1023.44/291.52 WORST_CASE(Omega(n^1), O(n^5)) 1023.44/291.54 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1023.44/291.54 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1023.44/291.54 1023.44/291.54 1023.44/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^5). 1023.44/291.54 1023.44/291.54 (0) CpxTRS 1023.44/291.54 (1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] 1023.44/291.54 (2) CpxTRS 1023.44/291.54 (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 1023.44/291.54 (4) CpxWeightedTrs 1023.44/291.54 (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1023.44/291.54 (6) CpxTypedWeightedTrs 1023.44/291.54 (7) CompletionProof [UPPER BOUND(ID), 0 ms] 1023.44/291.54 (8) CpxTypedWeightedCompleteTrs 1023.44/291.54 (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 1023.44/291.54 (10) CpxTypedWeightedCompleteTrs 1023.44/291.54 (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 1023.44/291.54 (12) CpxRNTS 1023.44/291.54 (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] 1023.44/291.54 (14) CpxRNTS 1023.44/291.54 (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] 1023.44/291.54 (16) CpxRNTS 1023.44/291.54 (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1023.44/291.54 (18) CpxRNTS 1023.44/291.54 (19) IntTrsBoundProof [UPPER BOUND(ID), 3575 ms] 1023.44/291.54 (20) CpxRNTS 1023.44/291.54 (21) IntTrsBoundProof [UPPER BOUND(ID), 3633 ms] 1023.44/291.54 (22) CpxRNTS 1023.44/291.54 (23) FinalProof [FINISHED, 0 ms] 1023.44/291.54 (24) BOUNDS(1, n^5) 1023.44/291.54 (25) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1023.44/291.54 (26) TRS for Loop Detection 1023.44/291.54 (27) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1023.44/291.54 (28) BEST 1023.44/291.54 (29) proven lower bound 1023.44/291.54 (30) LowerBoundPropagationProof [FINISHED, 0 ms] 1023.44/291.54 (31) BOUNDS(n^1, INF) 1023.44/291.54 (32) TRS for Loop Detection 1023.44/291.54 1023.44/291.54 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (0) 1023.44/291.54 Obligation: 1023.44/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^5). 1023.44/291.54 1023.44/291.54 1023.44/291.54 The TRS R consists of the following rules: 1023.44/291.54 1023.44/291.54 f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) 1023.44/291.54 f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) 1023.44/291.54 f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) 1023.44/291.54 f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) 1023.44/291.54 f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) 1023.44/291.54 f(0, 0, 0, 0, 0) -> 0 1023.44/291.54 1023.44/291.54 S is empty. 1023.44/291.54 Rewrite Strategy: FULL 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (1) RcToIrcProof (BOTH BOUNDS(ID, ID)) 1023.44/291.54 Converted rc-obligation to irc-obligation. 1023.44/291.54 1023.44/291.54 As the TRS does not nest defined symbols, we have rc = irc. 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (2) 1023.44/291.54 Obligation: 1023.44/291.54 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^5). 1023.44/291.54 1023.44/291.54 1023.44/291.54 The TRS R consists of the following rules: 1023.44/291.54 1023.44/291.54 f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) 1023.44/291.54 f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) 1023.44/291.54 f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) 1023.44/291.54 f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) 1023.44/291.54 f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) 1023.44/291.54 f(0, 0, 0, 0, 0) -> 0 1023.44/291.54 1023.44/291.54 S is empty. 1023.44/291.54 Rewrite Strategy: INNERMOST 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 1023.44/291.54 Transformed relative TRS to weighted TRS 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (4) 1023.44/291.54 Obligation: 1023.44/291.54 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^5). 1023.44/291.54 1023.44/291.54 1023.44/291.54 The TRS R consists of the following rules: 1023.44/291.54 1023.44/291.54 f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) [1] 1023.44/291.54 f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) [1] 1023.44/291.54 f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) [1] 1023.44/291.54 f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) [1] 1023.44/291.54 f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) [1] 1023.44/291.54 f(0, 0, 0, 0, 0) -> 0 [1] 1023.44/291.54 1023.44/291.54 Rewrite Strategy: INNERMOST 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1023.44/291.54 Infered types. 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (6) 1023.44/291.54 Obligation: 1023.44/291.54 Runtime Complexity Weighted TRS with Types. 1023.44/291.54 The TRS R consists of the following rules: 1023.44/291.54 1023.44/291.54 f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) [1] 1023.44/291.54 f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) [1] 1023.44/291.54 f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) [1] 1023.44/291.54 f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) [1] 1023.44/291.54 f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) [1] 1023.44/291.54 f(0, 0, 0, 0, 0) -> 0 [1] 1023.44/291.54 1023.44/291.54 The TRS has the following type information: 1023.44/291.54 f :: s:0 -> s:0 -> s:0 -> s:0 -> s:0 -> s:0 1023.44/291.54 s :: s:0 -> s:0 1023.44/291.54 0 :: s:0 1023.44/291.54 1023.44/291.54 Rewrite Strategy: INNERMOST 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (7) CompletionProof (UPPER BOUND(ID)) 1023.44/291.54 The transformation into a RNTS is sound, since: 1023.44/291.54 1023.44/291.54 (a) The obligation is a constructor system where every type has a constant constructor, 1023.44/291.54 1023.44/291.54 (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: 1023.44/291.54 1023.44/291.54 f_5 1023.44/291.54 1023.44/291.54 (c) The following functions are completely defined: 1023.44/291.54 none 1023.44/291.54 1023.44/291.54 Due to the following rules being added: 1023.44/291.54 none 1023.44/291.54 1023.44/291.54 And the following fresh constants: none 1023.44/291.54 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (8) 1023.44/291.54 Obligation: 1023.44/291.54 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 1023.44/291.54 1023.44/291.54 Runtime Complexity Weighted TRS with Types. 1023.44/291.54 The TRS R consists of the following rules: 1023.44/291.54 1023.44/291.54 f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) [1] 1023.44/291.54 f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) [1] 1023.44/291.54 f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) [1] 1023.44/291.54 f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) [1] 1023.44/291.54 f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) [1] 1023.44/291.54 f(0, 0, 0, 0, 0) -> 0 [1] 1023.44/291.54 1023.44/291.54 The TRS has the following type information: 1023.44/291.54 f :: s:0 -> s:0 -> s:0 -> s:0 -> s:0 -> s:0 1023.44/291.54 s :: s:0 -> s:0 1023.44/291.54 0 :: s:0 1023.44/291.54 1023.44/291.54 Rewrite Strategy: INNERMOST 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (9) NarrowingProof (BOTH BOUNDS(ID, ID)) 1023.44/291.54 Narrowed the inner basic terms of all right-hand sides by a single narrowing step. 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (10) 1023.44/291.54 Obligation: 1023.44/291.54 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 1023.44/291.54 1023.44/291.54 Runtime Complexity Weighted TRS with Types. 1023.44/291.54 The TRS R consists of the following rules: 1023.44/291.54 1023.44/291.54 f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) [1] 1023.44/291.54 f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) [1] 1023.44/291.54 f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) [1] 1023.44/291.54 f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) [1] 1023.44/291.54 f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) [1] 1023.44/291.54 f(0, 0, 0, 0, 0) -> 0 [1] 1023.44/291.54 1023.44/291.54 The TRS has the following type information: 1023.44/291.54 f :: s:0 -> s:0 -> s:0 -> s:0 -> s:0 -> s:0 1023.44/291.54 s :: s:0 -> s:0 1023.44/291.54 0 :: s:0 1023.44/291.54 1023.44/291.54 Rewrite Strategy: INNERMOST 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 1023.44/291.54 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 1023.44/291.54 The constant constructors are abstracted as follows: 1023.44/291.54 1023.44/291.54 0 => 0 1023.44/291.54 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (12) 1023.44/291.54 Obligation: 1023.44/291.54 Complexity RNTS consisting of the following rules: 1023.44/291.54 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(x1, x2, x3, x4, x5) :|: z' = x2, z1 = x4, x1 >= 0, x4 >= 0, x5 >= 0, z = 1 + x1, z'' = x3, z2 = x5, x2 >= 0, x3 >= 0 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(x2, x2, x3, x4, x5) :|: z1 = x4, x4 >= 0, x5 >= 0, z' = 1 + x2, z'' = x3, z = 0, z2 = x5, x2 >= 0, x3 >= 0 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(x3, x3, x3, x4, x5) :|: z1 = x4, x4 >= 0, x5 >= 0, z'' = 1 + x3, z = 0, z2 = x5, x3 >= 0, z' = 0 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(x4, x4, x4, x4, x5) :|: z'' = 0, x4 >= 0, x5 >= 0, z1 = 1 + x4, z = 0, z2 = x5, z' = 0 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(x5, x5, x5, x5, x5) :|: z'' = 0, z1 = 0, x5 >= 0, z = 0, z2 = 1 + x5, z' = 0 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0 1023.44/291.54 1023.44/291.54 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (13) SimplificationProof (BOTH BOUNDS(ID, ID)) 1023.44/291.54 Simplified the RNTS by moving equalities from the constraints into the right-hand sides. 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (14) 1023.44/291.54 Obligation: 1023.44/291.54 Complexity RNTS consisting of the following rules: 1023.44/291.54 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0 1023.44/291.54 1023.44/291.54 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) 1023.44/291.54 Found the following analysis order by SCC decomposition: 1023.44/291.54 1023.44/291.54 { f } 1023.44/291.54 1023.44/291.54 ---------------------------------------- 1023.44/291.54 1023.44/291.54 (16) 1023.44/291.54 Obligation: 1023.44/291.54 Complexity RNTS consisting of the following rules: 1023.44/291.54 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0 1023.44/291.54 f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0 1023.44/291.54 1023.44/291.54 Function symbols to be analyzed: {f} 1023.44/291.54 1023.44/291.54 ---------------------------------------- 1023.73/291.54 1023.73/291.54 (17) ResultPropagationProof (UPPER BOUND(ID)) 1023.73/291.54 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1023.73/291.54 ---------------------------------------- 1023.73/291.54 1023.73/291.54 (18) 1023.73/291.54 Obligation: 1023.73/291.54 Complexity RNTS consisting of the following rules: 1023.73/291.54 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0 1023.73/291.54 1023.73/291.54 Function symbols to be analyzed: {f} 1023.73/291.54 1023.73/291.54 ---------------------------------------- 1023.73/291.54 1023.73/291.54 (19) IntTrsBoundProof (UPPER BOUND(ID)) 1023.73/291.54 1023.73/291.54 Computed SIZE bound using CoFloCo for: f 1023.73/291.54 after applying outer abstraction to obtain an ITS, 1023.73/291.54 resulting in: O(1) with polynomial bound: 0 1023.73/291.54 1023.73/291.54 ---------------------------------------- 1023.73/291.54 1023.73/291.54 (20) 1023.73/291.54 Obligation: 1023.73/291.54 Complexity RNTS consisting of the following rules: 1023.73/291.54 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0 1023.73/291.54 1023.73/291.54 Function symbols to be analyzed: {f} 1023.73/291.54 Previous analysis results are: 1023.73/291.54 f: runtime: ?, size: O(1) [0] 1023.73/291.54 1023.73/291.54 ---------------------------------------- 1023.73/291.54 1023.73/291.54 (21) IntTrsBoundProof (UPPER BOUND(ID)) 1023.73/291.54 1023.73/291.54 Computed RUNTIME bound using KoAT for: f 1023.73/291.54 after applying outer abstraction to obtain an ITS, 1023.73/291.54 resulting in: O(n^5) with polynomial bound: 1 + z + z' + z'*z'' + z'*z''*z1 + z'*z''*z1*z2 + z'*z''*z1*z2^2 + z'*z''*z1^2 + z'*z''*z2 + z'*z''*z2^2 + z'*z''*z2^3 + z'*z''^2 + z'*z1 + z'*z1*z2 + 3*z'*z1*z2^2 + 2*z'*z1*z2^3 + z'*z1^2 + 2*z'*z1^2*z2 + z'*z1^2*z2^2 + z'*z1^3 + z'*z2 + z'*z2^2 + 2*z'*z2^3 + z'*z2^4 + z'^2 + z'' + 2*z''*z1 + 5*z''*z1*z2 + 7*z''*z1*z2^2 + 4*z''*z1*z2^3 + 4*z''*z1^2 + 4*z''*z1^2*z2 + 2*z''*z1^2*z2^2 + 2*z''*z1^3 + 2*z''*z2 + 4*z''*z2^2 + 5*z''*z2^3 + 2*z''*z2^4 + 2*z''^2 + 2*z''^2*z1 + z''^2*z1*z2 + z''^2*z1*z2^2 + z''^2*z1^2 + 2*z''^2*z2 + z''^2*z2^2 + z''^2*z2^3 + z''^3 + z1 + 3*z1*z2 + 9*z1*z2^2 + 9*z1*z2^3 + 3*z1*z2^4 + 3*z1^2 + 6*z1^2*z2 + 7*z1^2*z2^2 + 3*z1^2*z2^3 + 3*z1^3 + 3*z1^3*z2 + z1^3*z2^2 + z1^4 + z2 + 4*z2^2 + 6*z2^3 + 4*z2^4 + z2^5 1023.73/291.54 1023.73/291.54 ---------------------------------------- 1023.73/291.54 1023.73/291.54 (22) 1023.73/291.54 Obligation: 1023.73/291.54 Complexity RNTS consisting of the following rules: 1023.73/291.54 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0 1023.73/291.54 f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0 1023.73/291.54 1023.73/291.54 Function symbols to be analyzed: 1023.73/291.54 Previous analysis results are: 1023.73/291.54 f: runtime: O(n^5) [1 + z + z' + z'*z'' + z'*z''*z1 + z'*z''*z1*z2 + z'*z''*z1*z2^2 + z'*z''*z1^2 + z'*z''*z2 + z'*z''*z2^2 + z'*z''*z2^3 + z'*z''^2 + z'*z1 + z'*z1*z2 + 3*z'*z1*z2^2 + 2*z'*z1*z2^3 + z'*z1^2 + 2*z'*z1^2*z2 + z'*z1^2*z2^2 + z'*z1^3 + z'*z2 + z'*z2^2 + 2*z'*z2^3 + z'*z2^4 + z'^2 + z'' + 2*z''*z1 + 5*z''*z1*z2 + 7*z''*z1*z2^2 + 4*z''*z1*z2^3 + 4*z''*z1^2 + 4*z''*z1^2*z2 + 2*z''*z1^2*z2^2 + 2*z''*z1^3 + 2*z''*z2 + 4*z''*z2^2 + 5*z''*z2^3 + 2*z''*z2^4 + 2*z''^2 + 2*z''^2*z1 + z''^2*z1*z2 + z''^2*z1*z2^2 + z''^2*z1^2 + 2*z''^2*z2 + z''^2*z2^2 + z''^2*z2^3 + z''^3 + z1 + 3*z1*z2 + 9*z1*z2^2 + 9*z1*z2^3 + 3*z1*z2^4 + 3*z1^2 + 6*z1^2*z2 + 7*z1^2*z2^2 + 3*z1^2*z2^3 + 3*z1^3 + 3*z1^3*z2 + z1^3*z2^2 + z1^4 + z2 + 4*z2^2 + 6*z2^3 + 4*z2^4 + z2^5], size: O(1) [0] 1023.73/291.54 1023.73/291.54 ---------------------------------------- 1023.73/291.54 1023.73/291.54 (23) FinalProof (FINISHED) 1023.73/291.54 Computed overall runtime complexity 1023.73/291.54 ---------------------------------------- 1023.73/291.54 1023.73/291.54 (24) 1023.73/291.54 BOUNDS(1, n^5) 1023.73/291.54 1023.73/291.54 ---------------------------------------- 1023.73/291.54 1023.73/291.54 (25) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1023.73/291.54 Transformed a relative TRS into a decreasing-loop problem. 1023.73/291.54 ---------------------------------------- 1023.73/291.54 1023.73/291.54 (26) 1023.73/291.54 Obligation: 1023.73/291.54 Analyzing the following TRS for decreasing loops: 1023.73/291.54 1023.73/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^5). 1023.73/291.54 1023.73/291.54 1023.73/291.54 The TRS R consists of the following rules: 1023.73/291.54 1023.73/291.54 f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) 1023.73/291.54 f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) 1023.73/291.54 f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) 1023.73/291.54 f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) 1023.73/291.54 f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) 1023.73/291.54 f(0, 0, 0, 0, 0) -> 0 1023.73/291.54 1023.73/291.54 S is empty. 1023.73/291.54 Rewrite Strategy: FULL 1023.73/291.54 ---------------------------------------- 1023.73/291.54 1023.73/291.54 (27) DecreasingLoopProof (LOWER BOUND(ID)) 1023.73/291.54 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1023.73/291.54 1023.73/291.54 The rewrite sequence 1023.73/291.54 1023.73/291.54 f(s(x1), x2, x3, x4, x5) ->^+ f(x1, x2, x3, x4, x5) 1023.73/291.54 1023.73/291.54 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 1023.73/291.54 1023.73/291.54 The pumping substitution is [x1 / s(x1)]. 1023.73/291.54 1023.73/291.54 The result substitution is [ ]. 1023.73/291.54 1023.73/291.54 1023.73/291.54 1023.73/291.54 1023.73/291.54 ---------------------------------------- 1023.73/291.54 1023.73/291.54 (28) 1023.73/291.54 Complex Obligation (BEST) 1023.73/291.54 1023.73/291.54 ---------------------------------------- 1023.73/291.54 1023.73/291.54 (29) 1023.73/291.54 Obligation: 1023.73/291.54 Proved the lower bound n^1 for the following obligation: 1023.73/291.54 1023.73/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^5). 1023.73/291.54 1023.73/291.54 1023.73/291.54 The TRS R consists of the following rules: 1023.73/291.54 1023.73/291.54 f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) 1023.73/291.54 f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) 1023.73/291.54 f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) 1023.73/291.54 f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) 1023.73/291.54 f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) 1023.73/291.54 f(0, 0, 0, 0, 0) -> 0 1023.73/291.54 1023.73/291.54 S is empty. 1023.73/291.54 Rewrite Strategy: FULL 1023.73/291.54 ---------------------------------------- 1023.73/291.54 1023.73/291.54 (30) LowerBoundPropagationProof (FINISHED) 1023.73/291.54 Propagated lower bound. 1023.73/291.54 ---------------------------------------- 1023.73/291.54 1023.73/291.54 (31) 1023.73/291.54 BOUNDS(n^1, INF) 1023.73/291.54 1023.73/291.54 ---------------------------------------- 1023.73/291.54 1023.73/291.54 (32) 1023.73/291.54 Obligation: 1023.73/291.54 Analyzing the following TRS for decreasing loops: 1023.73/291.54 1023.73/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^5). 1023.73/291.54 1023.73/291.54 1023.73/291.54 The TRS R consists of the following rules: 1023.73/291.54 1023.73/291.54 f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) 1023.73/291.54 f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) 1023.73/291.54 f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) 1023.73/291.54 f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) 1023.73/291.54 f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) 1023.73/291.54 f(0, 0, 0, 0, 0) -> 0 1023.73/291.54 1023.73/291.54 S is empty. 1023.73/291.54 Rewrite Strategy: FULL 1023.79/291.58 EOF