, y) -> count(m, x) 1113.68/291.57 if(false, true, n, m, x, y) -> count(n, y) 1113.68/291.57 nrOfNodes(n) -> count(n, 0') 1113.68/291.57 1113.68/291.57 S is empty. 1113.68/291.57 Rewrite Strategy: INNERMOST 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1113.68/291.57 Infered types. 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (4) 1113.68/291.57 Obligation: 1113.68/291.57 Innermost TRS: 1113.68/291.57 Rules: 1113.68/291.57 isEmpty(empty) -> true 1113.68/291.57 isEmpty(node(l, r)) -> false 1113.68/291.57 left(empty) -> empty 1113.68/291.57 left(node(l, r)) -> l 1113.68/291.57 right(empty) -> empty 1113.68/291.57 right(node(l, r)) -> r 1113.68/291.57 inc(0') -> s(0') 1113.68/291.57 inc(s(x)) -> s(inc(x)) 1113.68/291.57 count(n, x) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), node(right(left(n)), right(n))), x, inc(x)) 1113.68/291.57 if(true, b, n, m, x, y) -> x 1113.68/291.57 if(false, false, n, m, x, y) -> count(m, x) 1113.68/291.57 if(false, true, n, m, x, y) -> count(n, y) 1113.68/291.57 nrOfNodes(n) -> count(n, 0') 1113.68/291.57 1113.68/291.57 Types: 1113.68/291.57 isEmpty :: empty:node -> true:false 1113.68/291.57 empty :: empty:node 1113.68/291.57 true :: true:false 1113.68/291.57 node :: empty:node -> empty:node -> empty:node 1113.68/291.57 false :: true:false 1113.68/291.57 left :: empty:node -> empty:node 1113.68/291.57 right :: empty:node -> empty:node 1113.68/291.57 inc :: 0':s -> 0':s 1113.68/291.57 0' :: 0':s 1113.68/291.57 s :: 0':s -> 0':s 1113.68/291.57 count :: empty:node -> 0':s -> 0':s 1113.68/291.57 if :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> 0':s 1113.68/291.57 nrOfNodes :: empty:node -> 0':s 1113.68/291.57 hole_true:false1_0 :: true:false 1113.68/291.57 hole_empty:node2_0 :: empty:node 1113.68/291.57 hole_0':s3_0 :: 0':s 1113.68/291.57 gen_empty:node4_0 :: Nat -> empty:node 1113.68/291.57 gen_0':s5_0 :: Nat -> 0':s 1113.68/291.57 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (5) OrderProof (LOWER BOUND(ID)) 1113.68/291.57 Heuristically decided to analyse the following defined symbols: 1113.68/291.57 inc, count 1113.68/291.57 1113.68/291.57 They will be analysed ascendingly in the following order: 1113.68/291.57 inc < count 1113.68/291.57 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (6) 1113.68/291.57 Obligation: 1113.68/291.57 Innermost TRS: 1113.68/291.57 Rules: 1113.68/291.57 isEmpty(empty) -> true 1113.68/291.57 isEmpty(node(l, r)) -> false 1113.68/291.57 left(empty) -> empty 1113.68/291.57 left(node(l, r)) -> l 1113.68/291.57 right(empty) -> empty 1113.68/291.57 right(node(l, r)) -> r 1113.68/291.57 inc(0') -> s(0') 1113.68/291.57 inc(s(x)) -> s(inc(x)) 1113.68/291.57 count(n, x) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), node(right(left(n)), right(n))), x, inc(x)) 1113.68/291.57 if(true, b, n, m, x, y) -> x 1113.68/291.57 if(false, false, n, m, x, y) -> count(m, x) 1113.68/291.57 if(false, true, n, m, x, y) -> count(n, y) 1113.68/291.57 nrOfNodes(n) -> count(n, 0') 1113.68/291.57 1113.68/291.57 Types: 1113.68/291.57 isEmpty :: empty:node -> true:false 1113.68/291.57 empty :: empty:node 1113.68/291.57 true :: true:false 1113.68/291.57 node :: empty:node -> empty:node -> empty:node 1113.68/291.57 false :: true:false 1113.68/291.57 left :: empty:node -> empty:node 1113.68/291.57 right :: empty:node -> empty:node 1113.68/291.57 inc :: 0':s -> 0':s 1113.68/291.57 0' :: 0':s 1113.68/291.57 s :: 0':s -> 0':s 1113.68/291.57 count :: empty:node -> 0':s -> 0':s 1113.68/291.57 if :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> 0':s 1113.68/291.57 nrOfNodes :: empty:node -> 0':s 1113.68/291.57 hole_true:false1_0 :: true:false 1113.68/291.57 hole_empty:node2_0 :: empty:node 1113.68/291.57 hole_0':s3_0 :: 0':s 1113.68/291.57 gen_empty:node4_0 :: Nat -> empty:node 1113.68/291.57 gen_0':s5_0 :: Nat -> 0':s 1113.68/291.57 1113.68/291.57 1113.68/291.57 Generator Equations: 1113.68/291.57 gen_empty:node4_0(0) <=> empty 1113.68/291.57 gen_empty:node4_0(+(x, 1)) <=> node(empty, gen_empty:node4_0(x)) 1113.68/291.57 gen_0':s5_0(0) <=> 0' 1113.68/291.57 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 1113.68/291.57 1113.68/291.57 1113.68/291.57 The following defined symbols remain to be analysed: 1113.68/291.57 inc, count 1113.68/291.57 1113.68/291.57 They will be analysed ascendingly in the following order: 1113.68/291.57 inc < count 1113.68/291.57 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (7) RewriteLemmaProof (LOWER BOUND(ID)) 1113.68/291.57 Proved the following rewrite lemma: 1113.68/291.57 inc(gen_0':s5_0(n7_0)) -> gen_0':s5_0(+(1, n7_0)), rt in Omega(1 + n7_0) 1113.68/291.57 1113.68/291.57 Induction Base: 1113.68/291.57 inc(gen_0':s5_0(0)) ->_R^Omega(1) 1113.68/291.57 s(0') 1113.68/291.57 1113.68/291.57 Induction Step: 1113.68/291.57 inc(gen_0':s5_0(+(n7_0, 1))) ->_R^Omega(1) 1113.68/291.57 s(inc(gen_0':s5_0(n7_0))) ->_IH 1113.68/291.57 s(gen_0':s5_0(+(1, c8_0))) 1113.68/291.57 1113.68/291.57 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (8) 1113.68/291.57 Complex Obligation (BEST) 1113.68/291.57 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (9) 1113.68/291.57 Obligation: 1113.68/291.57 Proved the lower bound n^1 for the following obligation: 1113.68/291.57 1113.68/291.57 Innermost TRS: 1113.68/291.57 Rules: 1113.68/291.57 isEmpty(empty) -> true 1113.68/291.57 isEmpty(node(l, r)) -> false 1113.68/291.57 left(empty) -> empty 1113.68/291.57 left(node(l, r)) -> l 1113.68/291.57 right(empty) -> empty 1113.68/291.57 right(node(l, r)) -> r 1113.68/291.57 inc(0') -> s(0') 1113.68/291.57 inc(s(x)) -> s(inc(x)) 1113.68/291.57 count(n, x) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), node(right(left(n)), right(n))), x, inc(x)) 1113.68/291.57 if(true, b, n, m, x, y) -> x 1113.68/291.57 if(false, false, n, m, x, y) -> count(m, x) 1113.68/291.57 if(false, true, n, m, x, y) -> count(n, y) 1113.68/291.57 nrOfNodes(n) -> count(n, 0') 1113.68/291.57 1113.68/291.57 Types: 1113.68/291.57 isEmpty :: empty:node -> true:false 1113.68/291.57 empty :: empty:node 1113.68/291.57 true :: true:false 1113.68/291.57 node :: empty:node -> empty:node -> empty:node 1113.68/291.57 false :: true:false 1113.68/291.57 left :: empty:node -> empty:node 1113.68/291.57 right :: empty:node -> empty:node 1113.68/291.57 inc :: 0':s -> 0':s 1113.68/291.57 0' :: 0':s 1113.68/291.57 s :: 0':s -> 0':s 1113.68/291.57 count :: empty:node -> 0':s -> 0':s 1113.68/291.57 if :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> 0':s 1113.68/291.57 nrOfNodes :: empty:node -> 0':s 1113.68/291.57 hole_true:false1_0 :: true:false 1113.68/291.57 hole_empty:node2_0 :: empty:node 1113.68/291.57 hole_0':s3_0 :: 0':s 1113.68/291.57 gen_empty:node4_0 :: Nat -> empty:node 1113.68/291.57 gen_0':s5_0 :: Nat -> 0':s 1113.68/291.57 1113.68/291.57 1113.68/291.57 Generator Equations: 1113.68/291.57 gen_empty:node4_0(0) <=> empty 1113.68/291.57 gen_empty:node4_0(+(x, 1)) <=> node(empty, gen_empty:node4_0(x)) 1113.68/291.57 gen_0':s5_0(0) <=> 0' 1113.68/291.57 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 1113.68/291.57 1113.68/291.57 1113.68/291.57 The following defined symbols remain to be analysed: 1113.68/291.57 inc, count 1113.68/291.57 1113.68/291.57 They will be analysed ascendingly in the following order: 1113.68/291.57 inc < count 1113.68/291.57 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (10) LowerBoundPropagationProof (FINISHED) 1113.68/291.57 Propagated lower bound. 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (11) 1113.68/291.57 BOUNDS(n^1, INF) 1113.68/291.57 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (12) 1113.68/291.57 Obligation: 1113.68/291.57 Innermost TRS: 1113.68/291.57 Rules: 1113.68/291.57 isEmpty(empty) -> true 1113.68/291.57 isEmpty(node(l, r)) -> false 1113.68/291.57 left(empty) -> empty 1113.68/291.57 left(node(l, r)) -> l 1113.68/291.57 right(empty) -> empty 1113.68/291.57 right(node(l, r)) -> r 1113.68/291.57 inc(0') -> s(0') 1113.68/291.57 inc(s(x)) -> s(inc(x)) 1113.68/291.57 count(n, x) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), node(right(left(n)), right(n))), x, inc(x)) 1113.68/291.57 if(true, b, n, m, x, y) -> x 1113.68/291.57 if(false, false, n, m, x, y) -> count(m, x) 1113.68/291.57 if(false, true, n, m, x, y) -> count(n, y) 1113.68/291.57 nrOfNodes(n) -> count(n, 0') 1113.68/291.57 1113.68/291.57 Types: 1113.68/291.57 isEmpty :: empty:node -> true:false 1113.68/291.57 empty :: empty:node 1113.68/291.57 true :: true:false 1113.68/291.57 node :: empty:node -> empty:node -> empty:node 1113.68/291.57 false :: true:false 1113.68/291.57 left :: empty:node -> empty:node 1113.68/291.57 right :: empty:node -> empty:node 1113.68/291.57 inc :: 0':s -> 0':s 1113.68/291.57 0' :: 0':s 1113.68/291.57 s :: 0':s -> 0':s 1113.68/291.57 count :: empty:node -> 0':s -> 0':s 1113.68/291.57 if :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> 0':s 1113.68/291.57 nrOfNodes :: empty:node -> 0':s 1113.68/291.57 hole_true:false1_0 :: true:false 1113.68/291.57 hole_empty:node2_0 :: empty:node 1113.68/291.57 hole_0':s3_0 :: 0':s 1113.68/291.57 gen_empty:node4_0 :: Nat -> empty:node 1113.68/291.57 gen_0':s5_0 :: Nat -> 0':s 1113.68/291.57 1113.68/291.57 1113.68/291.57 Lemmas: 1113.68/291.57 inc(gen_0':s5_0(n7_0)) -> gen_0':s5_0(+(1, n7_0)), rt in Omega(1 + n7_0) 1113.68/291.57 1113.68/291.57 1113.68/291.57 Generator Equations: 1113.68/291.57 gen_empty:node4_0(0) <=> empty 1113.68/291.57 gen_empty:node4_0(+(x, 1)) <=> node(empty, gen_empty:node4_0(x)) 1113.68/291.57 gen_0':s5_0(0) <=> 0' 1113.68/291.57 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 1113.68/291.57 1113.68/291.57 1113.68/291.57 The following defined symbols remain to be analysed: 1113.68/291.57 count 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (13) RewriteLemmaProof (LOWER BOUND(ID)) 1113.68/291.57 Proved the following rewrite lemma: 1113.68/291.57 count(gen_empty:node4_0(n291_0), gen_0':s5_0(b)) -> gen_0':s5_0(+(n291_0, b)), rt in Omega(1 + b + b*n291_0 + n291_0) 1113.68/291.57 1113.68/291.57 Induction Base: 1113.68/291.57 count(gen_empty:node4_0(0), gen_0':s5_0(b)) ->_R^Omega(1) 1113.68/291.57 if(isEmpty(gen_empty:node4_0(0)), isEmpty(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)), node(left(left(gen_empty:node4_0(0))), node(right(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(true, isEmpty(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)), node(left(left(gen_empty:node4_0(0))), node(right(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(true, isEmpty(empty), right(gen_empty:node4_0(0)), node(left(left(gen_empty:node4_0(0))), node(right(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(true, true, right(gen_empty:node4_0(0)), node(left(left(gen_empty:node4_0(0))), node(right(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(true, true, empty, node(left(left(gen_empty:node4_0(0))), node(right(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(true, true, empty, node(left(empty), node(right(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(true, true, empty, node(empty, node(right(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(true, true, empty, node(empty, node(right(empty), right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(true, true, empty, node(empty, node(empty, right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(true, true, empty, node(empty, node(empty, empty)), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_L^Omega(1 + b) 1113.68/291.57 if(true, true, empty, node(empty, node(empty, empty)), gen_0':s5_0(b), gen_0':s5_0(+(1, b))) ->_R^Omega(1) 1113.68/291.57 gen_0':s5_0(b) 1113.68/291.57 1113.68/291.57 Induction Step: 1113.68/291.57 count(gen_empty:node4_0(+(n291_0, 1)), gen_0':s5_0(b)) ->_R^Omega(1) 1113.68/291.57 if(isEmpty(gen_empty:node4_0(+(n291_0, 1))), isEmpty(left(gen_empty:node4_0(+(n291_0, 1)))), right(gen_empty:node4_0(+(n291_0, 1))), node(left(left(gen_empty:node4_0(+(n291_0, 1)))), node(right(left(gen_empty:node4_0(+(n291_0, 1)))), right(gen_empty:node4_0(+(n291_0, 1))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(false, isEmpty(left(gen_empty:node4_0(+(1, n291_0)))), right(gen_empty:node4_0(+(1, n291_0))), node(left(left(gen_empty:node4_0(+(1, n291_0)))), node(right(left(gen_empty:node4_0(+(1, n291_0)))), right(gen_empty:node4_0(+(1, n291_0))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(false, isEmpty(empty), right(gen_empty:node4_0(+(1, n291_0))), node(left(left(gen_empty:node4_0(+(1, n291_0)))), node(right(left(gen_empty:node4_0(+(1, n291_0)))), right(gen_empty:node4_0(+(1, n291_0))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(false, true, right(gen_empty:node4_0(+(1, n291_0))), node(left(left(gen_empty:node4_0(+(1, n291_0)))), node(right(left(gen_empty:node4_0(+(1, n291_0)))), right(gen_empty:node4_0(+(1, n291_0))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(false, true, gen_empty:node4_0(n291_0), node(left(left(gen_empty:node4_0(+(1, n291_0)))), node(right(left(gen_empty:node4_0(+(1, n291_0)))), right(gen_empty:node4_0(+(1, n291_0))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(false, true, gen_empty:node4_0(n291_0), node(left(empty), node(right(left(gen_empty:node4_0(+(1, n291_0)))), right(gen_empty:node4_0(+(1, n291_0))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(false, true, gen_empty:node4_0(n291_0), node(empty, node(right(left(gen_empty:node4_0(+(1, n291_0)))), right(gen_empty:node4_0(+(1, n291_0))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(false, true, gen_empty:node4_0(n291_0), node(empty, node(right(empty), right(gen_empty:node4_0(+(1, n291_0))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(false, true, gen_empty:node4_0(n291_0), node(empty, node(empty, right(gen_empty:node4_0(+(1, n291_0))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1) 1113.68/291.57 if(false, true, gen_empty:node4_0(n291_0), node(empty, node(empty, gen_empty:node4_0(n291_0))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_L^Omega(1 + b) 1113.68/291.57 if(false, true, gen_empty:node4_0(n291_0), node(empty, node(empty, gen_empty:node4_0(n291_0))), gen_0':s5_0(b), gen_0':s5_0(+(1, b))) ->_R^Omega(1) 1113.68/291.57 count(gen_empty:node4_0(n291_0), gen_0':s5_0(+(1, b))) ->_IH 1113.68/291.57 gen_0':s5_0(+(+(1, b), c292_0)) 1113.68/291.57 1113.68/291.57 We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (14) 1113.68/291.57 Obligation: 1113.68/291.57 Proved the lower bound n^2 for the following obligation: 1113.68/291.57 1113.68/291.57 Innermost TRS: 1113.68/291.57 Rules: 1113.68/291.57 isEmpty(empty) -> true 1113.68/291.57 isEmpty(node(l, r)) -> false 1113.68/291.57 left(empty) -> empty 1113.68/291.57 left(node(l, r)) -> l 1113.68/291.57 right(empty) -> empty 1113.68/291.57 right(node(l, r)) -> r 1113.68/291.57 inc(0') -> s(0') 1113.68/291.57 inc(s(x)) -> s(inc(x)) 1113.68/291.57 count(n, x) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), node(right(left(n)), right(n))), x, inc(x)) 1113.68/291.57 if(true, b, n, m, x, y) -> x 1113.68/291.57 if(false, false, n, m, x, y) -> count(m, x) 1113.68/291.57 if(false, true, n, m, x, y) -> count(n, y) 1113.68/291.57 nrOfNodes(n) -> count(n, 0') 1113.68/291.57 1113.68/291.57 Types: 1113.68/291.57 isEmpty :: empty:node -> true:false 1113.68/291.57 empty :: empty:node 1113.68/291.57 true :: true:false 1113.68/291.57 node :: empty:node -> empty:node -> empty:node 1113.68/291.57 false :: true:false 1113.68/291.57 left :: empty:node -> empty:node 1113.68/291.57 right :: empty:node -> empty:node 1113.68/291.57 inc :: 0':s -> 0':s 1113.68/291.57 0' :: 0':s 1113.68/291.57 s :: 0':s -> 0':s 1113.68/291.57 count :: empty:node -> 0':s -> 0':s 1113.68/291.57 if :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> 0':s 1113.68/291.57 nrOfNodes :: empty:node -> 0':s 1113.68/291.57 hole_true:false1_0 :: true:false 1113.68/291.57 hole_empty:node2_0 :: empty:node 1113.68/291.57 hole_0':s3_0 :: 0':s 1113.68/291.57 gen_empty:node4_0 :: Nat -> empty:node 1113.68/291.57 gen_0':s5_0 :: Nat -> 0':s 1113.68/291.57 1113.68/291.57 1113.68/291.57 Lemmas: 1113.68/291.57 inc(gen_0':s5_0(n7_0)) -> gen_0':s5_0(+(1, n7_0)), rt in Omega(1 + n7_0) 1113.68/291.57 1113.68/291.57 1113.68/291.57 Generator Equations: 1113.68/291.57 gen_empty:node4_0(0) <=> empty 1113.68/291.57 gen_empty:node4_0(+(x, 1)) <=> node(empty, gen_empty:node4_0(x)) 1113.68/291.57 gen_0':s5_0(0) <=> 0' 1113.68/291.57 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 1113.68/291.57 1113.68/291.57 1113.68/291.57 The following defined symbols remain to be analysed: 1113.68/291.57 count 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (15) LowerBoundPropagationProof (FINISHED) 1113.68/291.57 Propagated lower bound. 1113.68/291.57 ---------------------------------------- 1113.68/291.57 1113.68/291.57 (16) 1113.68/291.57 BOUNDS(n^2, INF) 1113.81/291.66 EOF