303.99/291.56 WORST_CASE(Omega(n^1), ?) 303.99/291.56 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 303.99/291.56 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 303.99/291.56 303.99/291.56 303.99/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 303.99/291.56 303.99/291.56 (0) CpxTRS 303.99/291.56 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 303.99/291.56 (2) TRS for Loop Detection 303.99/291.56 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 303.99/291.56 (4) BEST 303.99/291.56 (5) proven lower bound 303.99/291.56 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 303.99/291.56 (7) BOUNDS(n^1, INF) 303.99/291.56 (8) TRS for Loop Detection 303.99/291.56 303.99/291.56 303.99/291.56 ---------------------------------------- 303.99/291.56 303.99/291.56 (0) 303.99/291.56 Obligation: 303.99/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 303.99/291.56 303.99/291.56 303.99/291.56 The TRS R consists of the following rules: 303.99/291.56 303.99/291.56 lt(0, s(X)) -> true 303.99/291.56 lt(s(X), 0) -> false 303.99/291.56 lt(s(X), s(Y)) -> lt(X, Y) 303.99/291.56 append(nil, Y) -> Y 303.99/291.56 append(add(N, X), Y) -> add(N, append(X, Y)) 303.99/291.56 split(N, nil) -> pair(nil, nil) 303.99/291.56 split(N, add(M, Y)) -> f_1(split(N, Y), N, M, Y) 303.99/291.56 f_1(pair(X, Z), N, M, Y) -> f_2(lt(N, M), N, M, Y, X, Z) 303.99/291.56 f_2(true, N, M, Y, X, Z) -> pair(X, add(M, Z)) 303.99/291.56 f_2(false, N, M, Y, X, Z) -> pair(add(M, X), Z) 303.99/291.56 qsort(nil) -> nil 303.99/291.56 qsort(add(N, X)) -> f_3(split(N, X), N, X) 303.99/291.56 f_3(pair(Y, Z), N, X) -> append(qsort(Y), add(X, qsort(Z))) 303.99/291.56 303.99/291.56 S is empty. 303.99/291.56 Rewrite Strategy: FULL 303.99/291.56 ---------------------------------------- 303.99/291.56 303.99/291.56 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 303.99/291.56 Transformed a relative TRS into a decreasing-loop problem. 303.99/291.56 ---------------------------------------- 303.99/291.56 303.99/291.56 (2) 303.99/291.56 Obligation: 303.99/291.56 Analyzing the following TRS for decreasing loops: 303.99/291.56 303.99/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 303.99/291.56 303.99/291.56 303.99/291.56 The TRS R consists of the following rules: 303.99/291.56 303.99/291.56 lt(0, s(X)) -> true 303.99/291.56 lt(s(X), 0) -> false 303.99/291.56 lt(s(X), s(Y)) -> lt(X, Y) 303.99/291.56 append(nil, Y) -> Y 303.99/291.56 append(add(N, X), Y) -> add(N, append(X, Y)) 303.99/291.56 split(N, nil) -> pair(nil, nil) 303.99/291.56 split(N, add(M, Y)) -> f_1(split(N, Y), N, M, Y) 303.99/291.56 f_1(pair(X, Z), N, M, Y) -> f_2(lt(N, M), N, M, Y, X, Z) 303.99/291.56 f_2(true, N, M, Y, X, Z) -> pair(X, add(M, Z)) 303.99/291.56 f_2(false, N, M, Y, X, Z) -> pair(add(M, X), Z) 303.99/291.56 qsort(nil) -> nil 303.99/291.56 qsort(add(N, X)) -> f_3(split(N, X), N, X) 303.99/291.56 f_3(pair(Y, Z), N, X) -> append(qsort(Y), add(X, qsort(Z))) 303.99/291.56 303.99/291.56 S is empty. 303.99/291.56 Rewrite Strategy: FULL 303.99/291.56 ---------------------------------------- 303.99/291.56 303.99/291.56 (3) DecreasingLoopProof (LOWER BOUND(ID)) 303.99/291.56 The following loop(s) give(s) rise to the lower bound Omega(n^1): 303.99/291.56 303.99/291.56 The rewrite sequence 303.99/291.56 303.99/291.56 split(N, add(M, Y)) ->^+ f_1(split(N, Y), N, M, Y) 303.99/291.56 303.99/291.56 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 303.99/291.56 303.99/291.56 The pumping substitution is [Y / add(M, Y)]. 303.99/291.56 303.99/291.56 The result substitution is [ ]. 303.99/291.56 303.99/291.56 303.99/291.56 303.99/291.56 303.99/291.56 ---------------------------------------- 303.99/291.56 303.99/291.56 (4) 303.99/291.56 Complex Obligation (BEST) 303.99/291.56 303.99/291.56 ---------------------------------------- 303.99/291.56 303.99/291.56 (5) 303.99/291.56 Obligation: 303.99/291.56 Proved the lower bound n^1 for the following obligation: 303.99/291.56 303.99/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 303.99/291.56 303.99/291.56 303.99/291.56 The TRS R consists of the following rules: 303.99/291.56 303.99/291.56 lt(0, s(X)) -> true 303.99/291.56 lt(s(X), 0) -> false 303.99/291.56 lt(s(X), s(Y)) -> lt(X, Y) 303.99/291.56 append(nil, Y) -> Y 303.99/291.56 append(add(N, X), Y) -> add(N, append(X, Y)) 303.99/291.56 split(N, nil) -> pair(nil, nil) 303.99/291.56 split(N, add(M, Y)) -> f_1(split(N, Y), N, M, Y) 303.99/291.56 f_1(pair(X, Z), N, M, Y) -> f_2(lt(N, M), N, M, Y, X, Z) 303.99/291.56 f_2(true, N, M, Y, X, Z) -> pair(X, add(M, Z)) 303.99/291.56 f_2(false, N, M, Y, X, Z) -> pair(add(M, X), Z) 303.99/291.56 qsort(nil) -> nil 303.99/291.56 qsort(add(N, X)) -> f_3(split(N, X), N, X) 303.99/291.56 f_3(pair(Y, Z), N, X) -> append(qsort(Y), add(X, qsort(Z))) 303.99/291.56 303.99/291.56 S is empty. 303.99/291.56 Rewrite Strategy: FULL 303.99/291.56 ---------------------------------------- 303.99/291.56 303.99/291.56 (6) LowerBoundPropagationProof (FINISHED) 303.99/291.56 Propagated lower bound. 303.99/291.56 ---------------------------------------- 303.99/291.56 303.99/291.56 (7) 303.99/291.56 BOUNDS(n^1, INF) 303.99/291.56 303.99/291.56 ---------------------------------------- 303.99/291.56 303.99/291.56 (8) 303.99/291.56 Obligation: 303.99/291.56 Analyzing the following TRS for decreasing loops: 303.99/291.56 303.99/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 303.99/291.56 303.99/291.56 303.99/291.56 The TRS R consists of the following rules: 303.99/291.56 303.99/291.56 lt(0, s(X)) -> true 303.99/291.56 lt(s(X), 0) -> false 303.99/291.56 lt(s(X), s(Y)) -> lt(X, Y) 303.99/291.56 append(nil, Y) -> Y 303.99/291.56 append(add(N, X), Y) -> add(N, append(X, Y)) 303.99/291.56 split(N, nil) -> pair(nil, nil) 303.99/291.56 split(N, add(M, Y)) -> f_1(split(N, Y), N, M, Y) 303.99/291.56 f_1(pair(X, Z), N, M, Y) -> f_2(lt(N, M), N, M, Y, X, Z) 303.99/291.56 f_2(true, N, M, Y, X, Z) -> pair(X, add(M, Z)) 303.99/291.56 f_2(false, N, M, Y, X, Z) -> pair(add(M, X), Z) 303.99/291.56 qsort(nil) -> nil 303.99/291.56 qsort(add(N, X)) -> f_3(split(N, X), N, X) 303.99/291.56 f_3(pair(Y, Z), N, X) -> append(qsort(Y), add(X, qsort(Z))) 303.99/291.56 303.99/291.56 S is empty. 303.99/291.56 Rewrite Strategy: FULL 303.99/291.59 EOF