WORST_CASE(?,O(n^2)) proof of input_jtuZRE3Jdb.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2). (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), 29 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), 148 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 45 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 192 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 53 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 302 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 94 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 405 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 145 ms] (40) CpxRNTS (41) FinalProof [FINISHED, 0 ms] (42) BOUNDS(1, n^2) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: and(tt, X) -> activate(X) plus(N, 0) -> N plus(N, s(M)) -> s(plus(N, M)) x(N, 0) -> 0 x(N, s(M)) -> plus(x(N, M), N) activate(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^2). The TRS R consists of the following rules: and(tt, X) -> activate(X) [1] plus(N, 0) -> N [1] plus(N, s(M)) -> s(plus(N, M)) [1] x(N, 0) -> 0 [1] x(N, s(M)) -> plus(x(N, M), N) [1] activate(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: and(tt, X) -> activate(X) [1] plus(N, 0) -> N [1] plus(N, s(M)) -> s(plus(N, M)) [1] x(N, 0) -> 0 [1] x(N, s(M)) -> plus(x(N, M), N) [1] activate(X) -> X [1] The TRS has the following type information: and :: tt -> and:activate -> and:activate tt :: tt activate :: and:activate -> and:activate plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s x :: 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: and_2 activate_1 (c) The following functions are completely defined: x_2 plus_2 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: and(tt, X) -> activate(X) [1] plus(N, 0) -> N [1] plus(N, s(M)) -> s(plus(N, M)) [1] x(N, 0) -> 0 [1] x(N, s(M)) -> plus(x(N, M), N) [1] activate(X) -> X [1] The TRS has the following type information: and :: tt -> and:activate -> and:activate tt :: tt activate :: and:activate -> and:activate plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s x :: 0:s -> 0:s -> 0:s const :: and:activate 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: and(tt, X) -> activate(X) [1] plus(N, 0) -> N [1] plus(N, s(M)) -> s(plus(N, M)) [1] x(N, 0) -> 0 [1] x(N, s(0)) -> plus(0, N) [2] x(N, s(s(M'))) -> plus(plus(x(N, M'), N), N) [2] activate(X) -> X [1] The TRS has the following type information: and :: tt -> and:activate -> and:activate tt :: tt activate :: and:activate -> and:activate plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s x :: 0:s -> 0:s -> 0:s const :: and:activate 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: tt => 0 0 => 0 const => 0 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X and(z, z') -{ 1 }-> activate(X) :|: z' = X, X >= 0, z = 0 plus(z, z') -{ 1 }-> N :|: z = N, z' = 0, N >= 0 plus(z, z') -{ 1 }-> 1 + plus(N, M) :|: z' = 1 + M, z = N, M >= 0, N >= 0 x(z, z') -{ 2 }-> plus(plus(x(N, M'), N), N) :|: M' >= 0, z = N, z' = 1 + (1 + M'), N >= 0 x(z, z') -{ 2 }-> plus(0, N) :|: z = N, z' = 1 + 0, N >= 0 x(z, z') -{ 1 }-> 0 :|: z = N, z' = 0, N >= 0 ---------------------------------------- (11) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: activate(z) -{ 1 }-> X :|: X >= 0, z = X ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X and(z, z') -{ 2 }-> X' :|: z' = X, X >= 0, z = 0, X' >= 0, X = X' plus(z, z') -{ 1 }-> N :|: z = N, z' = 0, N >= 0 plus(z, z') -{ 1 }-> 1 + plus(N, M) :|: z' = 1 + M, z = N, M >= 0, N >= 0 x(z, z') -{ 2 }-> plus(plus(x(N, M'), N), N) :|: M' >= 0, z = N, z' = 1 + (1 + M'), N >= 0 x(z, z') -{ 2 }-> plus(0, N) :|: z = N, z' = 1 + 0, N >= 0 x(z, z') -{ 1 }-> 0 :|: z = N, z' = 0, N >= 0 ---------------------------------------- (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: activate(z) -{ 1 }-> z :|: z >= 0 and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0 x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { activate } { and } { plus } { x } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0 x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 Function symbols to be analyzed: {activate}, {and}, {plus}, {x} ---------------------------------------- (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: activate(z) -{ 1 }-> z :|: z >= 0 and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0 x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 Function symbols to be analyzed: {activate}, {and}, {plus}, {x} ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: activate after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0 x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 Function symbols to be analyzed: {activate}, {and}, {plus}, {x} Previous analysis results are: activate: runtime: ?, size: O(n^1) [z] ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: activate 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: activate(z) -{ 1 }-> z :|: z >= 0 and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0 x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 Function symbols to be analyzed: {and}, {plus}, {x} Previous analysis results are: activate: runtime: O(1) [1], size: O(n^1) [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: activate(z) -{ 1 }-> z :|: z >= 0 and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0 x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 Function symbols to be analyzed: {and}, {plus}, {x} Previous analysis results are: activate: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0 x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 Function symbols to be analyzed: {and}, {plus}, {x} Previous analysis results are: activate: runtime: O(1) [1], size: O(n^1) [z] and: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0 x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 Function symbols to be analyzed: {plus}, {x} Previous analysis results are: activate: runtime: O(1) [1], size: O(n^1) [z] and: runtime: O(1) [2], size: O(n^1) [z'] ---------------------------------------- (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: activate(z) -{ 1 }-> z :|: z >= 0 and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0 x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 Function symbols to be analyzed: {plus}, {x} Previous analysis results are: activate: runtime: O(1) [1], size: O(n^1) [z] and: runtime: O(1) [2], size: O(n^1) [z'] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0 x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 Function symbols to be analyzed: {plus}, {x} Previous analysis results are: activate: runtime: O(1) [1], size: O(n^1) [z] and: runtime: O(1) [2], size: O(n^1) [z'] plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0 x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0 x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 Function symbols to be analyzed: {x} Previous analysis results are: activate: runtime: O(1) [1], size: O(n^1) [z] and: runtime: O(1) [2], size: O(n^1) [z'] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (35) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z' - 1 >= 0, z >= 0 x(z, z') -{ 3 + z }-> s' :|: s' >= 0, s' <= 0 + z, z' = 1 + 0, z >= 0 x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0 x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 Function symbols to be analyzed: {x} Previous analysis results are: activate: runtime: O(1) [1], size: O(n^1) [z] and: runtime: O(1) [2], size: O(n^1) [z'] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: x after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: z + 2*z*z' ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z' - 1 >= 0, z >= 0 x(z, z') -{ 3 + z }-> s' :|: s' >= 0, s' <= 0 + z, z' = 1 + 0, z >= 0 x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0 x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 Function symbols to be analyzed: {x} Previous analysis results are: activate: runtime: O(1) [1], size: O(n^1) [z] and: runtime: O(1) [2], size: O(n^1) [z'] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] x: runtime: ?, size: O(n^2) [z + 2*z*z'] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: x after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + z + 2*z*z' + 4*z' ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z' - 1 >= 0, z >= 0 x(z, z') -{ 3 + z }-> s' :|: s' >= 0, s' <= 0 + z, z' = 1 + 0, z >= 0 x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0 x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0 Function symbols to be analyzed: Previous analysis results are: activate: runtime: O(1) [1], size: O(n^1) [z] and: runtime: O(1) [2], size: O(n^1) [z'] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] x: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (41) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (42) BOUNDS(1, n^2)