WORST_CASE(?,O(n^1)) proof of input_Zpk4DoQP4c.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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) 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), 565 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 185 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 200 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 96 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 118 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 52 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 209 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 84 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 241 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 83 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 139 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 53 ms] (50) CpxRNTS (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 138 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 52 ms] (56) CpxRNTS (57) FinalProof [FINISHED, 0 ms] (58) 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: append(@l1, @l2) -> append#1(@l1, @l2) append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) append#1(nil, @l2) -> @l2 appendAll(@l) -> appendAll#1(@l) appendAll#1(::(@l1, @ls)) -> append(@l1, appendAll(@ls)) appendAll#1(nil) -> nil appendAll2(@l) -> appendAll2#1(@l) appendAll2#1(::(@l1, @ls)) -> append(appendAll(@l1), appendAll2(@ls)) appendAll2#1(nil) -> nil appendAll3(@l) -> appendAll3#1(@l) appendAll3#1(::(@l1, @ls)) -> append(appendAll2(@l1), appendAll3(@ls)) appendAll3#1(nil) -> nil 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: append(@l1, @l2) -> append#1(@l1, @l2) [1] append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) [1] append#1(nil, @l2) -> @l2 [1] appendAll(@l) -> appendAll#1(@l) [1] appendAll#1(::(@l1, @ls)) -> append(@l1, appendAll(@ls)) [1] appendAll#1(nil) -> nil [1] appendAll2(@l) -> appendAll2#1(@l) [1] appendAll2#1(::(@l1, @ls)) -> append(appendAll(@l1), appendAll2(@ls)) [1] appendAll2#1(nil) -> nil [1] appendAll3(@l) -> appendAll3#1(@l) [1] appendAll3#1(::(@l1, @ls)) -> append(appendAll2(@l1), appendAll3(@ls)) [1] appendAll3#1(nil) -> nil [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: append(@l1, @l2) -> append#1(@l1, @l2) [1] append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) [1] append#1(nil, @l2) -> @l2 [1] appendAll(@l) -> appendAll#1(@l) [1] appendAll#1(::(@l1, @ls)) -> append(@l1, appendAll(@ls)) [1] appendAll#1(nil) -> nil [1] appendAll2(@l) -> appendAll2#1(@l) [1] appendAll2#1(::(@l1, @ls)) -> append(appendAll(@l1), appendAll2(@ls)) [1] appendAll2#1(nil) -> nil [1] appendAll3(@l) -> appendAll3#1(@l) [1] appendAll3#1(::(@l1, @ls)) -> append(appendAll2(@l1), appendAll3(@ls)) [1] appendAll3#1(nil) -> nil [1] The TRS has the following type information: append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil :: :: :::nil -> :::nil -> :::nil nil :: :::nil appendAll :: :::nil -> :::nil appendAll#1 :: :::nil -> :::nil appendAll2 :: :::nil -> :::nil appendAll2#1 :: :::nil -> :::nil appendAll3 :: :::nil -> :::nil appendAll3#1 :: :::nil -> :::nil 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: none (c) The following functions are completely defined: appendAll2_1 appendAll3_1 appendAll_1 appendAll#1_1 appendAll2#1_1 append_2 appendAll3#1_1 append#1_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (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: append(@l1, @l2) -> append#1(@l1, @l2) [1] append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) [1] append#1(nil, @l2) -> @l2 [1] appendAll(@l) -> appendAll#1(@l) [1] appendAll#1(::(@l1, @ls)) -> append(@l1, appendAll(@ls)) [1] appendAll#1(nil) -> nil [1] appendAll2(@l) -> appendAll2#1(@l) [1] appendAll2#1(::(@l1, @ls)) -> append(appendAll(@l1), appendAll2(@ls)) [1] appendAll2#1(nil) -> nil [1] appendAll3(@l) -> appendAll3#1(@l) [1] appendAll3#1(::(@l1, @ls)) -> append(appendAll2(@l1), appendAll3(@ls)) [1] appendAll3#1(nil) -> nil [1] The TRS has the following type information: append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil :: :: :::nil -> :::nil -> :::nil nil :: :::nil appendAll :: :::nil -> :::nil appendAll#1 :: :::nil -> :::nil appendAll2 :: :::nil -> :::nil appendAll2#1 :: :::nil -> :::nil appendAll3 :: :::nil -> :::nil appendAll3#1 :: :::nil -> :::nil 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: append(@l1, @l2) -> append#1(@l1, @l2) [1] append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) [1] append#1(nil, @l2) -> @l2 [1] appendAll(@l) -> appendAll#1(@l) [1] appendAll#1(::(@l1, @ls)) -> append(@l1, appendAll#1(@ls)) [2] appendAll#1(nil) -> nil [1] appendAll2(@l) -> appendAll2#1(@l) [1] appendAll2#1(::(@l1, @ls)) -> append(appendAll#1(@l1), appendAll2#1(@ls)) [3] appendAll2#1(nil) -> nil [1] appendAll3(@l) -> appendAll3#1(@l) [1] appendAll3#1(::(@l1, @ls)) -> append(appendAll2#1(@l1), appendAll3#1(@ls)) [3] appendAll3#1(nil) -> nil [1] The TRS has the following type information: append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil :: :: :::nil -> :::nil -> :::nil nil :: :::nil appendAll :: :::nil -> :::nil appendAll#1 :: :::nil -> :::nil appendAll2 :: :::nil -> :::nil appendAll2#1 :: :::nil -> :::nil appendAll3 :: :::nil -> :::nil appendAll3#1 :: :::nil -> :::nil 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 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> append#1(@l1, @l2) :|: @l1 >= 0, z' = @l2, @l2 >= 0, z = @l1 append#1(z, z') -{ 1 }-> @l2 :|: z' = @l2, @l2 >= 0, z = 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, @l2) :|: z' = @l2, @x >= 0, z = 1 + @x + @xs, @l2 >= 0, @xs >= 0 appendAll(z) -{ 1 }-> appendAll#1(@l) :|: z = @l, @l >= 0 appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 1 }-> appendAll2#1(@l) :|: z = @l, @l >= 0 appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(@l) :|: z = @l, @l >= 0 appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 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: append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 1 }-> appendAll#1(z) :|: z >= 0 appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0 appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 ---------------------------------------- (13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { append#1, append } { appendAll#1 } { appendAll } { appendAll2#1 } { appendAll3#1 } { appendAll2 } { appendAll3 } ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 1 }-> appendAll#1(z) :|: z >= 0 appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0 appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {append#1,append}, {appendAll#1}, {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3} ---------------------------------------- (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: append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 1 }-> appendAll#1(z) :|: z >= 0 appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0 appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {append#1,append}, {appendAll#1}, {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3} ---------------------------------------- (17) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: append#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' Computed SIZE bound using CoFloCo for: append 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: append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 1 }-> appendAll#1(z) :|: z >= 0 appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0 appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {append#1,append}, {appendAll#1}, {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3} Previous analysis results are: append#1: runtime: ?, size: O(n^1) [z + z'] append: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: append#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 2*z Computed RUNTIME bound using CoFloCo for: append after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + 2*z ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 1 }-> appendAll#1(z) :|: z >= 0 appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0 appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll#1}, {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*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: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 1 }-> appendAll#1(z) :|: z >= 0 appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0 appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll#1}, {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: appendAll#1 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: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 1 }-> appendAll#1(z) :|: z >= 0 appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0 appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll#1}, {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: ?, size: O(n^1) [z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: appendAll#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 6*z ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 1 }-> appendAll#1(z) :|: z >= 0 appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0 appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: O(n^1) [1 + 6*z], 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: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0 appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0 appendAll2#1(z) -{ 4 + 6*@l1 }-> append(s3, appendAll2#1(@ls)) :|: s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: appendAll 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: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0 appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0 appendAll2#1(z) -{ 4 + 6*@l1 }-> append(s3, appendAll2#1(@ls)) :|: s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z] appendAll: runtime: ?, size: O(n^1) [z] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: appendAll after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + 6*z ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0 appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0 appendAll2#1(z) -{ 4 + 6*@l1 }-> append(s3, appendAll2#1(@ls)) :|: s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z] appendAll: runtime: O(n^1) [2 + 6*z], 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: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0 appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0 appendAll2#1(z) -{ 4 + 6*@l1 }-> append(s3, appendAll2#1(@ls)) :|: s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z] appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: appendAll2#1 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: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0 appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0 appendAll2#1(z) -{ 4 + 6*@l1 }-> append(s3, appendAll2#1(@ls)) :|: s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z] appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z] appendAll2#1: runtime: ?, size: O(n^1) [z] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: appendAll2#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 8*z ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0 appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0 appendAll2#1(z) -{ 4 + 6*@l1 }-> append(s3, appendAll2#1(@ls)) :|: s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll3#1}, {appendAll2}, {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z] appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z] appendAll2#1: runtime: O(n^1) [1 + 8*z], 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: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0 appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0 appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 4 + 8*@l1 }-> append(s7, appendAll3#1(@ls)) :|: s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll3#1}, {appendAll2}, {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z] appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z] appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: appendAll3#1 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: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0 appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0 appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 4 + 8*@l1 }-> append(s7, appendAll3#1(@ls)) :|: s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll3#1}, {appendAll2}, {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z] appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z] appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z] appendAll3#1: runtime: ?, size: O(n^1) [z] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: appendAll3#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 10*z ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0 appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0 appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0 appendAll3#1(z) -{ 4 + 8*@l1 }-> append(s7, appendAll3#1(@ls)) :|: s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll2}, {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z] appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z] appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z] appendAll3#1: runtime: O(n^1) [1 + 10*z], size: O(n^1) [z] ---------------------------------------- (45) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0 appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0 appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 2 + 10*z }-> s8 :|: s8 >= 0, s8 <= z, z >= 0 appendAll3#1(z) -{ 9 + 8*@l1 + 10*@ls + 2*s7 }-> s10 :|: s9 >= 0, s9 <= @ls, s10 >= 0, s10 <= s7 + s9, s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll2}, {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z] appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z] appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z] appendAll3#1: runtime: O(n^1) [1 + 10*z], size: O(n^1) [z] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: appendAll2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0 appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0 appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 2 + 10*z }-> s8 :|: s8 >= 0, s8 <= z, z >= 0 appendAll3#1(z) -{ 9 + 8*@l1 + 10*@ls + 2*s7 }-> s10 :|: s9 >= 0, s9 <= @ls, s10 >= 0, s10 <= s7 + s9, s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll2}, {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z] appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z] appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z] appendAll3#1: runtime: O(n^1) [1 + 10*z], size: O(n^1) [z] appendAll2: runtime: ?, size: O(n^1) [z] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: appendAll2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + 8*z ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0 appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0 appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 2 + 10*z }-> s8 :|: s8 >= 0, s8 <= z, z >= 0 appendAll3#1(z) -{ 9 + 8*@l1 + 10*@ls + 2*s7 }-> s10 :|: s9 >= 0, s9 <= @ls, s10 >= 0, s10 <= s7 + s9, s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z] appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z] appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z] appendAll3#1: runtime: O(n^1) [1 + 10*z], size: O(n^1) [z] appendAll2: runtime: O(n^1) [2 + 8*z], size: O(n^1) [z] ---------------------------------------- (51) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0 appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0 appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 2 + 10*z }-> s8 :|: s8 >= 0, s8 <= z, z >= 0 appendAll3#1(z) -{ 9 + 8*@l1 + 10*@ls + 2*s7 }-> s10 :|: s9 >= 0, s9 <= @ls, s10 >= 0, s10 <= s7 + s9, s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z] appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z] appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z] appendAll3#1: runtime: O(n^1) [1 + 10*z], size: O(n^1) [z] appendAll2: runtime: O(n^1) [2 + 8*z], size: O(n^1) [z] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: appendAll3 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0 appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0 appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 2 + 10*z }-> s8 :|: s8 >= 0, s8 <= z, z >= 0 appendAll3#1(z) -{ 9 + 8*@l1 + 10*@ls + 2*s7 }-> s10 :|: s9 >= 0, s9 <= @ls, s10 >= 0, s10 <= s7 + s9, s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {appendAll3} Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z] appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z] appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z] appendAll3#1: runtime: O(n^1) [1 + 10*z], size: O(n^1) [z] appendAll2: runtime: O(n^1) [2 + 8*z], size: O(n^1) [z] appendAll3: runtime: ?, size: O(n^1) [z] ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: appendAll3 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + 10*z ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0 appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll#1(z) -{ 1 }-> 0 :|: z = 0 appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0 appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll2#1(z) -{ 1 }-> 0 :|: z = 0 appendAll3(z) -{ 2 + 10*z }-> s8 :|: s8 >= 0, s8 <= z, z >= 0 appendAll3#1(z) -{ 9 + 8*@l1 + 10*@ls + 2*s7 }-> s10 :|: s9 >= 0, s9 <= @ls, s10 >= 0, s10 <= s7 + s9, s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls appendAll3#1(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: Previous analysis results are: append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z] appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z] appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z] appendAll3#1: runtime: O(n^1) [1 + 10*z], size: O(n^1) [z] appendAll2: runtime: O(n^1) [2 + 8*z], size: O(n^1) [z] appendAll3: runtime: O(n^1) [2 + 10*z], size: O(n^1) [z] ---------------------------------------- (57) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (58) BOUNDS(1, n^1)