KILLED proof of input_ULrs3dyRu3.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 1461 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 374 ms] (24) CpxRNTS (25) CompletionProof [UPPER BOUND(ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (30) CdtProblem (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (42) 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: f(x, empty) -> x f(empty, cons(a, k)) -> f(cons(a, k), k) f(cons(a, k), y) -> f(y, k) 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: f(x, empty) -> x f(empty, cons(a, k)) -> f(cons(a, k), k) f(cons(a, k), y) -> f(y, k) 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: f(x, empty) -> x f(empty, cons(a, k)) -> f(cons(a, k), k) f(cons(a, k), y) -> f(y, k) 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: f(x, empty) -> x [1] f(empty, cons(a, k)) -> f(cons(a, k), k) [1] f(cons(a, k), y) -> f(y, k) [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: f(x, empty) -> x [1] f(empty, cons(a, k)) -> f(cons(a, k), k) [1] f(cons(a, k), y) -> f(y, k) [1] The TRS has the following type information: f :: empty:cons -> empty:cons -> empty:cons empty :: empty:cons cons :: a -> empty:cons -> empty:cons 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: f_2 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: const ---------------------------------------- (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: f(x, empty) -> x [1] f(empty, cons(a, k)) -> f(cons(a, k), k) [1] f(cons(a, k), y) -> f(y, k) [1] The TRS has the following type information: f :: empty:cons -> empty:cons -> empty:cons empty :: empty:cons cons :: a -> empty:cons -> empty:cons const :: a 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: f(x, empty) -> x [1] f(empty, cons(a, k)) -> f(cons(a, k), k) [1] f(cons(a, k), y) -> f(y, k) [1] The TRS has the following type information: f :: empty:cons -> empty:cons -> empty:cons empty :: empty:cons cons :: a -> empty:cons -> empty:cons const :: a 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: empty => 0 const => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 f(z, z') -{ 1 }-> f(y, k) :|: z = 1 + a + k, a >= 0, y >= 0, k >= 0, z' = y f(z, z') -{ 1 }-> f(1 + a + k, k) :|: z' = 1 + a + k, a >= 0, k >= 0, z = 0 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 f(z, z') -{ 1 }-> f(z', k) :|: z = 1 + a + k, a >= 0, z' >= 0, k >= 0 f(z, z') -{ 1 }-> f(1 + a + k, k) :|: z' = 1 + a + k, a >= 0, k >= 0, z = 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { f } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 f(z, z') -{ 1 }-> f(z', k) :|: z = 1 + a + k, a >= 0, z' >= 0, k >= 0 f(z, z') -{ 1 }-> f(1 + a + k, k) :|: z' = 1 + a + k, a >= 0, k >= 0, z = 0 Function symbols to be analyzed: {f} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 f(z, z') -{ 1 }-> f(z', k) :|: z = 1 + a + k, a >= 0, z' >= 0, k >= 0 f(z, z') -{ 1 }-> f(1 + a + k, k) :|: z' = 1 + a + k, a >= 0, k >= 0, z = 0 Function symbols to be analyzed: {f} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 f(z, z') -{ 1 }-> f(z', k) :|: z = 1 + a + k, a >= 0, z' >= 0, k >= 0 f(z, z') -{ 1 }-> f(1 + a + k, k) :|: z' = 1 + a + k, a >= 0, k >= 0, z = 0 Function symbols to be analyzed: {f} Previous analysis results are: f: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 f(z, z') -{ 1 }-> f(z', k) :|: z = 1 + a + k, a >= 0, z' >= 0, k >= 0 f(z, z') -{ 1 }-> f(1 + a + k, k) :|: z' = 1 + a + k, a >= 0, k >= 0, z = 0 Function symbols to be analyzed: {f} Previous analysis results are: f: runtime: INF, size: O(n^1) [z + z'] ---------------------------------------- (25) 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: const ---------------------------------------- (26) 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: f(x, empty) -> x [1] f(empty, cons(a, k)) -> f(cons(a, k), k) [1] f(cons(a, k), y) -> f(y, k) [1] The TRS has the following type information: f :: empty:cons -> empty:cons -> empty:cons empty :: empty:cons cons :: a -> empty:cons -> empty:cons const :: a Rewrite Strategy: INNERMOST ---------------------------------------- (27) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: empty => 0 const => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 f(z, z') -{ 1 }-> f(y, k) :|: z = 1 + a + k, a >= 0, y >= 0, k >= 0, z' = y f(z, z') -{ 1 }-> f(1 + a + k, k) :|: z' = 1 + a + k, a >= 0, k >= 0, z = 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (29) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, empty) -> z0 f(empty, cons(z0, z1)) -> f(cons(z0, z1), z1) f(cons(z0, z1), z2) -> f(z2, z1) Tuples: F(z0, empty) -> c F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1)) F(cons(z0, z1), z2) -> c2(F(z2, z1)) S tuples: F(z0, empty) -> c F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1)) F(cons(z0, z1), z2) -> c2(F(z2, z1)) K tuples:none Defined Rule Symbols: f_2 Defined Pair Symbols: F_2 Compound Symbols: c, c1_1, c2_1 ---------------------------------------- (31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(z0, empty) -> c ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: f(z0, empty) -> z0 f(empty, cons(z0, z1)) -> f(cons(z0, z1), z1) f(cons(z0, z1), z2) -> f(z2, z1) Tuples: F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1)) F(cons(z0, z1), z2) -> c2(F(z2, z1)) S tuples: F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1)) F(cons(z0, z1), z2) -> c2(F(z2, z1)) K tuples:none Defined Rule Symbols: f_2 Defined Pair Symbols: F_2 Compound Symbols: c1_1, c2_1 ---------------------------------------- (33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f(z0, empty) -> z0 f(empty, cons(z0, z1)) -> f(cons(z0, z1), z1) f(cons(z0, z1), z2) -> f(z2, z1) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1)) F(cons(z0, z1), z2) -> c2(F(z2, z1)) S tuples: F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1)) F(cons(z0, z1), z2) -> c2(F(z2, z1)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_2 Compound Symbols: c1_1, c2_1 ---------------------------------------- (35) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(cons(z0, z1), z2) -> c2(F(z2, z1)) by F(cons(z0, cons(y0, y1)), empty) -> c2(F(empty, cons(y0, y1))) F(cons(z0, z1), cons(y0, y1)) -> c2(F(cons(y0, y1), z1)) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1)) F(cons(z0, cons(y0, y1)), empty) -> c2(F(empty, cons(y0, y1))) F(cons(z0, z1), cons(y0, y1)) -> c2(F(cons(y0, y1), z1)) S tuples: F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1)) F(cons(z0, cons(y0, y1)), empty) -> c2(F(empty, cons(y0, y1))) F(cons(z0, z1), cons(y0, y1)) -> c2(F(cons(y0, y1), z1)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_2 Compound Symbols: c1_1, c2_1 ---------------------------------------- (37) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1)) by F(empty, cons(z0, cons(y2, y3))) -> c1(F(cons(z0, cons(y2, y3)), cons(y2, y3))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(cons(z0, cons(y0, y1)), empty) -> c2(F(empty, cons(y0, y1))) F(cons(z0, z1), cons(y0, y1)) -> c2(F(cons(y0, y1), z1)) F(empty, cons(z0, cons(y2, y3))) -> c1(F(cons(z0, cons(y2, y3)), cons(y2, y3))) S tuples: F(cons(z0, cons(y0, y1)), empty) -> c2(F(empty, cons(y0, y1))) F(cons(z0, z1), cons(y0, y1)) -> c2(F(cons(y0, y1), z1)) F(empty, cons(z0, cons(y2, y3))) -> c1(F(cons(z0, cons(y2, y3)), cons(y2, y3))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_2 Compound Symbols: c2_1, c1_1 ---------------------------------------- (39) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(cons(z0, cons(y0, y1)), empty) -> c2(F(empty, cons(y0, y1))) by F(cons(z0, cons(z1, cons(y1, y2))), empty) -> c2(F(empty, cons(z1, cons(y1, y2)))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(cons(z0, z1), cons(y0, y1)) -> c2(F(cons(y0, y1), z1)) F(empty, cons(z0, cons(y2, y3))) -> c1(F(cons(z0, cons(y2, y3)), cons(y2, y3))) F(cons(z0, cons(z1, cons(y1, y2))), empty) -> c2(F(empty, cons(z1, cons(y1, y2)))) S tuples: F(cons(z0, z1), cons(y0, y1)) -> c2(F(cons(y0, y1), z1)) F(empty, cons(z0, cons(y2, y3))) -> c1(F(cons(z0, cons(y2, y3)), cons(y2, y3))) F(cons(z0, cons(z1, cons(y1, y2))), empty) -> c2(F(empty, cons(z1, cons(y1, y2)))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_2 Compound Symbols: c2_1, c1_1 ---------------------------------------- (41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(cons(z0, z1), cons(y0, y1)) -> c2(F(cons(y0, y1), z1)) by F(cons(z0, cons(y2, y3)), cons(z2, z3)) -> c2(F(cons(z2, z3), cons(y2, y3))) F(cons(z0, empty), cons(z2, cons(y1, cons(y2, y3)))) -> c2(F(cons(z2, cons(y1, cons(y2, y3))), empty)) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(empty, cons(z0, cons(y2, y3))) -> c1(F(cons(z0, cons(y2, y3)), cons(y2, y3))) F(cons(z0, cons(z1, cons(y1, y2))), empty) -> c2(F(empty, cons(z1, cons(y1, y2)))) F(cons(z0, cons(y2, y3)), cons(z2, z3)) -> c2(F(cons(z2, z3), cons(y2, y3))) F(cons(z0, empty), cons(z2, cons(y1, cons(y2, y3)))) -> c2(F(cons(z2, cons(y1, cons(y2, y3))), empty)) S tuples: F(empty, cons(z0, cons(y2, y3))) -> c1(F(cons(z0, cons(y2, y3)), cons(y2, y3))) F(cons(z0, cons(z1, cons(y1, y2))), empty) -> c2(F(empty, cons(z1, cons(y1, y2)))) F(cons(z0, cons(y2, y3)), cons(z2, z3)) -> c2(F(cons(z2, z3), cons(y2, y3))) F(cons(z0, empty), cons(z2, cons(y1, cons(y2, y3)))) -> c2(F(cons(z2, cons(y1, cons(y2, y3))), empty)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_2 Compound Symbols: c1_1, c2_1