KILLED proof of input_6x6MVrGN0x.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) InliningProof [UPPER BOUND(ID), 23 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 109 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 46 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 206 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 64 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 1327 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 917 ms] (38) CpxRNTS (39) CompletionProof [UPPER BOUND(ID), 0 ms] (40) CpxTypedWeightedCompleteTrs (41) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (44) CdtProblem (45) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (46) CdtProblem (47) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (56) 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: immatcopy(Cons(x, xs)) -> Cons(Nil, immatcopy(xs)) nestimeql(Nil) -> number42(Nil) nestimeql(Cons(x, xs)) -> nestimeql(immatcopy(Cons(x, xs))) immatcopy(Nil) -> Nil number42(x) -> 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, 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, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) goal(x) -> nestimeql(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: immatcopy(Cons(x, xs)) -> Cons(Nil, immatcopy(xs)) nestimeql(Nil) -> number42(Nil) nestimeql(Cons(x, xs)) -> nestimeql(immatcopy(Cons(x, xs))) immatcopy(Nil) -> Nil number42(x) -> 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, 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, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) goal(x) -> nestimeql(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: immatcopy(Cons(x, xs)) -> Cons(Nil, immatcopy(xs)) nestimeql(Nil) -> number42(Nil) nestimeql(Cons(x, xs)) -> nestimeql(immatcopy(Cons(x, xs))) immatcopy(Nil) -> Nil number42(x) -> 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, 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, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) goal(x) -> nestimeql(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: immatcopy(Cons(x, xs)) -> Cons(Nil, immatcopy(xs)) [1] nestimeql(Nil) -> number42(Nil) [1] nestimeql(Cons(x, xs)) -> nestimeql(immatcopy(Cons(x, xs))) [1] immatcopy(Nil) -> Nil [1] number42(x) -> 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, 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, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] goal(x) -> nestimeql(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: immatcopy(Cons(x, xs)) -> Cons(Nil, immatcopy(xs)) [1] nestimeql(Nil) -> number42(Nil) [1] nestimeql(Cons(x, xs)) -> nestimeql(immatcopy(Cons(x, xs))) [1] immatcopy(Nil) -> Nil [1] number42(x) -> 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, 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, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] goal(x) -> nestimeql(x) [1] The TRS has the following type information: immatcopy :: Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil nestimeql :: Cons:Nil -> Cons:Nil number42 :: Cons:Nil -> Cons:Nil goal :: Cons:Nil -> Cons:Nil 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: nestimeql_1 number42_1 goal_1 (c) The following functions are completely defined: immatcopy_1 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (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: immatcopy(Cons(x, xs)) -> Cons(Nil, immatcopy(xs)) [1] nestimeql(Nil) -> number42(Nil) [1] nestimeql(Cons(x, xs)) -> nestimeql(immatcopy(Cons(x, xs))) [1] immatcopy(Nil) -> Nil [1] number42(x) -> 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, 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, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] goal(x) -> nestimeql(x) [1] The TRS has the following type information: immatcopy :: Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil nestimeql :: Cons:Nil -> Cons:Nil number42 :: Cons:Nil -> Cons:Nil goal :: Cons:Nil -> Cons:Nil 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: immatcopy(Cons(x, xs)) -> Cons(Nil, immatcopy(xs)) [1] nestimeql(Nil) -> number42(Nil) [1] nestimeql(Cons(x, xs)) -> nestimeql(Cons(Nil, immatcopy(xs))) [2] immatcopy(Nil) -> Nil [1] number42(x) -> 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, 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, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] goal(x) -> nestimeql(x) [1] The TRS has the following type information: immatcopy :: Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil nestimeql :: Cons:Nil -> Cons:Nil number42 :: Cons:Nil -> Cons:Nil goal :: Cons:Nil -> Cons:Nil 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 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> nestimeql(x) :|: x >= 0, z = x immatcopy(z) -{ 1 }-> 0 :|: z = 0 immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 1 }-> number42(0) :|: z = 0 nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 number42(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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: x >= 0, z = x ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: number42(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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: x >= 0, z = x ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> nestimeql(x) :|: x >= 0, z = x immatcopy(z) -{ 1 }-> 0 :|: z = 0 immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> 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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x number42(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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: x >= 0, z = x ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0 immatcopy(z) -{ 1 }-> 0 :|: z = 0 immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> 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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x number42(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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { number42 } { immatcopy } { nestimeql } { goal } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0 immatcopy(z) -{ 1 }-> 0 :|: z = 0 immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> 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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x number42(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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {number42}, {immatcopy}, {nestimeql}, {goal} ---------------------------------------- (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: goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0 immatcopy(z) -{ 1 }-> 0 :|: z = 0 immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> 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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x number42(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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {number42}, {immatcopy}, {nestimeql}, {goal} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: number42 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 42 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0 immatcopy(z) -{ 1 }-> 0 :|: z = 0 immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> 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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x number42(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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {number42}, {immatcopy}, {nestimeql}, {goal} Previous analysis results are: number42: runtime: ?, size: O(1) [42] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: number42 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0 immatcopy(z) -{ 1 }-> 0 :|: z = 0 immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> 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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x number42(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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {immatcopy}, {nestimeql}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] ---------------------------------------- (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: goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0 immatcopy(z) -{ 1 }-> 0 :|: z = 0 immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> 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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x number42(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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {immatcopy}, {nestimeql}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: immatcopy 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: goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0 immatcopy(z) -{ 1 }-> 0 :|: z = 0 immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> 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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x number42(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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {immatcopy}, {nestimeql}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] immatcopy: runtime: ?, size: O(n^1) [z] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: immatcopy after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0 immatcopy(z) -{ 1 }-> 0 :|: z = 0 immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> 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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x number42(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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {nestimeql}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] immatcopy: runtime: O(n^1) [1 + 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: goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0 immatcopy(z) -{ 1 }-> 0 :|: z = 0 immatcopy(z) -{ 2 + xs }-> 1 + 0 + s :|: s >= 0, s <= xs, z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 3 + xs }-> nestimeql(1 + 0 + s') :|: s' >= 0, s' <= xs, z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> 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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x number42(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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {nestimeql}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] immatcopy: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: nestimeql after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 42 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0 immatcopy(z) -{ 1 }-> 0 :|: z = 0 immatcopy(z) -{ 2 + xs }-> 1 + 0 + s :|: s >= 0, s <= xs, z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 3 + xs }-> nestimeql(1 + 0 + s') :|: s' >= 0, s' <= xs, z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> 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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x number42(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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {nestimeql}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] immatcopy: runtime: O(n^1) [1 + z], size: O(n^1) [z] nestimeql: runtime: ?, size: O(1) [42] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: nestimeql after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0 immatcopy(z) -{ 1 }-> 0 :|: z = 0 immatcopy(z) -{ 2 + xs }-> 1 + 0 + s :|: s >= 0, s <= xs, z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 3 + xs }-> nestimeql(1 + 0 + s') :|: s' >= 0, s' <= xs, z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 2 }-> 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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x number42(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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0 Function symbols to be analyzed: {nestimeql}, {goal} Previous analysis results are: number42: runtime: O(1) [1], size: O(1) [42] immatcopy: runtime: O(n^1) [1 + z], size: O(n^1) [z] nestimeql: runtime: INF, size: O(1) [42] ---------------------------------------- (39) 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: none ---------------------------------------- (40) 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: immatcopy(Cons(x, xs)) -> Cons(Nil, immatcopy(xs)) [1] nestimeql(Nil) -> number42(Nil) [1] nestimeql(Cons(x, xs)) -> nestimeql(immatcopy(Cons(x, xs))) [1] immatcopy(Nil) -> Nil [1] number42(x) -> 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, 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, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1] goal(x) -> nestimeql(x) [1] The TRS has the following type information: immatcopy :: Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil nestimeql :: Cons:Nil -> Cons:Nil number42 :: Cons:Nil -> Cons:Nil goal :: Cons:Nil -> Cons:Nil Rewrite Strategy: INNERMOST ---------------------------------------- (41) 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 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> nestimeql(x) :|: x >= 0, z = x immatcopy(z) -{ 1 }-> 0 :|: z = 0 immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestimeql(z) -{ 1 }-> number42(0) :|: z = 0 nestimeql(z) -{ 1 }-> nestimeql(immatcopy(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 number42(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 + (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 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: x >= 0, z = x Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (43) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: immatcopy(Cons(z0, z1)) -> Cons(Nil, immatcopy(z1)) immatcopy(Nil) -> Nil nestimeql(Nil) -> number42(Nil) nestimeql(Cons(z0, z1)) -> nestimeql(immatcopy(Cons(z0, z1))) number42(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, 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, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) goal(z0) -> nestimeql(z0) Tuples: IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1)) IMMATCOPY(Nil) -> c1 NESTIMEQL(Nil) -> c2(NUMBER42(Nil)) NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(immatcopy(Cons(z0, z1))), IMMATCOPY(Cons(z0, z1))) NUMBER42(z0) -> c4 GOAL(z0) -> c5(NESTIMEQL(z0)) S tuples: IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1)) IMMATCOPY(Nil) -> c1 NESTIMEQL(Nil) -> c2(NUMBER42(Nil)) NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(immatcopy(Cons(z0, z1))), IMMATCOPY(Cons(z0, z1))) NUMBER42(z0) -> c4 GOAL(z0) -> c5(NESTIMEQL(z0)) K tuples:none Defined Rule Symbols: immatcopy_1, nestimeql_1, number42_1, goal_1 Defined Pair Symbols: IMMATCOPY_1, NESTIMEQL_1, NUMBER42_1, GOAL_1 Compound Symbols: c_1, c1, c2_1, c3_2, c4, c5_1 ---------------------------------------- (45) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: GOAL(z0) -> c5(NESTIMEQL(z0)) Removed 3 trailing nodes: NUMBER42(z0) -> c4 NESTIMEQL(Nil) -> c2(NUMBER42(Nil)) IMMATCOPY(Nil) -> c1 ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: immatcopy(Cons(z0, z1)) -> Cons(Nil, immatcopy(z1)) immatcopy(Nil) -> Nil nestimeql(Nil) -> number42(Nil) nestimeql(Cons(z0, z1)) -> nestimeql(immatcopy(Cons(z0, z1))) number42(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, 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, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) goal(z0) -> nestimeql(z0) Tuples: IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1)) NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(immatcopy(Cons(z0, z1))), IMMATCOPY(Cons(z0, z1))) S tuples: IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1)) NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(immatcopy(Cons(z0, z1))), IMMATCOPY(Cons(z0, z1))) K tuples:none Defined Rule Symbols: immatcopy_1, nestimeql_1, number42_1, goal_1 Defined Pair Symbols: IMMATCOPY_1, NESTIMEQL_1 Compound Symbols: c_1, c3_2 ---------------------------------------- (47) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: nestimeql(Nil) -> number42(Nil) nestimeql(Cons(z0, z1)) -> nestimeql(immatcopy(Cons(z0, z1))) number42(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, 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, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) goal(z0) -> nestimeql(z0) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: immatcopy(Cons(z0, z1)) -> Cons(Nil, immatcopy(z1)) immatcopy(Nil) -> Nil Tuples: IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1)) NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(immatcopy(Cons(z0, z1))), IMMATCOPY(Cons(z0, z1))) S tuples: IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1)) NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(immatcopy(Cons(z0, z1))), IMMATCOPY(Cons(z0, z1))) K tuples:none Defined Rule Symbols: immatcopy_1 Defined Pair Symbols: IMMATCOPY_1, NESTIMEQL_1 Compound Symbols: c_1, c3_2 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(immatcopy(Cons(z0, z1))), IMMATCOPY(Cons(z0, z1))) by NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(z1))), IMMATCOPY(Cons(z0, z1))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: immatcopy(Cons(z0, z1)) -> Cons(Nil, immatcopy(z1)) immatcopy(Nil) -> Nil Tuples: IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1)) NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(z1))), IMMATCOPY(Cons(z0, z1))) S tuples: IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1)) NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(z1))), IMMATCOPY(Cons(z0, z1))) K tuples:none Defined Rule Symbols: immatcopy_1 Defined Pair Symbols: IMMATCOPY_1, NESTIMEQL_1 Compound Symbols: c_1, c3_2 ---------------------------------------- (51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1)) by IMMATCOPY(Cons(z0, Cons(y0, y1))) -> c(IMMATCOPY(Cons(y0, y1))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: immatcopy(Cons(z0, z1)) -> Cons(Nil, immatcopy(z1)) immatcopy(Nil) -> Nil Tuples: NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(z1))), IMMATCOPY(Cons(z0, z1))) IMMATCOPY(Cons(z0, Cons(y0, y1))) -> c(IMMATCOPY(Cons(y0, y1))) S tuples: NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(z1))), IMMATCOPY(Cons(z0, z1))) IMMATCOPY(Cons(z0, Cons(y0, y1))) -> c(IMMATCOPY(Cons(y0, y1))) K tuples:none Defined Rule Symbols: immatcopy_1 Defined Pair Symbols: NESTIMEQL_1, IMMATCOPY_1 Compound Symbols: c3_2, c_1 ---------------------------------------- (53) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(z1))), IMMATCOPY(Cons(z0, z1))) by NESTIMEQL(Cons(Nil, y0)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(y0))), IMMATCOPY(Cons(Nil, y0))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: immatcopy(Cons(z0, z1)) -> Cons(Nil, immatcopy(z1)) immatcopy(Nil) -> Nil Tuples: IMMATCOPY(Cons(z0, Cons(y0, y1))) -> c(IMMATCOPY(Cons(y0, y1))) NESTIMEQL(Cons(Nil, y0)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(y0))), IMMATCOPY(Cons(Nil, y0))) S tuples: IMMATCOPY(Cons(z0, Cons(y0, y1))) -> c(IMMATCOPY(Cons(y0, y1))) NESTIMEQL(Cons(Nil, y0)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(y0))), IMMATCOPY(Cons(Nil, y0))) K tuples:none Defined Rule Symbols: immatcopy_1 Defined Pair Symbols: IMMATCOPY_1, NESTIMEQL_1 Compound Symbols: c_1, c3_2 ---------------------------------------- (55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IMMATCOPY(Cons(z0, Cons(y0, y1))) -> c(IMMATCOPY(Cons(y0, y1))) by IMMATCOPY(Cons(z0, Cons(z1, Cons(y1, y2)))) -> c(IMMATCOPY(Cons(z1, Cons(y1, y2)))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: immatcopy(Cons(z0, z1)) -> Cons(Nil, immatcopy(z1)) immatcopy(Nil) -> Nil Tuples: NESTIMEQL(Cons(Nil, y0)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(y0))), IMMATCOPY(Cons(Nil, y0))) IMMATCOPY(Cons(z0, Cons(z1, Cons(y1, y2)))) -> c(IMMATCOPY(Cons(z1, Cons(y1, y2)))) S tuples: NESTIMEQL(Cons(Nil, y0)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(y0))), IMMATCOPY(Cons(Nil, y0))) IMMATCOPY(Cons(z0, Cons(z1, Cons(y1, y2)))) -> c(IMMATCOPY(Cons(z1, Cons(y1, y2)))) K tuples:none Defined Rule Symbols: immatcopy_1 Defined Pair Symbols: NESTIMEQL_1, IMMATCOPY_1 Compound Symbols: c3_2, c_1