WORST_CASE(?,O(n^1)) proof of input_sCZKvcjn5H.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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) InliningProof [UPPER BOUND(ID), 23 ms] (12) CpxRNTS (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxRNTS (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 326 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 114 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 131 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 45 ms] (28) CpxRNTS (29) FinalProof [FINISHED, 0 ms] (30) 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)) -> +(sqr(s(x)), sum(x)) sqr(x) -> *(x, x) sum(s(x)) -> +(*(s(x), s(x)), sum(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)) -> +(sqr(s(x)), sum(x)) [1] sqr(x) -> *(x, x) [1] sum(s(x)) -> +(*(s(x), s(x)), sum(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)) -> +(sqr(s(x)), sum(x)) [1] sqr(x) -> *(x, x) [1] sum(s(x)) -> +(*(s(x), s(x)), sum(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:+ sqr :: 0:s:+ -> * * :: 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 sqr_1 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: const ---------------------------------------- (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)) -> +(sqr(s(x)), sum(x)) [1] sqr(x) -> *(x, x) [1] sum(s(x)) -> +(*(s(x), s(x)), sum(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:+ sqr :: 0:s:+ -> * * :: 0:s:+ -> 0:s:+ -> * const :: * 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)) -> +(sqr(s(x)), sum(x)) [1] sqr(x) -> *(x, x) [1] sum(s(x)) -> +(*(s(x), s(x)), sum(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:+ sqr :: 0:s:+ -> * * :: 0:s:+ -> 0:s:+ -> * const :: * 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 const => 0 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: sqr(z) -{ 1 }-> 1 + x + x :|: x >= 0, z = x sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 1 }-> 1 + sqr(1 + x) + sum(x) :|: x >= 0, z = 1 + x sum(z) -{ 1 }-> 1 + (1 + (1 + x) + (1 + x)) + sum(x) :|: x >= 0, z = 1 + x ---------------------------------------- (11) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: sqr(z) -{ 1 }-> 1 + x + x :|: x >= 0, z = x ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: sqr(z) -{ 1 }-> 1 + x + x :|: x >= 0, z = x sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 2 }-> 1 + (1 + x' + x') + sum(x) :|: x >= 0, z = 1 + x, x' >= 0, 1 + x = x' sum(z) -{ 1 }-> 1 + (1 + (1 + x) + (1 + x)) + sum(x) :|: x >= 0, z = 1 + x ---------------------------------------- (13) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: sqr(z) -{ 1 }-> 1 + z + z :|: z >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 2 }-> 1 + (1 + x' + x') + sum(z - 1) :|: z - 1 >= 0, x' >= 0, 1 + (z - 1) = x' sum(z) -{ 1 }-> 1 + (1 + (1 + (z - 1)) + (1 + (z - 1))) + sum(z - 1) :|: z - 1 >= 0 ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { sum } { sqr } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: sqr(z) -{ 1 }-> 1 + z + z :|: z >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 2 }-> 1 + (1 + x' + x') + sum(z - 1) :|: z - 1 >= 0, x' >= 0, 1 + (z - 1) = x' sum(z) -{ 1 }-> 1 + (1 + (1 + (z - 1)) + (1 + (z - 1))) + sum(z - 1) :|: z - 1 >= 0 Function symbols to be analyzed: {sum}, {sqr} ---------------------------------------- (17) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: sqr(z) -{ 1 }-> 1 + z + z :|: z >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 2 }-> 1 + (1 + x' + x') + sum(z - 1) :|: z - 1 >= 0, x' >= 0, 1 + (z - 1) = x' sum(z) -{ 1 }-> 1 + (1 + (1 + (z - 1)) + (1 + (z - 1))) + sum(z - 1) :|: z - 1 >= 0 Function symbols to be analyzed: {sum}, {sqr} ---------------------------------------- (19) 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: 2*z + 2*z^2 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: sqr(z) -{ 1 }-> 1 + z + z :|: z >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 2 }-> 1 + (1 + x' + x') + sum(z - 1) :|: z - 1 >= 0, x' >= 0, 1 + (z - 1) = x' sum(z) -{ 1 }-> 1 + (1 + (1 + (z - 1)) + (1 + (z - 1))) + sum(z - 1) :|: z - 1 >= 0 Function symbols to be analyzed: {sum}, {sqr} Previous analysis results are: sum: runtime: ?, size: O(n^2) [2*z + 2*z^2] ---------------------------------------- (21) 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 + 2*z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: sqr(z) -{ 1 }-> 1 + z + z :|: z >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 2 }-> 1 + (1 + x' + x') + sum(z - 1) :|: z - 1 >= 0, x' >= 0, 1 + (z - 1) = x' sum(z) -{ 1 }-> 1 + (1 + (1 + (z - 1)) + (1 + (z - 1))) + sum(z - 1) :|: z - 1 >= 0 Function symbols to be analyzed: {sqr} Previous analysis results are: sum: runtime: O(n^1) [1 + 2*z], size: O(n^2) [2*z + 2*z^2] ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: sqr(z) -{ 1 }-> 1 + z + z :|: z >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 1 + 2*z }-> 1 + (1 + x' + x') + s' :|: s' >= 0, s' <= 2 * ((z - 1) * (z - 1)) + 2 * (z - 1), z - 1 >= 0, x' >= 0, 1 + (z - 1) = x' sum(z) -{ 2*z }-> 1 + (1 + (1 + (z - 1)) + (1 + (z - 1))) + s :|: s >= 0, s <= 2 * ((z - 1) * (z - 1)) + 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {sqr} Previous analysis results are: sum: runtime: O(n^1) [1 + 2*z], size: O(n^2) [2*z + 2*z^2] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: sqr after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 2*z ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: sqr(z) -{ 1 }-> 1 + z + z :|: z >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 1 + 2*z }-> 1 + (1 + x' + x') + s' :|: s' >= 0, s' <= 2 * ((z - 1) * (z - 1)) + 2 * (z - 1), z - 1 >= 0, x' >= 0, 1 + (z - 1) = x' sum(z) -{ 2*z }-> 1 + (1 + (1 + (z - 1)) + (1 + (z - 1))) + s :|: s >= 0, s <= 2 * ((z - 1) * (z - 1)) + 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: {sqr} Previous analysis results are: sum: runtime: O(n^1) [1 + 2*z], size: O(n^2) [2*z + 2*z^2] sqr: runtime: ?, size: O(n^1) [1 + 2*z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: sqr after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: sqr(z) -{ 1 }-> 1 + z + z :|: z >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 sum(z) -{ 1 + 2*z }-> 1 + (1 + x' + x') + s' :|: s' >= 0, s' <= 2 * ((z - 1) * (z - 1)) + 2 * (z - 1), z - 1 >= 0, x' >= 0, 1 + (z - 1) = x' sum(z) -{ 2*z }-> 1 + (1 + (1 + (z - 1)) + (1 + (z - 1))) + s :|: s >= 0, s <= 2 * ((z - 1) * (z - 1)) + 2 * (z - 1), z - 1 >= 0 Function symbols to be analyzed: Previous analysis results are: sum: runtime: O(n^1) [1 + 2*z], size: O(n^2) [2*z + 2*z^2] sqr: runtime: O(1) [1], size: O(n^1) [1 + 2*z] ---------------------------------------- (29) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (30) BOUNDS(1, n^1)