430.90/291.54 WORST_CASE(Omega(n^1), O(n^2)) 430.99/291.55 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 430.99/291.55 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 430.99/291.55 430.99/291.55 430.99/291.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 430.99/291.55 430.99/291.55 (0) CpxTRS 430.99/291.55 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 430.99/291.55 (2) CpxWeightedTrs 430.99/291.55 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 430.99/291.55 (4) CpxTypedWeightedTrs 430.99/291.55 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 430.99/291.55 (6) CpxTypedWeightedCompleteTrs 430.99/291.55 (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 430.99/291.55 (8) CpxTypedWeightedCompleteTrs 430.99/291.55 (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 430.99/291.55 (10) CpxRNTS 430.99/291.55 (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] 430.99/291.55 (12) CpxRNTS 430.99/291.55 (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 4 ms] 430.99/291.55 (14) CpxRNTS 430.99/291.55 (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 430.99/291.55 (16) CpxRNTS 430.99/291.55 (17) IntTrsBoundProof [UPPER BOUND(ID), 361 ms] 430.99/291.55 (18) CpxRNTS 430.99/291.55 (19) IntTrsBoundProof [UPPER BOUND(ID), 98 ms] 430.99/291.55 (20) CpxRNTS 430.99/291.55 (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 430.99/291.55 (22) CpxRNTS 430.99/291.55 (23) IntTrsBoundProof [UPPER BOUND(ID), 639 ms] 430.99/291.55 (24) CpxRNTS 430.99/291.55 (25) IntTrsBoundProof [UPPER BOUND(ID), 348 ms] 430.99/291.55 (26) CpxRNTS 430.99/291.55 (27) FinalProof [FINISHED, 0 ms] 430.99/291.55 (28) BOUNDS(1, n^2) 430.99/291.55 (29) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 430.99/291.55 (30) TRS for Loop Detection 430.99/291.55 (31) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 430.99/291.55 (32) BEST 430.99/291.55 (33) proven lower bound 430.99/291.55 (34) LowerBoundPropagationProof [FINISHED, 0 ms] 430.99/291.55 (35) BOUNDS(n^1, INF) 430.99/291.55 (36) TRS for Loop Detection 430.99/291.55 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (0) 430.99/291.55 Obligation: 430.99/291.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 430.99/291.55 430.99/291.55 430.99/291.55 The TRS R consists of the following rules: 430.99/291.55 430.99/291.55 quot(0, s(y), s(z)) -> 0 430.99/291.55 quot(s(x), s(y), z) -> quot(x, y, z) 430.99/291.55 plus(0, y) -> y 430.99/291.55 plus(s(x), y) -> s(plus(x, y)) 430.99/291.55 quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z))) 430.99/291.55 430.99/291.55 S is empty. 430.99/291.55 Rewrite Strategy: INNERMOST 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 430.99/291.55 Transformed relative TRS to weighted TRS 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (2) 430.99/291.55 Obligation: 430.99/291.55 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 430.99/291.55 430.99/291.55 430.99/291.55 The TRS R consists of the following rules: 430.99/291.55 430.99/291.55 quot(0, s(y), s(z)) -> 0 [1] 430.99/291.55 quot(s(x), s(y), z) -> quot(x, y, z) [1] 430.99/291.55 plus(0, y) -> y [1] 430.99/291.55 plus(s(x), y) -> s(plus(x, y)) [1] 430.99/291.55 quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z))) [1] 430.99/291.55 430.99/291.55 Rewrite Strategy: INNERMOST 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 430.99/291.55 Infered types. 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (4) 430.99/291.55 Obligation: 430.99/291.55 Runtime Complexity Weighted TRS with Types. 430.99/291.55 The TRS R consists of the following rules: 430.99/291.55 430.99/291.55 quot(0, s(y), s(z)) -> 0 [1] 430.99/291.55 quot(s(x), s(y), z) -> quot(x, y, z) [1] 430.99/291.55 plus(0, y) -> y [1] 430.99/291.55 plus(s(x), y) -> s(plus(x, y)) [1] 430.99/291.55 quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z))) [1] 430.99/291.55 430.99/291.55 The TRS has the following type information: 430.99/291.55 quot :: 0:s -> 0:s -> 0:s -> 0:s 430.99/291.55 0 :: 0:s 430.99/291.55 s :: 0:s -> 0:s 430.99/291.55 plus :: 0:s -> 0:s -> 0:s 430.99/291.55 430.99/291.55 Rewrite Strategy: INNERMOST 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (5) CompletionProof (UPPER BOUND(ID)) 430.99/291.55 The transformation into a RNTS is sound, since: 430.99/291.55 430.99/291.55 (a) The obligation is a constructor system where every type has a constant constructor, 430.99/291.55 430.99/291.55 (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: 430.99/291.55 430.99/291.55 quot_3 430.99/291.55 430.99/291.55 (c) The following functions are completely defined: 430.99/291.55 430.99/291.55 plus_2 430.99/291.55 430.99/291.55 Due to the following rules being added: 430.99/291.55 none 430.99/291.55 430.99/291.55 And the following fresh constants: none 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (6) 430.99/291.55 Obligation: 430.99/291.55 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 430.99/291.55 430.99/291.55 Runtime Complexity Weighted TRS with Types. 430.99/291.55 The TRS R consists of the following rules: 430.99/291.55 430.99/291.55 quot(0, s(y), s(z)) -> 0 [1] 430.99/291.55 quot(s(x), s(y), z) -> quot(x, y, z) [1] 430.99/291.55 plus(0, y) -> y [1] 430.99/291.55 plus(s(x), y) -> s(plus(x, y)) [1] 430.99/291.55 quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z))) [1] 430.99/291.55 430.99/291.55 The TRS has the following type information: 430.99/291.55 quot :: 0:s -> 0:s -> 0:s -> 0:s 430.99/291.55 0 :: 0:s 430.99/291.55 s :: 0:s -> 0:s 430.99/291.55 plus :: 0:s -> 0:s -> 0:s 430.99/291.55 430.99/291.55 Rewrite Strategy: INNERMOST 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (7) NarrowingProof (BOTH BOUNDS(ID, ID)) 430.99/291.55 Narrowed the inner basic terms of all right-hand sides by a single narrowing step. 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (8) 430.99/291.55 Obligation: 430.99/291.55 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 430.99/291.55 430.99/291.55 Runtime Complexity Weighted TRS with Types. 430.99/291.55 The TRS R consists of the following rules: 430.99/291.55 430.99/291.55 quot(0, s(y), s(z)) -> 0 [1] 430.99/291.55 quot(s(x), s(y), z) -> quot(x, y, z) [1] 430.99/291.55 plus(0, y) -> y [1] 430.99/291.55 plus(s(x), y) -> s(plus(x, y)) [1] 430.99/291.55 quot(x, 0, s(0)) -> s(quot(x, s(0), s(0))) [2] 430.99/291.55 quot(x, 0, s(s(x'))) -> s(quot(x, s(plus(x', s(0))), s(s(x')))) [2] 430.99/291.55 430.99/291.55 The TRS has the following type information: 430.99/291.55 quot :: 0:s -> 0:s -> 0:s -> 0:s 430.99/291.55 0 :: 0:s 430.99/291.55 s :: 0:s -> 0:s 430.99/291.55 plus :: 0:s -> 0:s -> 0:s 430.99/291.55 430.99/291.55 Rewrite Strategy: INNERMOST 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 430.99/291.55 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 430.99/291.55 The constant constructors are abstracted as follows: 430.99/291.55 430.99/291.55 0 => 0 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (10) 430.99/291.55 Obligation: 430.99/291.55 Complexity RNTS consisting of the following rules: 430.99/291.55 430.99/291.55 plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0 430.99/291.55 plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> quot(x, y, z) :|: z' = 1 + x, z1 = z, z >= 0, x >= 0, y >= 0, z'' = 1 + y 430.99/291.55 quot(z', z'', z1) -{ 1 }-> 0 :|: z >= 0, y >= 0, z'' = 1 + y, z1 = 1 + z, z' = 0 430.99/291.55 quot(z', z'', z1) -{ 2 }-> 1 + quot(x, 1 + plus(x', 1 + 0), 1 + (1 + x')) :|: z'' = 0, z' = x, x >= 0, x' >= 0, z1 = 1 + (1 + x') 430.99/291.55 quot(z', z'', z1) -{ 2 }-> 1 + quot(x, 1 + 0, 1 + 0) :|: z'' = 0, z' = x, z1 = 1 + 0, x >= 0 430.99/291.55 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (11) SimplificationProof (BOTH BOUNDS(ID, ID)) 430.99/291.55 Simplified the RNTS by moving equalities from the constraints into the right-hand sides. 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (12) 430.99/291.55 Obligation: 430.99/291.55 Complexity RNTS consisting of the following rules: 430.99/291.55 430.99/291.55 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 430.99/291.55 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0 430.99/291.55 quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + plus(z1 - 2, 1 + 0), 1 + (1 + (z1 - 2))) :|: z'' = 0, z' >= 0, z1 - 2 >= 0 430.99/291.55 quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0 430.99/291.55 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) 430.99/291.55 Found the following analysis order by SCC decomposition: 430.99/291.55 430.99/291.55 { plus } 430.99/291.55 { quot } 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (14) 430.99/291.55 Obligation: 430.99/291.55 Complexity RNTS consisting of the following rules: 430.99/291.55 430.99/291.55 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 430.99/291.55 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0 430.99/291.55 quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + plus(z1 - 2, 1 + 0), 1 + (1 + (z1 - 2))) :|: z'' = 0, z' >= 0, z1 - 2 >= 0 430.99/291.55 quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0 430.99/291.55 430.99/291.55 Function symbols to be analyzed: {plus}, {quot} 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (15) ResultPropagationProof (UPPER BOUND(ID)) 430.99/291.55 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (16) 430.99/291.55 Obligation: 430.99/291.55 Complexity RNTS consisting of the following rules: 430.99/291.55 430.99/291.55 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 430.99/291.55 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0 430.99/291.55 quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + plus(z1 - 2, 1 + 0), 1 + (1 + (z1 - 2))) :|: z'' = 0, z' >= 0, z1 - 2 >= 0 430.99/291.55 quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0 430.99/291.55 430.99/291.55 Function symbols to be analyzed: {plus}, {quot} 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (17) IntTrsBoundProof (UPPER BOUND(ID)) 430.99/291.55 430.99/291.55 Computed SIZE bound using CoFloCo for: plus 430.99/291.55 after applying outer abstraction to obtain an ITS, 430.99/291.55 resulting in: O(n^1) with polynomial bound: z' + z'' 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (18) 430.99/291.55 Obligation: 430.99/291.55 Complexity RNTS consisting of the following rules: 430.99/291.55 430.99/291.55 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 430.99/291.55 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0 430.99/291.55 quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + plus(z1 - 2, 1 + 0), 1 + (1 + (z1 - 2))) :|: z'' = 0, z' >= 0, z1 - 2 >= 0 430.99/291.55 quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0 430.99/291.55 430.99/291.55 Function symbols to be analyzed: {plus}, {quot} 430.99/291.55 Previous analysis results are: 430.99/291.55 plus: runtime: ?, size: O(n^1) [z' + z''] 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (19) IntTrsBoundProof (UPPER BOUND(ID)) 430.99/291.55 430.99/291.55 Computed RUNTIME bound using CoFloCo for: plus 430.99/291.55 after applying outer abstraction to obtain an ITS, 430.99/291.55 resulting in: O(n^1) with polynomial bound: 1 + z' 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (20) 430.99/291.55 Obligation: 430.99/291.55 Complexity RNTS consisting of the following rules: 430.99/291.55 430.99/291.55 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 430.99/291.55 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0 430.99/291.55 quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + plus(z1 - 2, 1 + 0), 1 + (1 + (z1 - 2))) :|: z'' = 0, z' >= 0, z1 - 2 >= 0 430.99/291.55 quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0 430.99/291.55 430.99/291.55 Function symbols to be analyzed: {quot} 430.99/291.55 Previous analysis results are: 430.99/291.55 plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (21) ResultPropagationProof (UPPER BOUND(ID)) 430.99/291.55 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (22) 430.99/291.55 Obligation: 430.99/291.55 Complexity RNTS consisting of the following rules: 430.99/291.55 430.99/291.55 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 430.99/291.55 plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0 430.99/291.55 quot(z', z'', z1) -{ 1 + z1 }-> 1 + quot(z', 1 + s', 1 + (1 + (z1 - 2))) :|: s' >= 0, s' <= z1 - 2 + (1 + 0), z'' = 0, z' >= 0, z1 - 2 >= 0 430.99/291.55 quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0 430.99/291.55 430.99/291.55 Function symbols to be analyzed: {quot} 430.99/291.55 Previous analysis results are: 430.99/291.55 plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (23) IntTrsBoundProof (UPPER BOUND(ID)) 430.99/291.55 430.99/291.55 Computed SIZE bound using KoAT for: quot 430.99/291.55 after applying outer abstraction to obtain an ITS, 430.99/291.55 resulting in: O(n^1) with polynomial bound: 2 + 2*z' 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (24) 430.99/291.55 Obligation: 430.99/291.55 Complexity RNTS consisting of the following rules: 430.99/291.55 430.99/291.55 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 430.99/291.55 plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0 430.99/291.55 quot(z', z'', z1) -{ 1 + z1 }-> 1 + quot(z', 1 + s', 1 + (1 + (z1 - 2))) :|: s' >= 0, s' <= z1 - 2 + (1 + 0), z'' = 0, z' >= 0, z1 - 2 >= 0 430.99/291.55 quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0 430.99/291.55 430.99/291.55 Function symbols to be analyzed: {quot} 430.99/291.55 Previous analysis results are: 430.99/291.55 plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] 430.99/291.55 quot: runtime: ?, size: O(n^1) [2 + 2*z'] 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (25) IntTrsBoundProof (UPPER BOUND(ID)) 430.99/291.55 430.99/291.55 Computed RUNTIME bound using KoAT for: quot 430.99/291.55 after applying outer abstraction to obtain an ITS, 430.99/291.55 resulting in: O(n^2) with polynomial bound: 5 + 5*z' + z'*z1 + z1 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (26) 430.99/291.55 Obligation: 430.99/291.55 Complexity RNTS consisting of the following rules: 430.99/291.55 430.99/291.55 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 430.99/291.55 plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 430.99/291.55 quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0 430.99/291.55 quot(z', z'', z1) -{ 1 + z1 }-> 1 + quot(z', 1 + s', 1 + (1 + (z1 - 2))) :|: s' >= 0, s' <= z1 - 2 + (1 + 0), z'' = 0, z' >= 0, z1 - 2 >= 0 430.99/291.55 quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0 430.99/291.55 430.99/291.55 Function symbols to be analyzed: 430.99/291.55 Previous analysis results are: 430.99/291.55 plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] 430.99/291.55 quot: runtime: O(n^2) [5 + 5*z' + z'*z1 + z1], size: O(n^1) [2 + 2*z'] 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (27) FinalProof (FINISHED) 430.99/291.55 Computed overall runtime complexity 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (28) 430.99/291.55 BOUNDS(1, n^2) 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (29) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 430.99/291.55 Transformed a relative TRS into a decreasing-loop problem. 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (30) 430.99/291.55 Obligation: 430.99/291.55 Analyzing the following TRS for decreasing loops: 430.99/291.55 430.99/291.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 430.99/291.55 430.99/291.55 430.99/291.55 The TRS R consists of the following rules: 430.99/291.55 430.99/291.55 quot(0, s(y), s(z)) -> 0 430.99/291.55 quot(s(x), s(y), z) -> quot(x, y, z) 430.99/291.55 plus(0, y) -> y 430.99/291.55 plus(s(x), y) -> s(plus(x, y)) 430.99/291.55 quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z))) 430.99/291.55 430.99/291.55 S is empty. 430.99/291.55 Rewrite Strategy: INNERMOST 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (31) DecreasingLoopProof (LOWER BOUND(ID)) 430.99/291.55 The following loop(s) give(s) rise to the lower bound Omega(n^1): 430.99/291.55 430.99/291.55 The rewrite sequence 430.99/291.55 430.99/291.55 plus(s(x), y) ->^+ s(plus(x, y)) 430.99/291.55 430.99/291.55 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 430.99/291.55 430.99/291.55 The pumping substitution is [x / s(x)]. 430.99/291.55 430.99/291.55 The result substitution is [ ]. 430.99/291.55 430.99/291.55 430.99/291.55 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (32) 430.99/291.55 Complex Obligation (BEST) 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (33) 430.99/291.55 Obligation: 430.99/291.55 Proved the lower bound n^1 for the following obligation: 430.99/291.55 430.99/291.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 430.99/291.55 430.99/291.55 430.99/291.55 The TRS R consists of the following rules: 430.99/291.55 430.99/291.55 quot(0, s(y), s(z)) -> 0 430.99/291.55 quot(s(x), s(y), z) -> quot(x, y, z) 430.99/291.55 plus(0, y) -> y 430.99/291.55 plus(s(x), y) -> s(plus(x, y)) 430.99/291.55 quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z))) 430.99/291.55 430.99/291.55 S is empty. 430.99/291.55 Rewrite Strategy: INNERMOST 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (34) LowerBoundPropagationProof (FINISHED) 430.99/291.55 Propagated lower bound. 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (35) 430.99/291.55 BOUNDS(n^1, INF) 430.99/291.55 430.99/291.55 ---------------------------------------- 430.99/291.55 430.99/291.55 (36) 430.99/291.55 Obligation: 430.99/291.55 Analyzing the following TRS for decreasing loops: 430.99/291.55 430.99/291.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 430.99/291.55 430.99/291.55 430.99/291.55 The TRS R consists of the following rules: 430.99/291.55 430.99/291.55 quot(0, s(y), s(z)) -> 0 430.99/291.55 quot(s(x), s(y), z) -> quot(x, y, z) 430.99/291.55 plus(0, y) -> y 430.99/291.55 plus(s(x), y) -> s(plus(x, y)) 430.99/291.55 quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z))) 430.99/291.55 430.99/291.55 S is empty. 430.99/291.55 Rewrite Strategy: INNERMOST 431.02/291.60 EOF