KILLED proof of input_kKrlP6xBH4.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 5 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), 115 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 26 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 228 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 75 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 1668 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 1187 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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtInstantiationProof [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: inc(Cons(x, xs)) -> Cons(Cons(Nil, Nil), inc(xs)) nestinc(Nil) -> number17(Nil) nestinc(Cons(x, xs)) -> nestinc(inc(Cons(x, xs))) inc(Nil) -> Cons(Nil, Nil) number17(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, Nil))))))))))))))))) goal(x) -> nestinc(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: inc(Cons(x, xs)) -> Cons(Cons(Nil, Nil), inc(xs)) nestinc(Nil) -> number17(Nil) nestinc(Cons(x, xs)) -> nestinc(inc(Cons(x, xs))) inc(Nil) -> Cons(Nil, Nil) number17(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, Nil))))))))))))))))) goal(x) -> nestinc(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: inc(Cons(x, xs)) -> Cons(Cons(Nil, Nil), inc(xs)) nestinc(Nil) -> number17(Nil) nestinc(Cons(x, xs)) -> nestinc(inc(Cons(x, xs))) inc(Nil) -> Cons(Nil, Nil) number17(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, Nil))))))))))))))))) goal(x) -> nestinc(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: inc(Cons(x, xs)) -> Cons(Cons(Nil, Nil), inc(xs)) [1] nestinc(Nil) -> number17(Nil) [1] nestinc(Cons(x, xs)) -> nestinc(inc(Cons(x, xs))) [1] inc(Nil) -> Cons(Nil, Nil) [1] number17(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, Nil))))))))))))))))) [1] goal(x) -> nestinc(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: inc(Cons(x, xs)) -> Cons(Cons(Nil, Nil), inc(xs)) [1] nestinc(Nil) -> number17(Nil) [1] nestinc(Cons(x, xs)) -> nestinc(inc(Cons(x, xs))) [1] inc(Nil) -> Cons(Nil, Nil) [1] number17(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, Nil))))))))))))))))) [1] goal(x) -> nestinc(x) [1] The TRS has the following type information: inc :: Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil nestinc :: Cons:Nil -> Cons:Nil number17 :: 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: nestinc_1 number17_1 goal_1 (c) The following functions are completely defined: inc_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: inc(Cons(x, xs)) -> Cons(Cons(Nil, Nil), inc(xs)) [1] nestinc(Nil) -> number17(Nil) [1] nestinc(Cons(x, xs)) -> nestinc(inc(Cons(x, xs))) [1] inc(Nil) -> Cons(Nil, Nil) [1] number17(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, Nil))))))))))))))))) [1] goal(x) -> nestinc(x) [1] The TRS has the following type information: inc :: Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil nestinc :: Cons:Nil -> Cons:Nil number17 :: 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: inc(Cons(x, xs)) -> Cons(Cons(Nil, Nil), inc(xs)) [1] nestinc(Nil) -> number17(Nil) [1] nestinc(Cons(x, xs)) -> nestinc(Cons(Cons(Nil, Nil), inc(xs))) [2] inc(Nil) -> Cons(Nil, Nil) [1] number17(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, Nil))))))))))))))))) [1] goal(x) -> nestinc(x) [1] The TRS has the following type information: inc :: Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil nestinc :: Cons:Nil -> Cons:Nil number17 :: 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 }-> nestinc(x) :|: x >= 0, z = x inc(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0 inc(z) -{ 1 }-> 1 + (1 + 0 + 0) + inc(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(z) -{ 1 }-> number17(0) :|: z = 0 nestinc(z) -{ 2 }-> nestinc(1 + (1 + 0 + 0) + inc(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 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)))))))))))))))) :|: x >= 0, z = x ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: 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)))))))))))))))) :|: x >= 0, z = x ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> nestinc(x) :|: x >= 0, z = x inc(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0 inc(z) -{ 1 }-> 1 + (1 + 0 + 0) + inc(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(z) -{ 2 }-> nestinc(1 + (1 + 0 + 0) + inc(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(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 + 0)))))))))))))))) :|: z = 0, x >= 0, 0 = x 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)))))))))))))))) :|: 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 }-> nestinc(z) :|: z >= 0 inc(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0 inc(z) -{ 1 }-> 1 + (1 + 0 + 0) + inc(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(z) -{ 2 }-> nestinc(1 + (1 + 0 + 0) + inc(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(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 + 0)))))))))))))))) :|: z = 0, x >= 0, 0 = x 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 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { number17 } { inc } { nestinc } { goal } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> nestinc(z) :|: z >= 0 inc(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0 inc(z) -{ 1 }-> 1 + (1 + 0 + 0) + inc(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(z) -{ 2 }-> nestinc(1 + (1 + 0 + 0) + inc(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(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 + 0)))))))))))))))) :|: z = 0, x >= 0, 0 = x 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}, {inc}, {nestinc}, {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 }-> nestinc(z) :|: z >= 0 inc(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0 inc(z) -{ 1 }-> 1 + (1 + 0 + 0) + inc(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(z) -{ 2 }-> nestinc(1 + (1 + 0 + 0) + inc(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(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 + 0)))))))))))))))) :|: z = 0, x >= 0, 0 = x 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}, {inc}, {nestinc}, {goal} ---------------------------------------- (23) 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 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> nestinc(z) :|: z >= 0 inc(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0 inc(z) -{ 1 }-> 1 + (1 + 0 + 0) + inc(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(z) -{ 2 }-> nestinc(1 + (1 + 0 + 0) + inc(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(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 + 0)))))))))))))))) :|: z = 0, x >= 0, 0 = x 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}, {inc}, {nestinc}, {goal} Previous analysis results are: number17: runtime: ?, size: O(1) [17] ---------------------------------------- (25) 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 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> nestinc(z) :|: z >= 0 inc(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0 inc(z) -{ 1 }-> 1 + (1 + 0 + 0) + inc(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(z) -{ 2 }-> nestinc(1 + (1 + 0 + 0) + inc(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(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 + 0)))))))))))))))) :|: z = 0, x >= 0, 0 = x 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: {inc}, {nestinc}, {goal} Previous analysis results are: number17: runtime: O(1) [1], size: O(1) [17] ---------------------------------------- (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 }-> nestinc(z) :|: z >= 0 inc(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0 inc(z) -{ 1 }-> 1 + (1 + 0 + 0) + inc(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(z) -{ 2 }-> nestinc(1 + (1 + 0 + 0) + inc(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(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 + 0)))))))))))))))) :|: z = 0, x >= 0, 0 = x 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: {inc}, {nestinc}, {goal} Previous analysis results are: number17: runtime: O(1) [1], size: O(1) [17] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: inc after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 2*z ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> nestinc(z) :|: z >= 0 inc(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0 inc(z) -{ 1 }-> 1 + (1 + 0 + 0) + inc(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(z) -{ 2 }-> nestinc(1 + (1 + 0 + 0) + inc(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(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 + 0)))))))))))))))) :|: z = 0, x >= 0, 0 = x 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: {inc}, {nestinc}, {goal} Previous analysis results are: number17: runtime: O(1) [1], size: O(1) [17] inc: runtime: ?, size: O(n^1) [1 + 2*z] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: inc 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 }-> nestinc(z) :|: z >= 0 inc(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0 inc(z) -{ 1 }-> 1 + (1 + 0 + 0) + inc(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(z) -{ 2 }-> nestinc(1 + (1 + 0 + 0) + inc(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(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 + 0)))))))))))))))) :|: z = 0, x >= 0, 0 = x 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: {nestinc}, {goal} Previous analysis results are: number17: runtime: O(1) [1], size: O(1) [17] inc: runtime: O(n^1) [1 + z], size: O(n^1) [1 + 2*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 }-> nestinc(z) :|: z >= 0 inc(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0 inc(z) -{ 2 + xs }-> 1 + (1 + 0 + 0) + s :|: s >= 0, s <= 2 * xs + 1, z = 1 + x + xs, xs >= 0, x >= 0 nestinc(z) -{ 3 + xs }-> nestinc(1 + (1 + 0 + 0) + s') :|: s' >= 0, s' <= 2 * xs + 1, z = 1 + x + xs, xs >= 0, x >= 0 nestinc(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 + 0)))))))))))))))) :|: z = 0, x >= 0, 0 = x 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: {nestinc}, {goal} Previous analysis results are: number17: runtime: O(1) [1], size: O(1) [17] inc: runtime: O(n^1) [1 + z], size: O(n^1) [1 + 2*z] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: nestinc after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 17 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: goal(z) -{ 1 }-> nestinc(z) :|: z >= 0 inc(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0 inc(z) -{ 2 + xs }-> 1 + (1 + 0 + 0) + s :|: s >= 0, s <= 2 * xs + 1, z = 1 + x + xs, xs >= 0, x >= 0 nestinc(z) -{ 3 + xs }-> nestinc(1 + (1 + 0 + 0) + s') :|: s' >= 0, s' <= 2 * xs + 1, z = 1 + x + xs, xs >= 0, x >= 0 nestinc(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 + 0)))))))))))))))) :|: z = 0, x >= 0, 0 = x 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: {nestinc}, {goal} Previous analysis results are: number17: runtime: O(1) [1], size: O(1) [17] inc: runtime: O(n^1) [1 + z], size: O(n^1) [1 + 2*z] nestinc: runtime: ?, size: O(1) [17] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: nestinc 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 }-> nestinc(z) :|: z >= 0 inc(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0 inc(z) -{ 2 + xs }-> 1 + (1 + 0 + 0) + s :|: s >= 0, s <= 2 * xs + 1, z = 1 + x + xs, xs >= 0, x >= 0 nestinc(z) -{ 3 + xs }-> nestinc(1 + (1 + 0 + 0) + s') :|: s' >= 0, s' <= 2 * xs + 1, z = 1 + x + xs, xs >= 0, x >= 0 nestinc(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 + 0)))))))))))))))) :|: z = 0, x >= 0, 0 = x 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: {nestinc}, {goal} Previous analysis results are: number17: runtime: O(1) [1], size: O(1) [17] inc: runtime: O(n^1) [1 + z], size: O(n^1) [1 + 2*z] nestinc: runtime: INF, size: O(1) [17] ---------------------------------------- (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: inc(Cons(x, xs)) -> Cons(Cons(Nil, Nil), inc(xs)) [1] nestinc(Nil) -> number17(Nil) [1] nestinc(Cons(x, xs)) -> nestinc(inc(Cons(x, xs))) [1] inc(Nil) -> Cons(Nil, Nil) [1] number17(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, Nil))))))))))))))))) [1] goal(x) -> nestinc(x) [1] The TRS has the following type information: inc :: Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil nestinc :: Cons:Nil -> Cons:Nil number17 :: 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 }-> nestinc(x) :|: x >= 0, z = x inc(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0 inc(z) -{ 1 }-> 1 + (1 + 0 + 0) + inc(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 nestinc(z) -{ 1 }-> number17(0) :|: z = 0 nestinc(z) -{ 1 }-> nestinc(inc(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 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)))))))))))))))) :|: 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: inc(Cons(z0, z1)) -> Cons(Cons(Nil, Nil), inc(z1)) inc(Nil) -> Cons(Nil, Nil) nestinc(Nil) -> number17(Nil) nestinc(Cons(z0, z1)) -> nestinc(inc(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) -> nestinc(z0) Tuples: INC(Cons(z0, z1)) -> c(INC(z1)) INC(Nil) -> c1 NESTINC(Nil) -> c2(NUMBER17(Nil)) NESTINC(Cons(z0, z1)) -> c3(NESTINC(inc(Cons(z0, z1))), INC(Cons(z0, z1))) NUMBER17(z0) -> c4 GOAL(z0) -> c5(NESTINC(z0)) S tuples: INC(Cons(z0, z1)) -> c(INC(z1)) INC(Nil) -> c1 NESTINC(Nil) -> c2(NUMBER17(Nil)) NESTINC(Cons(z0, z1)) -> c3(NESTINC(inc(Cons(z0, z1))), INC(Cons(z0, z1))) NUMBER17(z0) -> c4 GOAL(z0) -> c5(NESTINC(z0)) K tuples:none Defined Rule Symbols: inc_1, nestinc_1, number17_1, goal_1 Defined Pair Symbols: INC_1, NESTINC_1, NUMBER17_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(NESTINC(z0)) Removed 3 trailing nodes: NUMBER17(z0) -> c4 NESTINC(Nil) -> c2(NUMBER17(Nil)) INC(Nil) -> c1 ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: inc(Cons(z0, z1)) -> Cons(Cons(Nil, Nil), inc(z1)) inc(Nil) -> Cons(Nil, Nil) nestinc(Nil) -> number17(Nil) nestinc(Cons(z0, z1)) -> nestinc(inc(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) -> nestinc(z0) Tuples: INC(Cons(z0, z1)) -> c(INC(z1)) NESTINC(Cons(z0, z1)) -> c3(NESTINC(inc(Cons(z0, z1))), INC(Cons(z0, z1))) S tuples: INC(Cons(z0, z1)) -> c(INC(z1)) NESTINC(Cons(z0, z1)) -> c3(NESTINC(inc(Cons(z0, z1))), INC(Cons(z0, z1))) K tuples:none Defined Rule Symbols: inc_1, nestinc_1, number17_1, goal_1 Defined Pair Symbols: INC_1, NESTINC_1 Compound Symbols: c_1, c3_2 ---------------------------------------- (47) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: nestinc(Nil) -> number17(Nil) nestinc(Cons(z0, z1)) -> nestinc(inc(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) -> nestinc(z0) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: inc(Cons(z0, z1)) -> Cons(Cons(Nil, Nil), inc(z1)) inc(Nil) -> Cons(Nil, Nil) Tuples: INC(Cons(z0, z1)) -> c(INC(z1)) NESTINC(Cons(z0, z1)) -> c3(NESTINC(inc(Cons(z0, z1))), INC(Cons(z0, z1))) S tuples: INC(Cons(z0, z1)) -> c(INC(z1)) NESTINC(Cons(z0, z1)) -> c3(NESTINC(inc(Cons(z0, z1))), INC(Cons(z0, z1))) K tuples:none Defined Rule Symbols: inc_1 Defined Pair Symbols: INC_1, NESTINC_1 Compound Symbols: c_1, c3_2 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace NESTINC(Cons(z0, z1)) -> c3(NESTINC(inc(Cons(z0, z1))), INC(Cons(z0, z1))) by NESTINC(Cons(z0, z1)) -> c3(NESTINC(Cons(Cons(Nil, Nil), inc(z1))), INC(Cons(z0, z1))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: inc(Cons(z0, z1)) -> Cons(Cons(Nil, Nil), inc(z1)) inc(Nil) -> Cons(Nil, Nil) Tuples: INC(Cons(z0, z1)) -> c(INC(z1)) NESTINC(Cons(z0, z1)) -> c3(NESTINC(Cons(Cons(Nil, Nil), inc(z1))), INC(Cons(z0, z1))) S tuples: INC(Cons(z0, z1)) -> c(INC(z1)) NESTINC(Cons(z0, z1)) -> c3(NESTINC(Cons(Cons(Nil, Nil), inc(z1))), INC(Cons(z0, z1))) K tuples:none Defined Rule Symbols: inc_1 Defined Pair Symbols: INC_1, NESTINC_1 Compound Symbols: c_1, c3_2 ---------------------------------------- (51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace INC(Cons(z0, z1)) -> c(INC(z1)) by INC(Cons(z0, Cons(y0, y1))) -> c(INC(Cons(y0, y1))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: inc(Cons(z0, z1)) -> Cons(Cons(Nil, Nil), inc(z1)) inc(Nil) -> Cons(Nil, Nil) Tuples: NESTINC(Cons(z0, z1)) -> c3(NESTINC(Cons(Cons(Nil, Nil), inc(z1))), INC(Cons(z0, z1))) INC(Cons(z0, Cons(y0, y1))) -> c(INC(Cons(y0, y1))) S tuples: NESTINC(Cons(z0, z1)) -> c3(NESTINC(Cons(Cons(Nil, Nil), inc(z1))), INC(Cons(z0, z1))) INC(Cons(z0, Cons(y0, y1))) -> c(INC(Cons(y0, y1))) K tuples:none Defined Rule Symbols: inc_1 Defined Pair Symbols: NESTINC_1, INC_1 Compound Symbols: c3_2, c_1 ---------------------------------------- (53) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace INC(Cons(z0, Cons(y0, y1))) -> c(INC(Cons(y0, y1))) by INC(Cons(z0, Cons(z1, Cons(y1, y2)))) -> c(INC(Cons(z1, Cons(y1, y2)))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: inc(Cons(z0, z1)) -> Cons(Cons(Nil, Nil), inc(z1)) inc(Nil) -> Cons(Nil, Nil) Tuples: NESTINC(Cons(z0, z1)) -> c3(NESTINC(Cons(Cons(Nil, Nil), inc(z1))), INC(Cons(z0, z1))) INC(Cons(z0, Cons(z1, Cons(y1, y2)))) -> c(INC(Cons(z1, Cons(y1, y2)))) S tuples: NESTINC(Cons(z0, z1)) -> c3(NESTINC(Cons(Cons(Nil, Nil), inc(z1))), INC(Cons(z0, z1))) INC(Cons(z0, Cons(z1, Cons(y1, y2)))) -> c(INC(Cons(z1, Cons(y1, y2)))) K tuples:none Defined Rule Symbols: inc_1 Defined Pair Symbols: NESTINC_1, INC_1 Compound Symbols: c3_2, c_1 ---------------------------------------- (55) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace NESTINC(Cons(z0, z1)) -> c3(NESTINC(Cons(Cons(Nil, Nil), inc(z1))), INC(Cons(z0, z1))) by NESTINC(Cons(Cons(Nil, Nil), y0)) -> c3(NESTINC(Cons(Cons(Nil, Nil), inc(y0))), INC(Cons(Cons(Nil, Nil), y0))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: inc(Cons(z0, z1)) -> Cons(Cons(Nil, Nil), inc(z1)) inc(Nil) -> Cons(Nil, Nil) Tuples: INC(Cons(z0, Cons(z1, Cons(y1, y2)))) -> c(INC(Cons(z1, Cons(y1, y2)))) NESTINC(Cons(Cons(Nil, Nil), y0)) -> c3(NESTINC(Cons(Cons(Nil, Nil), inc(y0))), INC(Cons(Cons(Nil, Nil), y0))) S tuples: INC(Cons(z0, Cons(z1, Cons(y1, y2)))) -> c(INC(Cons(z1, Cons(y1, y2)))) NESTINC(Cons(Cons(Nil, Nil), y0)) -> c3(NESTINC(Cons(Cons(Nil, Nil), inc(y0))), INC(Cons(Cons(Nil, Nil), y0))) K tuples:none Defined Rule Symbols: inc_1 Defined Pair Symbols: INC_1, NESTINC_1 Compound Symbols: c_1, c3_2