KILLED proof of input_U6Hqx86bzm.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 683 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 279 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 [ComplexityIfPolyImplication, 0 ms] (32) CdtProblem (33) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (38) 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: letexp(x, y) -> letexp(Cons(Cons(Nil, Nil), x), y) goal(x, y) -> letexp(x, y) 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: letexp(x, y) -> letexp(Cons(Cons(Nil, Nil), x), y) goal(x, y) -> letexp(x, y) 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: letexp(x, y) -> letexp(Cons(Cons(Nil, Nil), x), y) goal(x, y) -> letexp(x, y) 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: letexp(x, y) -> letexp(Cons(Cons(Nil, Nil), x), y) [1] goal(x, y) -> letexp(x, y) [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: letexp(x, y) -> letexp(Cons(Cons(Nil, Nil), x), y) [1] goal(x, y) -> letexp(x, y) [1] The TRS has the following type information: letexp :: Nil:Cons -> a -> letexp:goal Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons Nil :: Nil:Cons goal :: Nil:Cons -> a -> letexp:goal 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: letexp_2 goal_2 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: const, const1 ---------------------------------------- (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: letexp(x, y) -> letexp(Cons(Cons(Nil, Nil), x), y) [1] goal(x, y) -> letexp(x, y) [1] The TRS has the following type information: letexp :: Nil:Cons -> a -> letexp:goal Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons Nil :: Nil:Cons goal :: Nil:Cons -> a -> letexp:goal const :: letexp:goal const1 :: 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: letexp(x, y) -> letexp(Cons(Cons(Nil, Nil), x), y) [1] goal(x, y) -> letexp(x, y) [1] The TRS has the following type information: letexp :: Nil:Cons -> a -> letexp:goal Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons Nil :: Nil:Cons goal :: Nil:Cons -> a -> letexp:goal const :: letexp:goal const1 :: 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: Nil => 0 const => 0 const1 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: goal(z, z') -{ 1 }-> letexp(x, y) :|: x >= 0, y >= 0, z = x, z' = y letexp(z, z') -{ 1 }-> letexp(1 + (1 + 0 + 0) + x, y) :|: x >= 0, y >= 0, z = x, z' = y ---------------------------------------- (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: goal(z, z') -{ 1 }-> letexp(z, z') :|: z >= 0, z' >= 0 letexp(z, z') -{ 1 }-> letexp(1 + (1 + 0 + 0) + z, z') :|: z >= 0, z' >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { letexp } { goal } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: goal(z, z') -{ 1 }-> letexp(z, z') :|: z >= 0, z' >= 0 letexp(z, z') -{ 1 }-> letexp(1 + (1 + 0 + 0) + z, z') :|: z >= 0, z' >= 0 Function symbols to be analyzed: {letexp}, {goal} ---------------------------------------- (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: goal(z, z') -{ 1 }-> letexp(z, z') :|: z >= 0, z' >= 0 letexp(z, z') -{ 1 }-> letexp(1 + (1 + 0 + 0) + z, z') :|: z >= 0, z' >= 0 Function symbols to be analyzed: {letexp}, {goal} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: letexp after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: goal(z, z') -{ 1 }-> letexp(z, z') :|: z >= 0, z' >= 0 letexp(z, z') -{ 1 }-> letexp(1 + (1 + 0 + 0) + z, z') :|: z >= 0, z' >= 0 Function symbols to be analyzed: {letexp}, {goal} Previous analysis results are: letexp: runtime: ?, size: O(1) [0] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: letexp after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: goal(z, z') -{ 1 }-> letexp(z, z') :|: z >= 0, z' >= 0 letexp(z, z') -{ 1 }-> letexp(1 + (1 + 0 + 0) + z, z') :|: z >= 0, z' >= 0 Function symbols to be analyzed: {letexp}, {goal} Previous analysis results are: letexp: runtime: INF, size: O(1) [0] ---------------------------------------- (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, const1 ---------------------------------------- (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: letexp(x, y) -> letexp(Cons(Cons(Nil, Nil), x), y) [1] goal(x, y) -> letexp(x, y) [1] The TRS has the following type information: letexp :: Nil:Cons -> a -> letexp:goal Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons Nil :: Nil:Cons goal :: Nil:Cons -> a -> letexp:goal const :: letexp:goal const1 :: 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: Nil => 0 const => 0 const1 => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: goal(z, z') -{ 1 }-> letexp(x, y) :|: x >= 0, y >= 0, z = x, z' = y letexp(z, z') -{ 1 }-> letexp(1 + (1 + 0 + 0) + x, y) :|: x >= 0, y >= 0, z = x, z' = y 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: letexp(z0, z1) -> letexp(Cons(Cons(Nil, Nil), z0), z1) goal(z0, z1) -> letexp(z0, z1) Tuples: LETEXP(z0, z1) -> c(LETEXP(Cons(Cons(Nil, Nil), z0), z1)) GOAL(z0, z1) -> c1(LETEXP(z0, z1)) S tuples: LETEXP(z0, z1) -> c(LETEXP(Cons(Cons(Nil, Nil), z0), z1)) GOAL(z0, z1) -> c1(LETEXP(z0, z1)) K tuples:none Defined Rule Symbols: letexp_2, goal_2 Defined Pair Symbols: LETEXP_2, GOAL_2 Compound Symbols: c_1, c1_1 ---------------------------------------- (31) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: GOAL(z0, z1) -> c1(LETEXP(z0, z1)) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: letexp(z0, z1) -> letexp(Cons(Cons(Nil, Nil), z0), z1) goal(z0, z1) -> letexp(z0, z1) Tuples: LETEXP(z0, z1) -> c(LETEXP(Cons(Cons(Nil, Nil), z0), z1)) S tuples: LETEXP(z0, z1) -> c(LETEXP(Cons(Cons(Nil, Nil), z0), z1)) K tuples:none Defined Rule Symbols: letexp_2, goal_2 Defined Pair Symbols: LETEXP_2 Compound Symbols: c_1 ---------------------------------------- (33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: letexp(z0, z1) -> letexp(Cons(Cons(Nil, Nil), z0), z1) goal(z0, z1) -> letexp(z0, z1) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LETEXP(z0, z1) -> c(LETEXP(Cons(Cons(Nil, Nil), z0), z1)) S tuples: LETEXP(z0, z1) -> c(LETEXP(Cons(Cons(Nil, Nil), z0), z1)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LETEXP_2 Compound Symbols: c_1 ---------------------------------------- (35) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace LETEXP(z0, z1) -> c(LETEXP(Cons(Cons(Nil, Nil), z0), z1)) by LETEXP(Cons(Cons(Nil, Nil), x0), x1) -> c(LETEXP(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x0)), x1)) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LETEXP(Cons(Cons(Nil, Nil), x0), x1) -> c(LETEXP(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x0)), x1)) S tuples: LETEXP(Cons(Cons(Nil, Nil), x0), x1) -> c(LETEXP(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x0)), x1)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LETEXP_2 Compound Symbols: c_1 ---------------------------------------- (37) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace LETEXP(Cons(Cons(Nil, Nil), x0), x1) -> c(LETEXP(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x0)), x1)) by LETEXP(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x0)), x1) -> c(LETEXP(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x0))), x1)) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LETEXP(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x0)), x1) -> c(LETEXP(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x0))), x1)) S tuples: LETEXP(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x0)), x1) -> c(LETEXP(Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), Cons(Cons(Nil, Nil), x0))), x1)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LETEXP_2 Compound Symbols: c_1