WORST_CASE(?,O(n^1)) proof of input_qzYGSgqPnU.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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), 453 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 67 ms] (20) CpxRNTS (21) FinalProof [FINISHED, 0 ms] (22) 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: D(t) -> 1 D(constant) -> 0 D(+(x, y)) -> +(D(x), D(y)) D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) 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: D(t) -> 1 [1] D(constant) -> 0 [1] D(+(x, y)) -> +(D(x), D(y)) [1] D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) [1] D(-(x, y)) -> -(D(x), D(y)) [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: D(t) -> 1 [1] D(constant) -> 0 [1] D(+(x, y)) -> +(D(x), D(y)) [1] D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) [1] D(-(x, y)) -> -(D(x), D(y)) [1] The TRS has the following type information: D :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- t :: t:1:constant:0:+:*:- 1 :: t:1:constant:0:+:*:- constant :: t:1:constant:0:+:*:- 0 :: t:1:constant:0:+:*:- + :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- * :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- - :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- 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: D_1 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: D(t) -> 1 [1] D(constant) -> 0 [1] D(+(x, y)) -> +(D(x), D(y)) [1] D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) [1] D(-(x, y)) -> -(D(x), D(y)) [1] The TRS has the following type information: D :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- t :: t:1:constant:0:+:*:- 1 :: t:1:constant:0:+:*:- constant :: t:1:constant:0:+:*:- 0 :: t:1:constant:0:+:*:- + :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- * :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- - :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- 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: D(t) -> 1 [1] D(constant) -> 0 [1] D(+(x, y)) -> +(D(x), D(y)) [1] D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) [1] D(-(x, y)) -> -(D(x), D(y)) [1] The TRS has the following type information: D :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- t :: t:1:constant:0:+:*:- 1 :: t:1:constant:0:+:*:- constant :: t:1:constant:0:+:*:- 0 :: t:1:constant:0:+:*:- + :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- * :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- - :: t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- -> t:1:constant:0:+:*:- 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: t => 3 1 => 1 constant => 2 0 => 0 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: D(z) -{ 1 }-> 1 :|: z = 3 D(z) -{ 1 }-> 0 :|: z = 2 D(z) -{ 1 }-> 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0 D(z) -{ 1 }-> 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0 ---------------------------------------- (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: D(z) -{ 1 }-> 1 :|: z = 3 D(z) -{ 1 }-> 0 :|: z = 2 D(z) -{ 1 }-> 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0 D(z) -{ 1 }-> 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0 ---------------------------------------- (13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { D } ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: D(z) -{ 1 }-> 1 :|: z = 3 D(z) -{ 1 }-> 0 :|: z = 2 D(z) -{ 1 }-> 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0 D(z) -{ 1 }-> 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {D} ---------------------------------------- (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: D(z) -{ 1 }-> 1 :|: z = 3 D(z) -{ 1 }-> 0 :|: z = 2 D(z) -{ 1 }-> 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0 D(z) -{ 1 }-> 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {D} ---------------------------------------- (17) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: D after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + 2*z + z^2 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: D(z) -{ 1 }-> 1 :|: z = 3 D(z) -{ 1 }-> 0 :|: z = 2 D(z) -{ 1 }-> 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0 D(z) -{ 1 }-> 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {D} Previous analysis results are: D: runtime: ?, size: O(n^2) [1 + 2*z + z^2] ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: D after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3*z ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: D(z) -{ 1 }-> 1 :|: z = 3 D(z) -{ 1 }-> 0 :|: z = 2 D(z) -{ 1 }-> 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0 D(z) -{ 1 }-> 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: Previous analysis results are: D: runtime: O(n^1) [3*z], size: O(n^2) [1 + 2*z + z^2] ---------------------------------------- (21) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (22) BOUNDS(1, n^1)