WORST_CASE(?,O(n^1)) proof of input_HwK14XjvGz.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). (0) CpxTRS (1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxWeightedTrs (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxTypedWeightedTrs (5) CompletionProof [UPPER BOUND(ID), 0 ms] (6) CpxTypedWeightedCompleteTrs (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (10) CpxRNTS (11) InliningProof [UPPER BOUND(ID), 77 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), 149 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 33 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 415 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 126 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 631 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 53 ms] (34) CpxRNTS (35) FinalProof [FINISHED, 0 ms] (36) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: a__f(b, X, c) -> a__f(X, a__c, X) a__c -> b mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) mark(c) -> a__c mark(b) -> b a__f(X1, X2, X3) -> f(X1, X2, X3) a__c -> c 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: a__f(b, X, c) -> a__f(X, a__c, X) [1] a__c -> b [1] mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) [1] mark(c) -> a__c [1] mark(b) -> b [1] a__f(X1, X2, X3) -> f(X1, X2, X3) [1] a__c -> c [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: a__f(b, X, c) -> a__f(X, a__c, X) [1] a__c -> b [1] mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) [1] mark(c) -> a__c [1] mark(b) -> b [1] a__f(X1, X2, X3) -> f(X1, X2, X3) [1] a__c -> c [1] The TRS has the following type information: a__f :: b:c:f -> b:c:f -> b:c:f -> b:c:f b :: b:c:f c :: b:c:f a__c :: b:c:f mark :: b:c:f -> b:c:f f :: b:c:f -> b:c:f -> b:c:f -> b:c:f 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: mark_1 a__c a__f_3 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: a__f(b, X, c) -> a__f(X, a__c, X) [1] a__c -> b [1] mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) [1] mark(c) -> a__c [1] mark(b) -> b [1] a__f(X1, X2, X3) -> f(X1, X2, X3) [1] a__c -> c [1] The TRS has the following type information: a__f :: b:c:f -> b:c:f -> b:c:f -> b:c:f b :: b:c:f c :: b:c:f a__c :: b:c:f mark :: b:c:f -> b:c:f f :: b:c:f -> b:c:f -> b:c:f -> b:c:f 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: a__f(b, X, c) -> a__f(X, b, X) [2] a__f(b, X, c) -> a__f(X, c, X) [2] a__c -> b [1] mark(f(X1, f(X1', X2', X3'), X3)) -> a__f(X1, a__f(X1', mark(X2'), X3'), X3) [2] mark(f(X1, c, X3)) -> a__f(X1, a__c, X3) [2] mark(f(X1, b, X3)) -> a__f(X1, b, X3) [2] mark(c) -> a__c [1] mark(b) -> b [1] a__f(X1, X2, X3) -> f(X1, X2, X3) [1] a__c -> c [1] The TRS has the following type information: a__f :: b:c:f -> b:c:f -> b:c:f -> b:c:f b :: b:c:f c :: b:c:f a__c :: b:c:f mark :: b:c:f -> b:c:f f :: b:c:f -> b:c:f -> b:c:f -> b:c:f 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: b => 0 c => 1 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: a__c -{ 1 }-> 1 :|: a__c -{ 1 }-> 0 :|: a__f(z, z', z'') -{ 2 }-> a__f(X, 1, X) :|: z' = X, X >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 2 }-> a__f(X, 0, X) :|: z' = X, X >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3 mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3 mark(z) -{ 2 }-> a__f(X1, a__c, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0 mark(z) -{ 1 }-> a__c :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 ---------------------------------------- (11) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: a__c -{ 1 }-> 0 :|: a__c -{ 1 }-> 1 :|: ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: a__c -{ 1 }-> 1 :|: a__c -{ 1 }-> 0 :|: a__f(z, z', z'') -{ 2 }-> a__f(X, 1, X) :|: z' = X, X >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 2 }-> a__f(X, 0, X) :|: z' = X, X >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3 mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3 mark(z) -{ 3 }-> a__f(X1, 1, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0 mark(z) -{ 3 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 2 }-> 0 :|: z = 1 ---------------------------------------- (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: a__c -{ 1 }-> 1 :|: a__c -{ 1 }-> 0 :|: a__f(z, z', z'') -{ 2 }-> a__f(z', 1, z') :|: z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 2 }-> a__f(z', 0, z') :|: z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0 mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3 mark(z) -{ 3 }-> a__f(X1, 1, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0 mark(z) -{ 3 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 2 }-> 0 :|: z = 1 ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { a__c } { a__f } { mark } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: a__c -{ 1 }-> 1 :|: a__c -{ 1 }-> 0 :|: a__f(z, z', z'') -{ 2 }-> a__f(z', 1, z') :|: z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 2 }-> a__f(z', 0, z') :|: z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0 mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3 mark(z) -{ 3 }-> a__f(X1, 1, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0 mark(z) -{ 3 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 2 }-> 0 :|: z = 1 Function symbols to be analyzed: {a__c}, {a__f}, {mark} ---------------------------------------- (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: a__c -{ 1 }-> 1 :|: a__c -{ 1 }-> 0 :|: a__f(z, z', z'') -{ 2 }-> a__f(z', 1, z') :|: z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 2 }-> a__f(z', 0, z') :|: z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0 mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3 mark(z) -{ 3 }-> a__f(X1, 1, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0 mark(z) -{ 3 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 2 }-> 0 :|: z = 1 Function symbols to be analyzed: {a__c}, {a__f}, {mark} ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: a__c after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: a__c -{ 1 }-> 1 :|: a__c -{ 1 }-> 0 :|: a__f(z, z', z'') -{ 2 }-> a__f(z', 1, z') :|: z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 2 }-> a__f(z', 0, z') :|: z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0 mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3 mark(z) -{ 3 }-> a__f(X1, 1, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0 mark(z) -{ 3 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 2 }-> 0 :|: z = 1 Function symbols to be analyzed: {a__c}, {a__f}, {mark} Previous analysis results are: a__c: runtime: ?, size: O(1) [1] ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: a__c 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: a__c -{ 1 }-> 1 :|: a__c -{ 1 }-> 0 :|: a__f(z, z', z'') -{ 2 }-> a__f(z', 1, z') :|: z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 2 }-> a__f(z', 0, z') :|: z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0 mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3 mark(z) -{ 3 }-> a__f(X1, 1, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0 mark(z) -{ 3 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 2 }-> 0 :|: z = 1 Function symbols to be analyzed: {a__f}, {mark} Previous analysis results are: a__c: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (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: a__c -{ 1 }-> 1 :|: a__c -{ 1 }-> 0 :|: a__f(z, z', z'') -{ 2 }-> a__f(z', 1, z') :|: z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 2 }-> a__f(z', 0, z') :|: z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0 mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3 mark(z) -{ 3 }-> a__f(X1, 1, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0 mark(z) -{ 3 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 2 }-> 0 :|: z = 1 Function symbols to be analyzed: {a__f}, {mark} Previous analysis results are: a__c: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: a__f after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z + 2*z' + z'' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: a__c -{ 1 }-> 1 :|: a__c -{ 1 }-> 0 :|: a__f(z, z', z'') -{ 2 }-> a__f(z', 1, z') :|: z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 2 }-> a__f(z', 0, z') :|: z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0 mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3 mark(z) -{ 3 }-> a__f(X1, 1, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0 mark(z) -{ 3 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 2 }-> 0 :|: z = 1 Function symbols to be analyzed: {a__f}, {mark} Previous analysis results are: a__c: runtime: O(1) [1], size: O(1) [1] a__f: runtime: ?, size: O(n^1) [2 + z + 2*z' + z''] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: a__f after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: a__c -{ 1 }-> 1 :|: a__c -{ 1 }-> 0 :|: a__f(z, z', z'') -{ 2 }-> a__f(z', 1, z') :|: z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 2 }-> a__f(z', 0, z') :|: z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0 mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3 mark(z) -{ 3 }-> a__f(X1, 1, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0 mark(z) -{ 3 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 2 }-> 0 :|: z = 1 Function symbols to be analyzed: {mark} Previous analysis results are: a__c: runtime: O(1) [1], size: O(1) [1] a__f: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z' + 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: a__c -{ 1 }-> 1 :|: a__c -{ 1 }-> 0 :|: a__f(z, z', z'') -{ 5 }-> s :|: s >= 0, s <= z' + 2 * 0 + z' + 2, z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 5 }-> s' :|: s' >= 0, s' <= z' + 2 * 1 + z' + 2, z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0 mark(z) -{ 5 }-> s'' :|: s'' >= 0, s'' <= X1 + 2 * 0 + X3 + 2, X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0 mark(z) -{ 6 }-> s1 :|: s1 >= 0, s1 <= X1 + 2 * 0 + X3 + 2, X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 6 }-> s2 :|: s2 >= 0, s2 <= X1 + 2 * 1 + X3 + 2, X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3 mark(z) -{ 2 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 2 }-> 0 :|: z = 1 Function symbols to be analyzed: {mark} Previous analysis results are: a__c: runtime: O(1) [1], size: O(1) [1] a__f: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z' + z''] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: mark after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: a__c -{ 1 }-> 1 :|: a__c -{ 1 }-> 0 :|: a__f(z, z', z'') -{ 5 }-> s :|: s >= 0, s <= z' + 2 * 0 + z' + 2, z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 5 }-> s' :|: s' >= 0, s' <= z' + 2 * 1 + z' + 2, z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0 mark(z) -{ 5 }-> s'' :|: s'' >= 0, s'' <= X1 + 2 * 0 + X3 + 2, X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0 mark(z) -{ 6 }-> s1 :|: s1 >= 0, s1 <= X1 + 2 * 0 + X3 + 2, X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 6 }-> s2 :|: s2 >= 0, s2 <= X1 + 2 * 1 + X3 + 2, X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3 mark(z) -{ 2 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 2 }-> 0 :|: z = 1 Function symbols to be analyzed: {mark} Previous analysis results are: a__c: runtime: O(1) [1], size: O(1) [1] a__f: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z' + z''] mark: runtime: ?, size: EXP ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: mark after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 20 + 8*z ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: a__c -{ 1 }-> 1 :|: a__c -{ 1 }-> 0 :|: a__f(z, z', z'') -{ 5 }-> s :|: s >= 0, s <= z' + 2 * 0 + z' + 2, z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 5 }-> s' :|: s' >= 0, s' <= z' + 2 * 1 + z' + 2, z' >= 0, z = 0, z'' = 1 a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0 mark(z) -{ 5 }-> s'' :|: s'' >= 0, s'' <= X1 + 2 * 0 + X3 + 2, X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0 mark(z) -{ 6 }-> s1 :|: s1 >= 0, s1 <= X1 + 2 * 0 + X3 + 2, X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 6 }-> s2 :|: s2 >= 0, s2 <= X1 + 2 * 1 + X3 + 2, X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0 mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3 mark(z) -{ 2 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 2 }-> 0 :|: z = 1 Function symbols to be analyzed: Previous analysis results are: a__c: runtime: O(1) [1], size: O(1) [1] a__f: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z' + z''] mark: runtime: O(n^1) [20 + 8*z], size: EXP ---------------------------------------- (35) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (36) BOUNDS(1, n^1)