WORST_CASE(?,O(n^3)) proof of input_vST2183VuA.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3). (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), 11 ms] (10) CpxRNTS (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxRNTS (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 1 ms] (14) CpxRNTS (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) IntTrsBoundProof [UPPER BOUND(ID), 387 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 111 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 413 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 106 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 385 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 16 ms] (32) CpxRNTS (33) FinalProof [FINISHED, 0 ms] (34) BOUNDS(1, n^3) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: app(nil, y) -> y app(add(n, x), y) -> add(n, app(x, y)) reverse(nil) -> nil reverse(add(n, x)) -> app(reverse(x), add(n, nil)) shuffle(nil) -> nil shuffle(add(n, x)) -> add(n, shuffle(reverse(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^3). The TRS R consists of the following rules: app(nil, y) -> y [1] app(add(n, x), y) -> add(n, app(x, y)) [1] reverse(nil) -> nil [1] reverse(add(n, x)) -> app(reverse(x), add(n, nil)) [1] shuffle(nil) -> nil [1] shuffle(add(n, x)) -> add(n, shuffle(reverse(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: app(nil, y) -> y [1] app(add(n, x), y) -> add(n, app(x, y)) [1] reverse(nil) -> nil [1] reverse(add(n, x)) -> app(reverse(x), add(n, nil)) [1] shuffle(nil) -> nil [1] shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) [1] The TRS has the following type information: app :: nil:add -> nil:add -> nil:add nil :: nil:add add :: a -> nil:add -> nil:add reverse :: nil:add -> nil:add shuffle :: nil:add -> nil:add 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: shuffle_1 (c) The following functions are completely defined: reverse_1 app_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: app(nil, y) -> y [1] app(add(n, x), y) -> add(n, app(x, y)) [1] reverse(nil) -> nil [1] reverse(add(n, x)) -> app(reverse(x), add(n, nil)) [1] shuffle(nil) -> nil [1] shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) [1] The TRS has the following type information: app :: nil:add -> nil:add -> nil:add nil :: nil:add add :: a -> nil:add -> nil:add reverse :: nil:add -> nil:add shuffle :: nil:add -> nil:add const :: a 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: app(nil, y) -> y [1] app(add(n, x), y) -> add(n, app(x, y)) [1] reverse(nil) -> nil [1] reverse(add(n, nil)) -> app(nil, add(n, nil)) [2] reverse(add(n, add(n', x'))) -> app(app(reverse(x'), add(n', nil)), add(n, nil)) [2] shuffle(nil) -> nil [1] shuffle(add(n, nil)) -> add(n, shuffle(nil)) [2] shuffle(add(n, add(n'', x''))) -> add(n, shuffle(app(reverse(x''), add(n'', nil)))) [2] The TRS has the following type information: app :: nil:add -> nil:add -> nil:add nil :: nil:add add :: a -> nil:add -> nil:add reverse :: nil:add -> nil:add shuffle :: nil:add -> nil:add const :: a 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: nil => 0 const => 0 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y app(z, z') -{ 1 }-> 1 + n + app(x, y) :|: n >= 0, x >= 0, y >= 0, z = 1 + n + x, z' = y reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0 reverse(z) -{ 2 }-> app(0, 1 + n + 0) :|: z = 1 + n + 0, n >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'') shuffle(z) -{ 2 }-> 1 + n + shuffle(0) :|: z = 1 + n + 0, n >= 0 ---------------------------------------- (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: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0 reverse(z) -{ 2 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'') shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0 ---------------------------------------- (13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { app } { reverse } { shuffle } ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0 reverse(z) -{ 2 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'') shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0 Function symbols to be analyzed: {app}, {reverse}, {shuffle} ---------------------------------------- (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: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0 reverse(z) -{ 2 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'') shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0 Function symbols to be analyzed: {app}, {reverse}, {shuffle} ---------------------------------------- (17) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: app after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0 reverse(z) -{ 2 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'') shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0 Function symbols to be analyzed: {app}, {reverse}, {shuffle} Previous analysis results are: app: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: app 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: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0 reverse(z) -{ 2 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'') shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0 Function symbols to be analyzed: {reverse}, {shuffle} Previous analysis results are: app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (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: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x reverse(z) -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z - 1) + 0), z - 1 >= 0 reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'') shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0 Function symbols to be analyzed: {reverse}, {shuffle} Previous analysis results are: app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: reverse after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x reverse(z) -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z - 1) + 0), z - 1 >= 0 reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'') shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0 Function symbols to be analyzed: {reverse}, {shuffle} Previous analysis results are: app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] reverse: runtime: ?, size: O(n^1) [z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: reverse after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + 3*z + 2*z^2 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x reverse(z) -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z - 1) + 0), z - 1 >= 0 reverse(z) -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 2 }-> 1 + n + shuffle(app(reverse(x''), 1 + n'' + 0)) :|: n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'') shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0 Function symbols to be analyzed: {shuffle} Previous analysis results are: app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] reverse: runtime: O(n^2) [4 + 3*z + 2*z^2], size: O(n^1) [z] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x reverse(z) -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z - 1) + 0), z - 1 >= 0 reverse(z) -{ 8 + s'' + s1 + 3*x' + 2*x'^2 }-> s2 :|: s'' >= 0, s'' <= x', s1 >= 0, s1 <= s'' + (1 + n' + 0), s2 >= 0, s2 <= s1 + (1 + n + 0), n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 7 + s3 + 3*x'' + 2*x''^2 }-> 1 + n + shuffle(s4) :|: s3 >= 0, s3 <= x'', s4 >= 0, s4 <= s3 + (1 + n'' + 0), n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'') shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0 Function symbols to be analyzed: {shuffle} Previous analysis results are: app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] reverse: runtime: O(n^2) [4 + 3*z + 2*z^2], size: O(n^1) [z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: shuffle after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x reverse(z) -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z - 1) + 0), z - 1 >= 0 reverse(z) -{ 8 + s'' + s1 + 3*x' + 2*x'^2 }-> s2 :|: s'' >= 0, s'' <= x', s1 >= 0, s1 <= s'' + (1 + n' + 0), s2 >= 0, s2 <= s1 + (1 + n + 0), n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 7 + s3 + 3*x'' + 2*x''^2 }-> 1 + n + shuffle(s4) :|: s3 >= 0, s3 <= x'', s4 >= 0, s4 <= s3 + (1 + n'' + 0), n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'') shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0 Function symbols to be analyzed: {shuffle} Previous analysis results are: app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] reverse: runtime: O(n^2) [4 + 3*z + 2*z^2], size: O(n^1) [z] shuffle: runtime: ?, size: O(n^1) [z] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: shuffle after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 1 + 9*z + 4*z^2 + 2*z^3 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 app(z, z') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x reverse(z) -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z - 1) + 0), z - 1 >= 0 reverse(z) -{ 8 + s'' + s1 + 3*x' + 2*x'^2 }-> s2 :|: s'' >= 0, s'' <= x', s1 >= 0, s1 <= s'' + (1 + n' + 0), s2 >= 0, s2 <= s1 + (1 + n + 0), n >= 0, z = 1 + n + (1 + n' + x'), x' >= 0, n' >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 1 }-> 0 :|: z = 0 shuffle(z) -{ 7 + s3 + 3*x'' + 2*x''^2 }-> 1 + n + shuffle(s4) :|: s3 >= 0, s3 <= x'', s4 >= 0, s4 <= s3 + (1 + n'' + 0), n >= 0, n'' >= 0, x'' >= 0, z = 1 + n + (1 + n'' + x'') shuffle(z) -{ 2 }-> 1 + (z - 1) + shuffle(0) :|: z - 1 >= 0 Function symbols to be analyzed: Previous analysis results are: app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] reverse: runtime: O(n^2) [4 + 3*z + 2*z^2], size: O(n^1) [z] shuffle: runtime: O(n^3) [1 + 9*z + 4*z^2 + 2*z^3], size: O(n^1) [z] ---------------------------------------- (33) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (34) BOUNDS(1, n^3)