WORST_CASE(?,O(n^2)) proof of input_n2d1NVwynU.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).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) 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), 318 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 131 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 122 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 46 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 139 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 44 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 129 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 45 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 275 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 137 ms] (44) CpxRNTS (45) FinalProof [FINISHED, 0 ms] (46) 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: rev(nil) -> nil rev(.(x, y)) -> ++(rev(y), .(x, nil)) car(.(x, y)) -> x cdr(.(x, y)) -> y null(nil) -> true null(.(x, y)) -> false ++(nil, y) -> y ++(.(x, y), z) -> .(x, ++(y, z)) 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: rev(nil) -> nil [1] rev(.(x, y)) -> ++(rev(y), .(x, nil)) [1] car(.(x, y)) -> x [1] cdr(.(x, y)) -> y [1] null(nil) -> true [1] null(.(x, y)) -> false [1] ++(nil, y) -> y [1] ++(.(x, y), z) -> .(x, ++(y, z)) [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: rev(nil) -> nil [1] rev(.(x, y)) -> ++(rev(y), .(x, nil)) [1] car(.(x, y)) -> x [1] cdr(.(x, y)) -> y [1] null(nil) -> true [1] null(.(x, y)) -> false [1] ++(nil, y) -> y [1] ++(.(x, y), z) -> .(x, ++(y, z)) [1] The TRS has the following type information: rev :: nil:. -> nil:. nil :: nil:. . :: car -> nil:. -> nil:. ++ :: nil:. -> nil:. -> nil:. car :: nil:. -> car cdr :: nil:. -> nil:. null :: nil:. -> true:false true :: true:false false :: true:false 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: car_1 cdr_1 null_1 (c) The following functions are completely defined: rev_1 ++_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: rev(nil) -> nil [1] rev(.(x, y)) -> ++(rev(y), .(x, nil)) [1] car(.(x, y)) -> x [1] cdr(.(x, y)) -> y [1] null(nil) -> true [1] null(.(x, y)) -> false [1] ++(nil, y) -> y [1] ++(.(x, y), z) -> .(x, ++(y, z)) [1] The TRS has the following type information: rev :: nil:. -> nil:. nil :: nil:. . :: car -> nil:. -> nil:. ++ :: nil:. -> nil:. -> nil:. car :: nil:. -> car cdr :: nil:. -> nil:. null :: nil:. -> true:false true :: true:false false :: true:false const :: car 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: rev(nil) -> nil [1] rev(.(x, nil)) -> ++(nil, .(x, nil)) [2] rev(.(x, .(x', y'))) -> ++(++(rev(y'), .(x', nil)), .(x, nil)) [2] car(.(x, y)) -> x [1] cdr(.(x, y)) -> y [1] null(nil) -> true [1] null(.(x, y)) -> false [1] ++(nil, y) -> y [1] ++(.(x, y), z) -> .(x, ++(y, z)) [1] The TRS has the following type information: rev :: nil:. -> nil:. nil :: nil:. . :: car -> nil:. -> nil:. ++ :: nil:. -> nil:. -> nil:. car :: nil:. -> car cdr :: nil:. -> nil:. null :: nil:. -> true:false true :: true:false false :: true:false const :: car 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 true => 1 false => 0 const => 0 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: ++(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0 ++(z', z'') -{ 1 }-> 1 + x + ++(y, z) :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(0, 1 + x + 0) :|: x >= 0, z' = 1 + x + 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 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: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 1 }-> 1 + x + ++(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 ---------------------------------------- (13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { ++ } { null } { cdr } { car } { rev } ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 1 }-> 1 + x + ++(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 Function symbols to be analyzed: {++}, {null}, {cdr}, {car}, {rev} ---------------------------------------- (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: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 1 }-> 1 + x + ++(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 Function symbols to be analyzed: {++}, {null}, {cdr}, {car}, {rev} ---------------------------------------- (17) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: ++ 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: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 1 }-> 1 + x + ++(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 Function symbols to be analyzed: {++}, {null}, {cdr}, {car}, {rev} Previous analysis results are: ++: runtime: ?, size: O(n^1) [z' + z''] ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: ++ 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: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 1 }-> 1 + x + ++(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 Function symbols to be analyzed: {null}, {cdr}, {car}, {rev} Previous analysis results are: ++: 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: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 Function symbols to be analyzed: {null}, {cdr}, {car}, {rev} Previous analysis results are: ++: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: null after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 Function symbols to be analyzed: {null}, {cdr}, {car}, {rev} Previous analysis results are: ++: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] null: runtime: ?, size: O(1) [1] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: null after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 Function symbols to be analyzed: {cdr}, {car}, {rev} Previous analysis results are: ++: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] null: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (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: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 Function symbols to be analyzed: {cdr}, {car}, {rev} Previous analysis results are: ++: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] null: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: cdr 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: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 Function symbols to be analyzed: {cdr}, {car}, {rev} Previous analysis results are: ++: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] null: runtime: O(1) [1], size: O(1) [1] cdr: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: cdr after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 Function symbols to be analyzed: {car}, {rev} Previous analysis results are: ++: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] null: runtime: O(1) [1], size: O(1) [1] cdr: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 Function symbols to be analyzed: {car}, {rev} Previous analysis results are: ++: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] null: runtime: O(1) [1], size: O(1) [1] cdr: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: car after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 Function symbols to be analyzed: {car}, {rev} Previous analysis results are: ++: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] null: runtime: O(1) [1], size: O(1) [1] cdr: runtime: O(1) [1], size: O(n^1) [z'] car: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: car after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 Function symbols to be analyzed: {rev} Previous analysis results are: ++: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] null: runtime: O(1) [1], size: O(1) [1] cdr: runtime: O(1) [1], size: O(n^1) [z'] car: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 Function symbols to be analyzed: {rev} Previous analysis results are: ++: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] null: runtime: O(1) [1], size: O(1) [1] cdr: runtime: O(1) [1], size: O(n^1) [z'] car: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: rev after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 Function symbols to be analyzed: {rev} Previous analysis results are: ++: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] null: runtime: O(1) [1], size: O(1) [1] cdr: runtime: O(1) [1], size: O(n^1) [z'] car: runtime: O(1) [1], size: O(n^1) [z'] rev: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: rev after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + 3*z' + 2*z'^2 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0 car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0 cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0 null(z') -{ 1 }-> 1 :|: z' = 0 null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0 rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0 rev(z') -{ 1 }-> 0 :|: z' = 0 rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0 Function symbols to be analyzed: Previous analysis results are: ++: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] null: runtime: O(1) [1], size: O(1) [1] cdr: runtime: O(1) [1], size: O(n^1) [z'] car: runtime: O(1) [1], size: O(n^1) [z'] rev: runtime: O(n^2) [4 + 3*z' + 2*z'^2], size: O(n^1) [z'] ---------------------------------------- (45) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (46) BOUNDS(1, n^2)