WORST_CASE(?,O(n^1)) proof of input_LGULMv8iDU.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).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) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxWeightedTrs (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxTypedWeightedTrs (7) CompletionProof [UPPER BOUND(ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxRNTS (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 216 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 55 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 284 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 106 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 244 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 107 ms] (34) CpxRNTS (35) FinalProof [FINISHED, 0 ms] (36) 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: +(X, 0) -> X +(X, s(Y)) -> s(+(X, Y)) f(0, s(0), X) -> f(X, +(X, X), X) g(X, Y) -> X g(X, Y) -> 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: +(X, 0) -> X [1] +(X, s(Y)) -> s(+(X, Y)) [1] f(0, s(0), X) -> f(X, +(X, X), X) [1] g(X, Y) -> X [1] g(X, Y) -> Y [1] Rewrite Strategy: INNERMOST ---------------------------------------- (3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: + => plus ---------------------------------------- (4) 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: plus(X, 0) -> X [1] plus(X, s(Y)) -> s(plus(X, Y)) [1] f(0, s(0), X) -> f(X, plus(X, X), X) [1] g(X, Y) -> X [1] g(X, Y) -> Y [1] Rewrite Strategy: INNERMOST ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: plus(X, 0) -> X [1] plus(X, s(Y)) -> s(plus(X, Y)) [1] f(0, s(0), X) -> f(X, plus(X, X), X) [1] g(X, Y) -> X [1] g(X, Y) -> Y [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s f :: 0:s -> 0:s -> 0:s -> f g :: g -> g -> g Rewrite Strategy: INNERMOST ---------------------------------------- (7) 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: f_3 g_2 (c) The following functions are completely defined: plus_2 Due to the following rules being added: none And the following fresh constants: const, const1 ---------------------------------------- (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: plus(X, 0) -> X [1] plus(X, s(Y)) -> s(plus(X, Y)) [1] f(0, s(0), X) -> f(X, plus(X, X), X) [1] g(X, Y) -> X [1] g(X, Y) -> Y [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s f :: 0:s -> 0:s -> 0:s -> f g :: g -> g -> g const :: f const1 :: g Rewrite Strategy: INNERMOST ---------------------------------------- (9) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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: plus(X, 0) -> X [1] plus(X, s(Y)) -> s(plus(X, Y)) [1] f(0, s(0), 0) -> f(0, 0, 0) [2] f(0, s(0), s(Y')) -> f(s(Y'), s(plus(s(Y'), Y')), s(Y')) [2] g(X, Y) -> X [1] g(X, Y) -> Y [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s f :: 0:s -> 0:s -> 0:s -> f g :: g -> g -> g const :: f const1 :: g Rewrite Strategy: INNERMOST ---------------------------------------- (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 const => 0 const1 => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: f(z, z', z'') -{ 2 }-> f(0, 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0 f(z, z', z'') -{ 2 }-> f(1 + Y', 1 + plus(1 + Y', Y'), 1 + Y') :|: z'' = 1 + Y', Y' >= 0, z' = 1 + 0, z = 0 g(z, z') -{ 1 }-> X :|: z' = Y, Y >= 0, X >= 0, z = X g(z, z') -{ 1 }-> Y :|: z' = Y, Y >= 0, X >= 0, z = X plus(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(X, Y) :|: Y >= 0, z' = 1 + Y, X >= 0, z = X ---------------------------------------- (13) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: f(z, z', z'') -{ 2 }-> f(0, 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0 f(z, z', z'') -{ 2 }-> f(1 + (z'' - 1), 1 + plus(1 + (z'' - 1), z'' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 1 + 0, z = 0 g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0 g(z, z') -{ 1 }-> z' :|: z' >= 0, z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { g } { plus } { f } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: f(z, z', z'') -{ 2 }-> f(0, 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0 f(z, z', z'') -{ 2 }-> f(1 + (z'' - 1), 1 + plus(1 + (z'' - 1), z'' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 1 + 0, z = 0 g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0 g(z, z') -{ 1 }-> z' :|: z' >= 0, z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 Function symbols to be analyzed: {g}, {plus}, {f} ---------------------------------------- (17) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z, z', z'') -{ 2 }-> f(0, 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0 f(z, z', z'') -{ 2 }-> f(1 + (z'' - 1), 1 + plus(1 + (z'' - 1), z'' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 1 + 0, z = 0 g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0 g(z, z') -{ 1 }-> z' :|: z' >= 0, z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 Function symbols to be analyzed: {g}, {plus}, {f} ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z, z', z'') -{ 2 }-> f(0, 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0 f(z, z', z'') -{ 2 }-> f(1 + (z'' - 1), 1 + plus(1 + (z'' - 1), z'' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 1 + 0, z = 0 g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0 g(z, z') -{ 1 }-> z' :|: z' >= 0, z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 Function symbols to be analyzed: {g}, {plus}, {f} Previous analysis results are: g: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z, z', z'') -{ 2 }-> f(0, 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0 f(z, z', z'') -{ 2 }-> f(1 + (z'' - 1), 1 + plus(1 + (z'' - 1), z'' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 1 + 0, z = 0 g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0 g(z, z') -{ 1 }-> z' :|: z' >= 0, z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 Function symbols to be analyzed: {plus}, {f} Previous analysis results are: g: runtime: O(1) [1], size: O(n^1) [z + z'] ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z, z', z'') -{ 2 }-> f(0, 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0 f(z, z', z'') -{ 2 }-> f(1 + (z'' - 1), 1 + plus(1 + (z'' - 1), z'' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 1 + 0, z = 0 g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0 g(z, z') -{ 1 }-> z' :|: z' >= 0, z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 Function symbols to be analyzed: {plus}, {f} Previous analysis results are: g: runtime: O(1) [1], size: O(n^1) [z + z'] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: f(z, z', z'') -{ 2 }-> f(0, 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0 f(z, z', z'') -{ 2 }-> f(1 + (z'' - 1), 1 + plus(1 + (z'' - 1), z'' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 1 + 0, z = 0 g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0 g(z, z') -{ 1 }-> z' :|: z' >= 0, z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 Function symbols to be analyzed: {plus}, {f} Previous analysis results are: g: runtime: O(1) [1], size: O(n^1) [z + z'] plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z, z', z'') -{ 2 }-> f(0, 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0 f(z, z', z'') -{ 2 }-> f(1 + (z'' - 1), 1 + plus(1 + (z'' - 1), z'' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 1 + 0, z = 0 g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0 g(z, z') -{ 1 }-> z' :|: z' >= 0, z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0 Function symbols to be analyzed: {f} Previous analysis results are: g: runtime: O(1) [1], size: O(n^1) [z + z'] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: f(z, z', z'') -{ 2 }-> f(0, 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0 f(z, z', z'') -{ 2 + z'' }-> f(1 + (z'' - 1), 1 + s', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1 + (z'' - 1) + (z'' - 1), z'' - 1 >= 0, z' = 1 + 0, z = 0 g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0 g(z, z') -{ 1 }-> z' :|: z' >= 0, z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z' - 1 >= 0, z >= 0 Function symbols to be analyzed: {f} Previous analysis results are: g: runtime: O(1) [1], size: O(n^1) [z + z'] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: f(z, z', z'') -{ 2 }-> f(0, 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0 f(z, z', z'') -{ 2 + z'' }-> f(1 + (z'' - 1), 1 + s', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1 + (z'' - 1) + (z'' - 1), z'' - 1 >= 0, z' = 1 + 0, z = 0 g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0 g(z, z') -{ 1 }-> z' :|: z' >= 0, z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z' - 1 >= 0, z >= 0 Function symbols to be analyzed: {f} Previous analysis results are: g: runtime: O(1) [1], size: O(n^1) [z + z'] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] f: runtime: ?, size: O(1) [0] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z'' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: f(z, z', z'') -{ 2 }-> f(0, 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0 f(z, z', z'') -{ 2 + z'' }-> f(1 + (z'' - 1), 1 + s', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1 + (z'' - 1) + (z'' - 1), z'' - 1 >= 0, z' = 1 + 0, z = 0 g(z, z') -{ 1 }-> z :|: z' >= 0, z >= 0 g(z, z') -{ 1 }-> z' :|: z' >= 0, z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z' - 1 >= 0, z >= 0 Function symbols to be analyzed: Previous analysis results are: g: runtime: O(1) [1], size: O(n^1) [z + z'] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] f: runtime: O(n^1) [2 + z''], size: O(1) [0] ---------------------------------------- (35) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (36) BOUNDS(1, n^1)