WORST_CASE(?,O(n^1)) proof of input_Aej0q87UuZ.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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), 38 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), 191 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 55 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 363 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 129 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 539 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 66 ms] (34) CpxRNTS (35) FinalProof [FINISHED, 0 ms] (36) 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: prime(0) -> false prime(s(0)) -> false prime(s(s(x))) -> prime1(s(s(x)), s(x)) prime1(x, 0) -> false prime1(x, s(0)) -> true prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y))) divp(x, y) -> =(rem(x, y), 0) 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: prime(0) -> false [1] prime(s(0)) -> false [1] prime(s(s(x))) -> prime1(s(s(x)), s(x)) [1] prime1(x, 0) -> false [1] prime1(x, s(0)) -> true [1] prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y))) [1] divp(x, y) -> =(rem(x, y), 0) [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: prime(0) -> false [1] prime(s(0)) -> false [1] prime(s(s(x))) -> prime1(s(s(x)), s(x)) [1] prime1(x, 0) -> false [1] prime1(x, s(0)) -> true [1] prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y))) [1] divp(x, y) -> =(rem(x, y), 0) [1] The TRS has the following type information: prime :: 0:s -> false:true:and 0 :: 0:s false :: false:true:and s :: 0:s -> 0:s prime1 :: 0:s -> 0:s -> false:true:and true :: false:true:and and :: not -> false:true:and -> false:true:and not :: = -> not divp :: 0:s -> 0:s -> = = :: rem -> 0:s -> = rem :: 0:s -> 0:s -> rem 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: prime_1 prime1_2 divp_2 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: const, const1, const2 ---------------------------------------- (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: prime(0) -> false [1] prime(s(0)) -> false [1] prime(s(s(x))) -> prime1(s(s(x)), s(x)) [1] prime1(x, 0) -> false [1] prime1(x, s(0)) -> true [1] prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y))) [1] divp(x, y) -> =(rem(x, y), 0) [1] The TRS has the following type information: prime :: 0:s -> false:true:and 0 :: 0:s false :: false:true:and s :: 0:s -> 0:s prime1 :: 0:s -> 0:s -> false:true:and true :: false:true:and and :: not -> false:true:and -> false:true:and not :: = -> not divp :: 0:s -> 0:s -> = = :: rem -> 0:s -> = rem :: 0:s -> 0:s -> rem const :: not const1 :: = const2 :: rem 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: prime(0) -> false [1] prime(s(0)) -> false [1] prime(s(s(x))) -> prime1(s(s(x)), s(x)) [1] prime1(x, 0) -> false [1] prime1(x, s(0)) -> true [1] prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y))) [1] divp(x, y) -> =(rem(x, y), 0) [1] The TRS has the following type information: prime :: 0:s -> false:true:and 0 :: 0:s false :: false:true:and s :: 0:s -> 0:s prime1 :: 0:s -> 0:s -> false:true:and true :: false:true:and and :: not -> false:true:and -> false:true:and not :: = -> not divp :: 0:s -> 0:s -> = = :: rem -> 0:s -> = rem :: 0:s -> 0:s -> rem const :: not const1 :: = const2 :: rem 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 false => 0 true => 1 const => 0 const1 => 0 const2 => 0 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: divp(z, z') -{ 1 }-> 1 + (1 + x + y) + 0 :|: x >= 0, y >= 0, z = x, z' = y prime(z) -{ 1 }-> prime1(1 + (1 + x), 1 + x) :|: x >= 0, z = 1 + (1 + x) prime(z) -{ 1 }-> 0 :|: z = 0 prime(z) -{ 1 }-> 0 :|: z = 1 + 0 prime1(z, z') -{ 1 }-> 1 :|: x >= 0, z' = 1 + 0, z = x prime1(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 prime1(z, z') -{ 1 }-> 1 + (1 + divp(1 + (1 + y), x)) + prime1(x, 1 + y) :|: z' = 1 + (1 + y), x >= 0, y >= 0, z = x ---------------------------------------- (11) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: divp(z, z') -{ 1 }-> 1 + (1 + x + y) + 0 :|: x >= 0, y >= 0, z = x, z' = y ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: divp(z, z') -{ 1 }-> 1 + (1 + x + y) + 0 :|: x >= 0, y >= 0, z = x, z' = y prime(z) -{ 1 }-> prime1(1 + (1 + x), 1 + x) :|: x >= 0, z = 1 + (1 + x) prime(z) -{ 1 }-> 0 :|: z = 0 prime(z) -{ 1 }-> 0 :|: z = 1 + 0 prime1(z, z') -{ 1 }-> 1 :|: x >= 0, z' = 1 + 0, z = x prime1(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + y') + 0)) + prime1(x, 1 + y) :|: z' = 1 + (1 + y), x >= 0, y >= 0, z = x, x' >= 0, y' >= 0, 1 + (1 + y) = x', x = y' ---------------------------------------- (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: divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0 prime(z) -{ 1 }-> prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0 prime(z) -{ 1 }-> 0 :|: z = 0 prime(z) -{ 1 }-> 0 :|: z = 1 + 0 prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0 prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x' ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { divp } { prime1 } { prime } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0 prime(z) -{ 1 }-> prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0 prime(z) -{ 1 }-> 0 :|: z = 0 prime(z) -{ 1 }-> 0 :|: z = 1 + 0 prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0 prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x' Function symbols to be analyzed: {divp}, {prime1}, {prime} ---------------------------------------- (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: divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0 prime(z) -{ 1 }-> prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0 prime(z) -{ 1 }-> 0 :|: z = 0 prime(z) -{ 1 }-> 0 :|: z = 1 + 0 prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0 prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x' Function symbols to be analyzed: {divp}, {prime1}, {prime} ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: divp after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z + z' ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0 prime(z) -{ 1 }-> prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0 prime(z) -{ 1 }-> 0 :|: z = 0 prime(z) -{ 1 }-> 0 :|: z = 1 + 0 prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0 prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x' Function symbols to be analyzed: {divp}, {prime1}, {prime} Previous analysis results are: divp: runtime: ?, size: O(n^1) [2 + z + z'] ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: divp after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0 prime(z) -{ 1 }-> prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0 prime(z) -{ 1 }-> 0 :|: z = 0 prime(z) -{ 1 }-> 0 :|: z = 1 + 0 prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0 prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x' Function symbols to be analyzed: {prime1}, {prime} Previous analysis results are: divp: runtime: O(1) [1], size: O(n^1) [2 + z + z'] ---------------------------------------- (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: divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0 prime(z) -{ 1 }-> prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0 prime(z) -{ 1 }-> 0 :|: z = 0 prime(z) -{ 1 }-> 0 :|: z = 1 + 0 prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0 prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x' Function symbols to be analyzed: {prime1}, {prime} Previous analysis results are: divp: runtime: O(1) [1], size: O(n^1) [2 + z + z'] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: prime1 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + z*z' + 4*z' + z'^2 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0 prime(z) -{ 1 }-> prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0 prime(z) -{ 1 }-> 0 :|: z = 0 prime(z) -{ 1 }-> 0 :|: z = 1 + 0 prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0 prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x' Function symbols to be analyzed: {prime1}, {prime} Previous analysis results are: divp: runtime: O(1) [1], size: O(n^1) [2 + z + z'] prime1: runtime: ?, size: O(n^2) [1 + z*z' + 4*z' + z'^2] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: prime1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + 2*z' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0 prime(z) -{ 1 }-> prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0 prime(z) -{ 1 }-> 0 :|: z = 0 prime(z) -{ 1 }-> 0 :|: z = 1 + 0 prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0 prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x' Function symbols to be analyzed: {prime} Previous analysis results are: divp: runtime: O(1) [1], size: O(n^1) [2 + z + z'] prime1: runtime: O(n^1) [2 + 2*z'], size: O(n^2) [1 + z*z' + 4*z' + z'^2] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0 prime(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= (1 + (z - 2)) * (1 + (z - 2)) + (1 + (z - 2)) * (1 + (1 + (z - 2))) + 4 * (1 + (z - 2)) + 1, z - 2 >= 0 prime(z) -{ 1 }-> 0 :|: z = 0 prime(z) -{ 1 }-> 0 :|: z = 1 + 0 prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0 prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 prime1(z, z') -{ 2 + 2*z' }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * (1 + (z' - 2)) + (1 + (z' - 2)) * z + 4 * (1 + (z' - 2)) + 1, z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x' Function symbols to be analyzed: {prime} Previous analysis results are: divp: runtime: O(1) [1], size: O(n^1) [2 + z + z'] prime1: runtime: O(n^1) [2 + 2*z'], size: O(n^2) [1 + z*z' + 4*z' + z'^2] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: prime after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2 + z + 2*z^2 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0 prime(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= (1 + (z - 2)) * (1 + (z - 2)) + (1 + (z - 2)) * (1 + (1 + (z - 2))) + 4 * (1 + (z - 2)) + 1, z - 2 >= 0 prime(z) -{ 1 }-> 0 :|: z = 0 prime(z) -{ 1 }-> 0 :|: z = 1 + 0 prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0 prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 prime1(z, z') -{ 2 + 2*z' }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * (1 + (z' - 2)) + (1 + (z' - 2)) * z + 4 * (1 + (z' - 2)) + 1, z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x' Function symbols to be analyzed: {prime} Previous analysis results are: divp: runtime: O(1) [1], size: O(n^1) [2 + z + z'] prime1: runtime: O(n^1) [2 + 2*z'], size: O(n^2) [1 + z*z' + 4*z' + z'^2] prime: runtime: ?, size: O(n^2) [2 + z + 2*z^2] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: prime after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 2*z ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0 prime(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= (1 + (z - 2)) * (1 + (z - 2)) + (1 + (z - 2)) * (1 + (1 + (z - 2))) + 4 * (1 + (z - 2)) + 1, z - 2 >= 0 prime(z) -{ 1 }-> 0 :|: z = 0 prime(z) -{ 1 }-> 0 :|: z = 1 + 0 prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0 prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 prime1(z, z') -{ 2 + 2*z' }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * (1 + (z' - 2)) + (1 + (z' - 2)) * z + 4 * (1 + (z' - 2)) + 1, z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x' Function symbols to be analyzed: Previous analysis results are: divp: runtime: O(1) [1], size: O(n^1) [2 + z + z'] prime1: runtime: O(n^1) [2 + 2*z'], size: O(n^2) [1 + z*z' + 4*z' + z'^2] prime: runtime: O(n^1) [1 + 2*z], size: O(n^2) [2 + z + 2*z^2] ---------------------------------------- (35) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (36) BOUNDS(1, n^1)