MAYBE proof of input_zeoOKBlUEA.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 818 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 218 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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 38 ms] (36) CdtProblem (37) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (50) 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: from(X) -> cons(X, from(s(X))) length(nil) -> 0 length(cons(X, Y)) -> s(length1(Y)) length1(X) -> length(X) 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: from(X) -> cons(X, from(s(X))) length(nil) -> 0' length(cons(X, Y)) -> s(length1(Y)) length1(X) -> length(X) 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: from(X) -> cons(X, from(s(X))) length(nil) -> 0 length(cons(X, Y)) -> s(length1(Y)) length1(X) -> length(X) 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: from(X) -> cons(X, from(s(X))) [1] length(nil) -> 0 [1] length(cons(X, Y)) -> s(length1(Y)) [1] length1(X) -> length(X) [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: from(X) -> cons(X, from(s(X))) [1] length(nil) -> 0 [1] length(cons(X, Y)) -> s(length1(Y)) [1] length1(X) -> length(X) [1] The TRS has the following type information: from :: s:0 -> cons:nil cons :: s:0 -> cons:nil -> cons:nil s :: s:0 -> s:0 length :: cons:nil -> s:0 nil :: cons:nil 0 :: s:0 length1 :: cons:nil -> s:0 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: from_1 length_1 length1_1 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (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: from(X) -> cons(X, from(s(X))) [1] length(nil) -> 0 [1] length(cons(X, Y)) -> s(length1(Y)) [1] length1(X) -> length(X) [1] The TRS has the following type information: from :: s:0 -> cons:nil cons :: s:0 -> cons:nil -> cons:nil s :: s:0 -> s:0 length :: cons:nil -> s:0 nil :: cons:nil 0 :: s:0 length1 :: cons:nil -> s:0 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: from(X) -> cons(X, from(s(X))) [1] length(nil) -> 0 [1] length(cons(X, Y)) -> s(length1(Y)) [1] length1(X) -> length(X) [1] The TRS has the following type information: from :: s:0 -> cons:nil cons :: s:0 -> cons:nil -> cons:nil s :: s:0 -> s:0 length :: cons:nil -> s:0 nil :: cons:nil 0 :: s:0 length1 :: cons:nil -> s:0 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 0 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 1 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length1(z) -{ 1 }-> length(X) :|: X >= 0, z = X ---------------------------------------- (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: from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 1 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length1(z) -{ 1 }-> length(z) :|: z >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { from } { length1, length } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 1 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length1(z) -{ 1 }-> length(z) :|: z >= 0 Function symbols to be analyzed: {from}, {length1,length} ---------------------------------------- (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: from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 1 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length1(z) -{ 1 }-> length(z) :|: z >= 0 Function symbols to be analyzed: {from}, {length1,length} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: from 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: from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 1 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length1(z) -{ 1 }-> length(z) :|: z >= 0 Function symbols to be analyzed: {from}, {length1,length} Previous analysis results are: from: runtime: ?, size: O(1) [0] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: from after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 1 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length1(z) -{ 1 }-> length(z) :|: z >= 0 Function symbols to be analyzed: {from}, {length1,length} Previous analysis results are: from: 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: none ---------------------------------------- (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: from(X) -> cons(X, from(s(X))) [1] length(nil) -> 0 [1] length(cons(X, Y)) -> s(length1(Y)) [1] length1(X) -> length(X) [1] The TRS has the following type information: from :: s:0 -> cons:nil cons :: s:0 -> cons:nil -> cons:nil s :: s:0 -> s:0 length :: cons:nil -> s:0 nil :: cons:nil 0 :: s:0 length1 :: cons:nil -> s:0 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 0 => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 1 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length1(z) -{ 1 }-> length(X) :|: X >= 0, z = X 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: from(z0) -> cons(z0, from(s(z0))) length(nil) -> 0 length(cons(z0, z1)) -> s(length1(z1)) length1(z0) -> length(z0) Tuples: FROM(z0) -> c(FROM(s(z0))) LENGTH(nil) -> c1 LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) LENGTH1(z0) -> c3(LENGTH(z0)) S tuples: FROM(z0) -> c(FROM(s(z0))) LENGTH(nil) -> c1 LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) LENGTH1(z0) -> c3(LENGTH(z0)) K tuples:none Defined Rule Symbols: from_1, length_1, length1_1 Defined Pair Symbols: FROM_1, LENGTH_1, LENGTH1_1 Compound Symbols: c_1, c1, c2_1, c3_1 ---------------------------------------- (31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: LENGTH(nil) -> c1 ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: from(z0) -> cons(z0, from(s(z0))) length(nil) -> 0 length(cons(z0, z1)) -> s(length1(z1)) length1(z0) -> length(z0) Tuples: FROM(z0) -> c(FROM(s(z0))) LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) LENGTH1(z0) -> c3(LENGTH(z0)) S tuples: FROM(z0) -> c(FROM(s(z0))) LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) LENGTH1(z0) -> c3(LENGTH(z0)) K tuples:none Defined Rule Symbols: from_1, length_1, length1_1 Defined Pair Symbols: FROM_1, LENGTH_1, LENGTH1_1 Compound Symbols: c_1, c2_1, c3_1 ---------------------------------------- (33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: from(z0) -> cons(z0, from(s(z0))) length(nil) -> 0 length(cons(z0, z1)) -> s(length1(z1)) length1(z0) -> length(z0) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(z0) -> c(FROM(s(z0))) LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) LENGTH1(z0) -> c3(LENGTH(z0)) S tuples: FROM(z0) -> c(FROM(s(z0))) LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) LENGTH1(z0) -> c3(LENGTH(z0)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: FROM_1, LENGTH_1, LENGTH1_1 Compound Symbols: c_1, c2_1, c3_1 ---------------------------------------- (35) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) We considered the (Usable) Rules:none And the Tuples: FROM(z0) -> c(FROM(s(z0))) LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) LENGTH1(z0) -> c3(LENGTH(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(FROM(x_1)) = x_1 POL(LENGTH(x_1)) = [1] + x_1 POL(LENGTH1(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(s(x_1)) = x_1 ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(z0) -> c(FROM(s(z0))) LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) LENGTH1(z0) -> c3(LENGTH(z0)) S tuples: FROM(z0) -> c(FROM(s(z0))) LENGTH1(z0) -> c3(LENGTH(z0)) K tuples: LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, LENGTH_1, LENGTH1_1 Compound Symbols: c_1, c2_1, c3_1 ---------------------------------------- (37) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: LENGTH1(z0) -> c3(LENGTH(z0)) LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(z0) -> c(FROM(s(z0))) LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) LENGTH1(z0) -> c3(LENGTH(z0)) S tuples: FROM(z0) -> c(FROM(s(z0))) K tuples: LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) LENGTH1(z0) -> c3(LENGTH(z0)) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, LENGTH_1, LENGTH1_1 Compound Symbols: c_1, c2_1, c3_1 ---------------------------------------- (39) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FROM(z0) -> c(FROM(s(z0))) by FROM(s(x0)) -> c(FROM(s(s(x0)))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) LENGTH1(z0) -> c3(LENGTH(z0)) FROM(s(x0)) -> c(FROM(s(s(x0)))) S tuples: FROM(s(x0)) -> c(FROM(s(s(x0)))) K tuples: LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) LENGTH1(z0) -> c3(LENGTH(z0)) Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1, FROM_1 Compound Symbols: c2_1, c3_1, c_1 ---------------------------------------- (41) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FROM(s(x0)) -> c(FROM(s(s(x0)))) by FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) LENGTH1(z0) -> c3(LENGTH(z0)) FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) S tuples: FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) K tuples: LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) LENGTH1(z0) -> c3(LENGTH(z0)) Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1, FROM_1 Compound Symbols: c2_1, c3_1, c_1 ---------------------------------------- (43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH1(z0) -> c3(LENGTH(z0)) by LENGTH1(cons(y0, y1)) -> c3(LENGTH(cons(y0, y1))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) LENGTH1(cons(y0, y1)) -> c3(LENGTH(cons(y0, y1))) S tuples: FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) K tuples: LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) LENGTH1(cons(y0, y1)) -> c3(LENGTH(cons(y0, y1))) Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, FROM_1, LENGTH1_1 Compound Symbols: c2_1, c_1, c3_1 ---------------------------------------- (45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1)) by LENGTH(cons(z0, cons(y0, y1))) -> c2(LENGTH1(cons(y0, y1))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) LENGTH1(cons(y0, y1)) -> c3(LENGTH(cons(y0, y1))) LENGTH(cons(z0, cons(y0, y1))) -> c2(LENGTH1(cons(y0, y1))) S tuples: FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) K tuples: LENGTH1(cons(y0, y1)) -> c3(LENGTH(cons(y0, y1))) LENGTH(cons(z0, cons(y0, y1))) -> c2(LENGTH1(cons(y0, y1))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, LENGTH1_1, LENGTH_1 Compound Symbols: c_1, c3_1, c2_1 ---------------------------------------- (47) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH1(cons(y0, y1)) -> c3(LENGTH(cons(y0, y1))) by LENGTH1(cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) LENGTH(cons(z0, cons(y0, y1))) -> c2(LENGTH1(cons(y0, y1))) LENGTH1(cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) S tuples: FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) K tuples: LENGTH(cons(z0, cons(y0, y1))) -> c2(LENGTH1(cons(y0, y1))) LENGTH1(cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, LENGTH_1, LENGTH1_1 Compound Symbols: c_1, c2_1, c3_1 ---------------------------------------- (49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(cons(z0, cons(y0, y1))) -> c2(LENGTH1(cons(y0, y1))) by LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LENGTH1(cons(z1, cons(y1, y2)))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) LENGTH1(cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LENGTH1(cons(z1, cons(y1, y2)))) S tuples: FROM(s(s(x0))) -> c(FROM(s(s(s(x0))))) K tuples: LENGTH1(cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LENGTH1(cons(z1, cons(y1, y2)))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, LENGTH1_1, LENGTH_1 Compound Symbols: c_1, c3_1, c2_1