MAYBE proof of input_ePmXzAqTqo.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 136 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 1307 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 617 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (38) CdtProblem (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 62 ms] (42) CdtProblem (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) nesteql(Cons(x, xs)) -> nesteql(eql(Cons(x, xs))) eql(Nil) -> Nil eql(Cons(x, xs)) -> eql(Cons(x, xs)) number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) goal(x) -> nesteql(x) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) nesteql(Cons(x, xs)) -> nesteql(eql(Cons(x, xs))) eql(Nil) -> Nil eql(Cons(x, xs)) -> eql(Cons(x, xs)) number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) goal(x) -> nesteql(x) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) nesteql(Cons(x, xs)) -> nesteql(eql(Cons(x, xs))) eql(Nil) -> Nil eql(Cons(x, xs)) -> eql(Cons(x, xs)) number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) goal(x) -> nesteql(x) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) [1] nesteql(Cons(x, xs)) -> nesteql(eql(Cons(x, xs))) [1] eql(Nil) -> Nil [1] eql(Cons(x, xs)) -> eql(Cons(x, xs)) [1] number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) [1] goal(x) -> nesteql(x) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) [1] nesteql(Cons(x, xs)) -> nesteql(eql(Cons(x, xs))) [1] eql(Nil) -> Nil [1] eql(Cons(x, xs)) -> eql(Cons(x, xs)) [1] number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) [1] goal(x) -> nesteql(x) [1] The TRS has the following type information: nesteql :: Nil:Cons -> Nil:Cons Nil :: Nil:Cons Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons eql :: Nil:Cons -> Nil:Cons number17 :: a -> Nil:Cons goal :: Nil:Cons -> Nil:Cons Rewrite Strategy: INNERMOST ---------------------------------------- (9) 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: nesteql_1 number17_1 goal_1 (c) The following functions are completely defined: eql_1 Due to the following rules being added: none And the following fresh constants: const ---------------------------------------- (10) 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: nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) [1] nesteql(Cons(x, xs)) -> nesteql(eql(Cons(x, xs))) [1] eql(Nil) -> Nil [1] eql(Cons(x, xs)) -> eql(Cons(x, xs)) [1] number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) [1] goal(x) -> nesteql(x) [1] The TRS has the following type information: nesteql :: Nil:Cons -> Nil:Cons Nil :: Nil:Cons Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons eql :: Nil:Cons -> Nil:Cons number17 :: a -> Nil:Cons goal :: Nil:Cons -> Nil:Cons const :: a Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) 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: nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) [1] nesteql(Cons(x, xs)) -> nesteql(eql(Cons(x, xs))) [2] eql(Nil) -> Nil [1] eql(Cons(x, xs)) -> eql(Cons(x, xs)) [1] number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) [1] goal(x) -> nesteql(x) [1] The TRS has the following type information: nesteql :: Nil:Cons -> Nil:Cons Nil :: Nil:Cons Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons eql :: Nil:Cons -> Nil:Cons number17 :: a -> Nil:Cons goal :: Nil:Cons -> Nil:Cons const :: a Rewrite Strategy: INNERMOST ---------------------------------------- (13) 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 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 eql(z) -{ 1 }-> 0 :|: z = 0 goal(z) -{ 1 }-> nesteql(x) :|: x >= 0, z = x nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nesteql(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0 number17(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: n >= 0, z = n ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 eql(z) -{ 1 }-> 0 :|: z = 0 goal(z) -{ 1 }-> nesteql(z) :|: z >= 0 nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nesteql(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0 number17(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { number17 } { eql } { nesteql } { goal } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 eql(z) -{ 1 }-> 0 :|: z = 0 goal(z) -{ 1 }-> nesteql(z) :|: z >= 0 nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nesteql(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0 number17(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {number17}, {eql}, {nesteql}, {goal} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 eql(z) -{ 1 }-> 0 :|: z = 0 goal(z) -{ 1 }-> nesteql(z) :|: z >= 0 nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nesteql(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0 number17(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {number17}, {eql}, {nesteql}, {goal} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: number17 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 17 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 eql(z) -{ 1 }-> 0 :|: z = 0 goal(z) -{ 1 }-> nesteql(z) :|: z >= 0 nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nesteql(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0 number17(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {number17}, {eql}, {nesteql}, {goal} Previous analysis results are: number17: runtime: ?, size: O(1) [17] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: number17 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: eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 eql(z) -{ 1 }-> 0 :|: z = 0 goal(z) -{ 1 }-> nesteql(z) :|: z >= 0 nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nesteql(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0 number17(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {eql}, {nesteql}, {goal} Previous analysis results are: number17: runtime: O(1) [1], size: O(1) [17] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 eql(z) -{ 1 }-> 0 :|: z = 0 goal(z) -{ 1 }-> nesteql(z) :|: z >= 0 nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nesteql(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0 number17(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {eql}, {nesteql}, {goal} Previous analysis results are: number17: runtime: O(1) [1], size: O(1) [17] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: eql after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 eql(z) -{ 1 }-> 0 :|: z = 0 goal(z) -{ 1 }-> nesteql(z) :|: z >= 0 nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nesteql(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0 number17(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {eql}, {nesteql}, {goal} Previous analysis results are: number17: runtime: O(1) [1], size: O(1) [17] eql: runtime: ?, size: O(1) [0] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: eql after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 eql(z) -{ 1 }-> 0 :|: z = 0 goal(z) -{ 1 }-> nesteql(z) :|: z >= 0 nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nesteql(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0 number17(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {eql}, {nesteql}, {goal} Previous analysis results are: number17: runtime: O(1) [1], size: O(1) [17] eql: runtime: INF, size: O(1) [0] ---------------------------------------- (31) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: const ---------------------------------------- (32) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) [1] nesteql(Cons(x, xs)) -> nesteql(eql(Cons(x, xs))) [1] eql(Nil) -> Nil [1] eql(Cons(x, xs)) -> eql(Cons(x, xs)) [1] number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) [1] goal(x) -> nesteql(x) [1] The TRS has the following type information: nesteql :: Nil:Cons -> Nil:Cons Nil :: Nil:Cons Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons eql :: Nil:Cons -> Nil:Cons number17 :: a -> Nil:Cons goal :: Nil:Cons -> Nil:Cons const :: a Rewrite Strategy: INNERMOST ---------------------------------------- (33) 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 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 eql(z) -{ 1 }-> 0 :|: z = 0 goal(z) -{ 1 }-> nesteql(x) :|: x >= 0, z = x nesteql(z) -{ 1 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nesteql(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: z = 0 number17(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0)))))))))))))))) :|: n >= 0, z = n Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) nesteql(Cons(z0, z1)) -> nesteql(eql(Cons(z0, z1))) eql(Nil) -> Nil eql(Cons(z0, z1)) -> eql(Cons(z0, z1)) number17(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) goal(z0) -> nesteql(z0) Tuples: NESTEQL(Nil) -> c NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) EQL(Nil) -> c2 EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) NUMBER17(z0) -> c4 GOAL(z0) -> c5(NESTEQL(z0)) S tuples: NESTEQL(Nil) -> c NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) EQL(Nil) -> c2 EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) NUMBER17(z0) -> c4 GOAL(z0) -> c5(NESTEQL(z0)) K tuples:none Defined Rule Symbols: nesteql_1, eql_1, number17_1, goal_1 Defined Pair Symbols: NESTEQL_1, EQL_1, NUMBER17_1, GOAL_1 Compound Symbols: c, c1_2, c2, c3_1, c4, c5_1 ---------------------------------------- (37) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: GOAL(z0) -> c5(NESTEQL(z0)) Removed 3 trailing nodes: NESTEQL(Nil) -> c NUMBER17(z0) -> c4 EQL(Nil) -> c2 ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) nesteql(Cons(z0, z1)) -> nesteql(eql(Cons(z0, z1))) eql(Nil) -> Nil eql(Cons(z0, z1)) -> eql(Cons(z0, z1)) number17(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) goal(z0) -> nesteql(z0) Tuples: NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) S tuples: NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) K tuples:none Defined Rule Symbols: nesteql_1, eql_1, number17_1, goal_1 Defined Pair Symbols: NESTEQL_1, EQL_1 Compound Symbols: c1_2, c3_1 ---------------------------------------- (39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) nesteql(Cons(z0, z1)) -> nesteql(eql(Cons(z0, z1))) eql(Nil) -> Nil number17(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))) goal(z0) -> nesteql(z0) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: eql(Cons(z0, z1)) -> eql(Cons(z0, z1)) Tuples: NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) S tuples: NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) K tuples:none Defined Rule Symbols: eql_1 Defined Pair Symbols: NESTEQL_1, EQL_1 Compound Symbols: c1_2, c3_1 ---------------------------------------- (41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) We considered the (Usable) Rules: eql(Cons(z0, z1)) -> eql(Cons(z0, z1)) And the Tuples: NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(Cons(x_1, x_2)) = [1] + x_1 + x_2 POL(EQL(x_1)) = 0 POL(NESTEQL(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(eql(x_1)) = 0 ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: eql(Cons(z0, z1)) -> eql(Cons(z0, z1)) Tuples: NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) S tuples: EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) K tuples: NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) Defined Rule Symbols: eql_1 Defined Pair Symbols: NESTEQL_1, EQL_1 Compound Symbols: c1_2, c3_1 ---------------------------------------- (43) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) by NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: eql(Cons(z0, z1)) -> eql(Cons(z0, z1)) Tuples: EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) S tuples: EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) K tuples: NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) Defined Rule Symbols: eql_1 Defined Pair Symbols: EQL_1, NESTEQL_1 Compound Symbols: c3_1, c1_2 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) by NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: eql(Cons(z0, z1)) -> eql(Cons(z0, z1)) Tuples: EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) S tuples: EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) K tuples: NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) Defined Rule Symbols: eql_1 Defined Pair Symbols: EQL_1, NESTEQL_1 Compound Symbols: c3_1, c1_2 ---------------------------------------- (47) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) by NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: eql(Cons(z0, z1)) -> eql(Cons(z0, z1)) Tuples: EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) S tuples: EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) K tuples: NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) Defined Rule Symbols: eql_1 Defined Pair Symbols: EQL_1, NESTEQL_1 Compound Symbols: c3_1, c1_2 ---------------------------------------- (49) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) by NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: eql(Cons(z0, z1)) -> eql(Cons(z0, z1)) Tuples: EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) S tuples: EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) K tuples: NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) Defined Rule Symbols: eql_1 Defined Pair Symbols: EQL_1, NESTEQL_1 Compound Symbols: c3_1, c1_2