WORST_CASE(?,O(n^1)) proof of input_16iVqFmmiO.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). (0) CpxTRS (1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxWeightedTrs (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxTypedWeightedTrs (5) CompletionProof [UPPER BOUND(ID), 0 ms] (6) CpxTypedWeightedCompleteTrs (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (10) CpxRNTS (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxRNTS (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxRNTS (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) IntTrsBoundProof [UPPER BOUND(ID), 239 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 79 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 245 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 64 ms] (26) CpxRNTS (27) FinalProof [FINISHED, 0 ms] (28) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: sum(0) -> 0 sum(s(x)) -> +(sum(x), s(x)) sum1(0) -> 0 sum1(s(x)) -> s(+(sum1(x), +(x, x))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: sum(0) -> 0 [1] sum(s(x)) -> +(sum(x), s(x)) [1] sum1(0) -> 0 [1] sum1(s(x)) -> s(+(sum1(x), +(x, x))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: sum(0) -> 0 [1] sum(s(x)) -> +(sum(x), s(x)) [1] sum1(0) -> 0 [1] sum1(s(x)) -> s(+(sum1(x), +(x, x))) [1] The TRS has the following type information: sum :: 0:s:+ -> 0:s:+ 0 :: 0:s:+ s :: 0:s:+ -> 0:s:+ + :: 0:s:+ -> 0:s:+ -> 0:s:+ sum1 :: 0:s:+ -> 0:s:+ Rewrite Strategy: INNERMOST ---------------------------------------- (5) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: sum_1 sum1_1 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: sum(0) -> 0 [1] sum(s(x)) -> +(sum(x), s(x)) [1] sum1(0) -> 0 [1] sum1(s(x)) -> s(+(sum1(x), +(x, x))) [1] The TRS has the following type information: sum :: 0:s:+ -> 0:s:+ 0 :: 0:s:+ s :: 0:s:+ -> 0:s:+ + :: 0:s:+ -> 0:s:+ -> 0:s:+ sum1 :: 0:s:+ -> 0:s:+ Rewrite Strategy: INNERMOST ---------------------------------------- (7) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: sum(0) -> 0 [1] sum(s(x)) -> +(sum(x), s(x)) [1] sum1(0) -> 0 [1] sum1(s(x)) -> s(+(sum1(x), +(x, x))) [1] The TRS has the following type information: sum :: 0:s:+ -> 0:s:+ 0 :: 0:s:+ s :: 0:s:+ -> 0:s:+ + :: 0:s:+ -> 0:s:+ -> 0:s:+ sum1 :: 0:s:+ -> 0:s:+ Rewrite Strategy: INNERMOST ---------------------------------------- (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 1 }-> 1 + sum(x) + (1 + x) :|: x >= 0, z = 1 + x sum1(z) -{ 1 }-> 0 :|: z = 0 sum1(z) -{ 1 }-> 1 + (1 + sum1(x) + (1 + x + x)) :|: x >= 0, z = 1 + x ---------------------------------------- (11) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 1 }-> 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0 sum1(z) -{ 1 }-> 0 :|: z = 0 sum1(z) -{ 1 }-> 1 + (1 + sum1(z - 1) + (1 + (z - 1) + (z - 1))) :|: z - 1 >= 0 ---------------------------------------- (13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { sum1 } { sum } ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 1 }-> 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0 sum1(z) -{ 1 }-> 0 :|: z = 0 sum1(z) -{ 1 }-> 1 + (1 + sum1(z - 1) + (1 + (z - 1) + (z - 1))) :|: z - 1 >= 0 Function symbols to be analyzed: {sum1}, {sum} ---------------------------------------- (15) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 1 }-> 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0 sum1(z) -{ 1 }-> 0 :|: z = 0 sum1(z) -{ 1 }-> 1 + (1 + sum1(z - 1) + (1 + (z - 1) + (z - 1))) :|: z - 1 >= 0 Function symbols to be analyzed: {sum1}, {sum} ---------------------------------------- (17) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: sum1 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: z + 2*z^2 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 1 }-> 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0 sum1(z) -{ 1 }-> 0 :|: z = 0 sum1(z) -{ 1 }-> 1 + (1 + sum1(z - 1) + (1 + (z - 1) + (z - 1))) :|: z - 1 >= 0 Function symbols to be analyzed: {sum1}, {sum} Previous analysis results are: sum1: runtime: ?, size: O(n^2) [z + 2*z^2] ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: sum1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 1 }-> 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0 sum1(z) -{ 1 }-> 0 :|: z = 0 sum1(z) -{ 1 }-> 1 + (1 + sum1(z - 1) + (1 + (z - 1) + (z - 1))) :|: z - 1 >= 0 Function symbols to be analyzed: {sum} Previous analysis results are: sum1: runtime: O(n^1) [1 + z], size: O(n^2) [z + 2*z^2] ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 1 }-> 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0 sum1(z) -{ 1 }-> 0 :|: z = 0 sum1(z) -{ 1 + z }-> 1 + (1 + s + (1 + (z - 1) + (z - 1))) :|: s >= 0, s <= 2 * ((z - 1) * (z - 1)) + (z - 1), z - 1 >= 0 Function symbols to be analyzed: {sum} Previous analysis results are: sum1: runtime: O(n^1) [1 + z], size: O(n^2) [z + 2*z^2] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: sum after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: z + z^2 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 1 }-> 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0 sum1(z) -{ 1 }-> 0 :|: z = 0 sum1(z) -{ 1 + z }-> 1 + (1 + s + (1 + (z - 1) + (z - 1))) :|: s >= 0, s <= 2 * ((z - 1) * (z - 1)) + (z - 1), z - 1 >= 0 Function symbols to be analyzed: {sum} Previous analysis results are: sum1: runtime: O(n^1) [1 + z], size: O(n^2) [z + 2*z^2] sum: runtime: ?, size: O(n^2) [z + z^2] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: sum after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 1 }-> 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0 sum1(z) -{ 1 }-> 0 :|: z = 0 sum1(z) -{ 1 + z }-> 1 + (1 + s + (1 + (z - 1) + (z - 1))) :|: s >= 0, s <= 2 * ((z - 1) * (z - 1)) + (z - 1), z - 1 >= 0 Function symbols to be analyzed: Previous analysis results are: sum1: runtime: O(n^1) [1 + z], size: O(n^2) [z + 2*z^2] sum: runtime: O(n^1) [1 + z], size: O(n^2) [z + z^2] ---------------------------------------- (27) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (28) BOUNDS(1, n^1)