WORST_CASE(?,O(n^1)) proof of input_B5sEyI8DQe.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). (0) CpxTRS (1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxWeightedTrs (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxTypedWeightedTrs (5) CompletionProof [UPPER BOUND(ID), 0 ms] (6) CpxTypedWeightedCompleteTrs (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (10) CpxRNTS (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxRNTS (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxRNTS (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) IntTrsBoundProof [UPPER BOUND(ID), 232 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 90 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 122 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 44 ms] (26) CpxRNTS (27) FinalProof [FINISHED, 0 ms] (28) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs))) duplicate(Nil) -> Nil goal(x) -> duplicate(x) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs))) [1] duplicate(Nil) -> Nil [1] goal(x) -> duplicate(x) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs))) [1] duplicate(Nil) -> Nil [1] goal(x) -> duplicate(x) [1] The TRS has the following type information: duplicate :: Cons:Nil -> Cons:Nil Cons :: a -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil Rewrite Strategy: INNERMOST ---------------------------------------- (5) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: duplicate_1 goal_1 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: const ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs))) [1] duplicate(Nil) -> Nil [1] goal(x) -> duplicate(x) [1] The TRS has the following type information: duplicate :: Cons:Nil -> Cons:Nil Cons :: a -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil const :: a Rewrite Strategy: INNERMOST ---------------------------------------- (7) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs))) [1] duplicate(Nil) -> Nil [1] goal(x) -> duplicate(x) [1] The TRS has the following type information: duplicate :: Cons:Nil -> Cons:Nil Cons :: a -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil goal :: Cons:Nil -> Cons:Nil const :: a Rewrite Strategy: INNERMOST ---------------------------------------- (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: Nil => 0 const => 0 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: duplicate(z) -{ 1 }-> 0 :|: z = 0 duplicate(z) -{ 1 }-> 1 + x + (1 + x + duplicate(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 goal(z) -{ 1 }-> duplicate(x) :|: x >= 0, z = x ---------------------------------------- (11) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: duplicate(z) -{ 1 }-> 0 :|: z = 0 duplicate(z) -{ 1 }-> 1 + x + (1 + x + duplicate(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 goal(z) -{ 1 }-> duplicate(z) :|: z >= 0 ---------------------------------------- (13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { duplicate } { goal } ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: duplicate(z) -{ 1 }-> 0 :|: z = 0 duplicate(z) -{ 1 }-> 1 + x + (1 + x + duplicate(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 goal(z) -{ 1 }-> duplicate(z) :|: z >= 0 Function symbols to be analyzed: {duplicate}, {goal} ---------------------------------------- (15) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: duplicate(z) -{ 1 }-> 0 :|: z = 0 duplicate(z) -{ 1 }-> 1 + x + (1 + x + duplicate(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 goal(z) -{ 1 }-> duplicate(z) :|: z >= 0 Function symbols to be analyzed: {duplicate}, {goal} ---------------------------------------- (17) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: duplicate after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: duplicate(z) -{ 1 }-> 0 :|: z = 0 duplicate(z) -{ 1 }-> 1 + x + (1 + x + duplicate(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 goal(z) -{ 1 }-> duplicate(z) :|: z >= 0 Function symbols to be analyzed: {duplicate}, {goal} Previous analysis results are: duplicate: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: duplicate after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: duplicate(z) -{ 1 }-> 0 :|: z = 0 duplicate(z) -{ 1 }-> 1 + x + (1 + x + duplicate(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 goal(z) -{ 1 }-> duplicate(z) :|: z >= 0 Function symbols to be analyzed: {goal} Previous analysis results are: duplicate: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: duplicate(z) -{ 1 }-> 0 :|: z = 0 duplicate(z) -{ 2 + xs }-> 1 + x + (1 + x + s) :|: s >= 0, s <= 2 * xs, z = 1 + x + xs, xs >= 0, x >= 0 goal(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= 2 * z, z >= 0 Function symbols to be analyzed: {goal} Previous analysis results are: duplicate: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: goal after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: duplicate(z) -{ 1 }-> 0 :|: z = 0 duplicate(z) -{ 2 + xs }-> 1 + x + (1 + x + s) :|: s >= 0, s <= 2 * xs, z = 1 + x + xs, xs >= 0, x >= 0 goal(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= 2 * z, z >= 0 Function symbols to be analyzed: {goal} Previous analysis results are: duplicate: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] goal: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: goal after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: duplicate(z) -{ 1 }-> 0 :|: z = 0 duplicate(z) -{ 2 + xs }-> 1 + x + (1 + x + s) :|: s >= 0, s <= 2 * xs, z = 1 + x + xs, xs >= 0, x >= 0 goal(z) -{ 2 + z }-> s' :|: s' >= 0, s' <= 2 * z, z >= 0 Function symbols to be analyzed: Previous analysis results are: duplicate: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] goal: runtime: O(n^1) [2 + z], size: O(n^1) [2*z] ---------------------------------------- (27) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (28) BOUNDS(1, n^1)