MAYBE proof of input_UoNUC5qTdL.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 92 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 45 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 156 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 65 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 141 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 54 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 149 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 42 ms] (42) CpxRNTS (43) CompletionProof [UPPER BOUND(ID), 0 ms] (44) CpxTypedWeightedCompleteTrs (45) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (48) CdtProblem (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (52) 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: eq -> true eq -> eq eq -> false inf(X) -> cons take(0, X) -> nil take(s, cons) -> cons length(nil) -> 0 length(cons) -> s 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: eq -> true eq -> eq eq -> false inf(X) -> cons take(0', X) -> nil take(s, cons) -> cons length(nil) -> 0' length(cons) -> s 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: eq -> true eq -> eq eq -> false inf(X) -> cons take(0, X) -> nil take(s, cons) -> cons length(nil) -> 0 length(cons) -> s 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: eq -> true [1] eq -> eq [1] eq -> false [1] inf(X) -> cons [1] take(0, X) -> nil [1] take(s, cons) -> cons [1] length(nil) -> 0 [1] length(cons) -> s [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: eq -> true [1] eq -> eq [1] eq -> false [1] inf(X) -> cons [1] take(0, X) -> nil [1] take(s, cons) -> cons [1] length(nil) -> 0 [1] length(cons) -> s [1] The TRS has the following type information: eq :: true:false true :: true:false false :: true:false inf :: a -> cons:nil cons :: cons:nil take :: 0:s -> cons:nil -> cons:nil 0 :: 0:s nil :: cons:nil s :: 0:s length :: cons:nil -> 0:s 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: eq inf_1 take_2 length_1 (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: eq -> true [1] eq -> eq [1] eq -> false [1] inf(X) -> cons [1] take(0, X) -> nil [1] take(s, cons) -> cons [1] length(nil) -> 0 [1] length(cons) -> s [1] The TRS has the following type information: eq :: true:false true :: true:false false :: true:false inf :: a -> cons:nil cons :: cons:nil take :: 0:s -> cons:nil -> cons:nil 0 :: 0:s nil :: cons:nil s :: 0:s length :: cons:nil -> 0:s 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: eq -> true [1] eq -> eq [1] eq -> false [1] inf(X) -> cons [1] take(0, X) -> nil [1] take(s, cons) -> cons [1] length(nil) -> 0 [1] length(cons) -> s [1] The TRS has the following type information: eq :: true:false true :: true:false false :: true:false inf :: a -> cons:nil cons :: cons:nil take :: 0:s -> cons:nil -> cons:nil 0 :: 0:s nil :: cons:nil s :: 0:s length :: cons:nil -> 0:s 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: true => 1 false => 0 cons => 0 0 => 0 nil => 1 s => 1 const => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: eq -{ 1 }-> eq :|: eq -{ 1 }-> 1 :|: eq -{ 1 }-> 0 :|: inf(z) -{ 1 }-> 0 :|: X >= 0, z = X length(z) -{ 1 }-> 1 :|: z = 0 length(z) -{ 1 }-> 0 :|: z = 1 take(z, z') -{ 1 }-> 1 :|: z' = X, X >= 0, z = 0 take(z, z') -{ 1 }-> 0 :|: z = 1, 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: eq -{ 1 }-> eq :|: eq -{ 1 }-> 1 :|: eq -{ 1 }-> 0 :|: inf(z) -{ 1 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 :|: z = 0 length(z) -{ 1 }-> 0 :|: z = 1 take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { inf } { take } { length } { eq } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: eq -{ 1 }-> eq :|: eq -{ 1 }-> 1 :|: eq -{ 1 }-> 0 :|: inf(z) -{ 1 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 :|: z = 0 length(z) -{ 1 }-> 0 :|: z = 1 take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0 Function symbols to be analyzed: {inf}, {take}, {length}, {eq} ---------------------------------------- (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: eq -{ 1 }-> eq :|: eq -{ 1 }-> 1 :|: eq -{ 1 }-> 0 :|: inf(z) -{ 1 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 :|: z = 0 length(z) -{ 1 }-> 0 :|: z = 1 take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0 Function symbols to be analyzed: {inf}, {take}, {length}, {eq} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: inf 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: eq -{ 1 }-> eq :|: eq -{ 1 }-> 1 :|: eq -{ 1 }-> 0 :|: inf(z) -{ 1 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 :|: z = 0 length(z) -{ 1 }-> 0 :|: z = 1 take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0 Function symbols to be analyzed: {inf}, {take}, {length}, {eq} Previous analysis results are: inf: runtime: ?, size: O(1) [0] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: inf after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: eq -{ 1 }-> eq :|: eq -{ 1 }-> 1 :|: eq -{ 1 }-> 0 :|: inf(z) -{ 1 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 :|: z = 0 length(z) -{ 1 }-> 0 :|: z = 1 take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0 Function symbols to be analyzed: {take}, {length}, {eq} Previous analysis results are: inf: runtime: O(1) [1], size: O(1) [0] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: eq -{ 1 }-> eq :|: eq -{ 1 }-> 1 :|: eq -{ 1 }-> 0 :|: inf(z) -{ 1 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 :|: z = 0 length(z) -{ 1 }-> 0 :|: z = 1 take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0 Function symbols to be analyzed: {take}, {length}, {eq} Previous analysis results are: inf: runtime: O(1) [1], size: O(1) [0] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: take after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: eq -{ 1 }-> eq :|: eq -{ 1 }-> 1 :|: eq -{ 1 }-> 0 :|: inf(z) -{ 1 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 :|: z = 0 length(z) -{ 1 }-> 0 :|: z = 1 take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0 Function symbols to be analyzed: {take}, {length}, {eq} Previous analysis results are: inf: runtime: O(1) [1], size: O(1) [0] take: runtime: ?, size: O(1) [1] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: take after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: eq -{ 1 }-> eq :|: eq -{ 1 }-> 1 :|: eq -{ 1 }-> 0 :|: inf(z) -{ 1 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 :|: z = 0 length(z) -{ 1 }-> 0 :|: z = 1 take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0 Function symbols to be analyzed: {length}, {eq} Previous analysis results are: inf: runtime: O(1) [1], size: O(1) [0] take: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: eq -{ 1 }-> eq :|: eq -{ 1 }-> 1 :|: eq -{ 1 }-> 0 :|: inf(z) -{ 1 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 :|: z = 0 length(z) -{ 1 }-> 0 :|: z = 1 take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0 Function symbols to be analyzed: {length}, {eq} Previous analysis results are: inf: runtime: O(1) [1], size: O(1) [0] take: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: length after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: eq -{ 1 }-> eq :|: eq -{ 1 }-> 1 :|: eq -{ 1 }-> 0 :|: inf(z) -{ 1 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 :|: z = 0 length(z) -{ 1 }-> 0 :|: z = 1 take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0 Function symbols to be analyzed: {length}, {eq} Previous analysis results are: inf: runtime: O(1) [1], size: O(1) [0] take: runtime: O(1) [1], size: O(1) [1] length: runtime: ?, size: O(1) [1] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: length after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: eq -{ 1 }-> eq :|: eq -{ 1 }-> 1 :|: eq -{ 1 }-> 0 :|: inf(z) -{ 1 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 :|: z = 0 length(z) -{ 1 }-> 0 :|: z = 1 take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0 Function symbols to be analyzed: {eq} Previous analysis results are: inf: runtime: O(1) [1], size: O(1) [0] take: runtime: O(1) [1], size: O(1) [1] length: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: eq -{ 1 }-> eq :|: eq -{ 1 }-> 1 :|: eq -{ 1 }-> 0 :|: inf(z) -{ 1 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 :|: z = 0 length(z) -{ 1 }-> 0 :|: z = 1 take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0 Function symbols to be analyzed: {eq} Previous analysis results are: inf: runtime: O(1) [1], size: O(1) [0] take: runtime: O(1) [1], size: O(1) [1] length: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: eq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: eq -{ 1 }-> eq :|: eq -{ 1 }-> 1 :|: eq -{ 1 }-> 0 :|: inf(z) -{ 1 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 :|: z = 0 length(z) -{ 1 }-> 0 :|: z = 1 take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0 Function symbols to be analyzed: {eq} Previous analysis results are: inf: runtime: O(1) [1], size: O(1) [0] take: runtime: O(1) [1], size: O(1) [1] length: runtime: O(1) [1], size: O(1) [1] eq: runtime: ?, size: O(1) [1] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: eq after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: eq -{ 1 }-> eq :|: eq -{ 1 }-> 1 :|: eq -{ 1 }-> 0 :|: inf(z) -{ 1 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 :|: z = 0 length(z) -{ 1 }-> 0 :|: z = 1 take(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 take(z, z') -{ 1 }-> 0 :|: z = 1, z' = 0 Function symbols to be analyzed: {eq} Previous analysis results are: inf: runtime: O(1) [1], size: O(1) [0] take: runtime: O(1) [1], size: O(1) [1] length: runtime: O(1) [1], size: O(1) [1] eq: runtime: INF, size: O(1) [1] ---------------------------------------- (43) 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: take(v0, v1) -> null_take [0] length(v0) -> null_length [0] And the following fresh constants: null_take, null_length, const ---------------------------------------- (44) 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: eq -> true [1] eq -> eq [1] eq -> false [1] inf(X) -> cons [1] take(0, X) -> nil [1] take(s, cons) -> cons [1] length(nil) -> 0 [1] length(cons) -> s [1] take(v0, v1) -> null_take [0] length(v0) -> null_length [0] The TRS has the following type information: eq :: true:false true :: true:false false :: true:false inf :: a -> cons:nil:null_take cons :: cons:nil:null_take take :: 0:s:null_length -> cons:nil:null_take -> cons:nil:null_take 0 :: 0:s:null_length nil :: cons:nil:null_take s :: 0:s:null_length length :: cons:nil:null_take -> 0:s:null_length null_take :: cons:nil:null_take null_length :: 0:s:null_length const :: a Rewrite Strategy: INNERMOST ---------------------------------------- (45) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: true => 1 false => 0 cons => 1 0 => 1 nil => 2 s => 2 null_take => 0 null_length => 0 const => 0 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: eq -{ 1 }-> eq :|: eq -{ 1 }-> 1 :|: eq -{ 1 }-> 0 :|: inf(z) -{ 1 }-> 1 :|: X >= 0, z = X length(z) -{ 1 }-> 2 :|: z = 1 length(z) -{ 1 }-> 1 :|: z = 2 length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 take(z, z') -{ 1 }-> 2 :|: z' = X, z = 1, X >= 0 take(z, z') -{ 1 }-> 1 :|: z = 2, z' = 1 take(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (47) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: eq -> true eq -> eq eq -> false inf(z0) -> cons take(0, z0) -> nil take(s, cons) -> cons length(nil) -> 0 length(cons) -> s Tuples: EQ -> c EQ -> c1(EQ) EQ -> c2 INF(z0) -> c3 TAKE(0, z0) -> c4 TAKE(s, cons) -> c5 LENGTH(nil) -> c6 LENGTH(cons) -> c7 S tuples: EQ -> c EQ -> c1(EQ) EQ -> c2 INF(z0) -> c3 TAKE(0, z0) -> c4 TAKE(s, cons) -> c5 LENGTH(nil) -> c6 LENGTH(cons) -> c7 K tuples:none Defined Rule Symbols: eq, inf_1, take_2, length_1 Defined Pair Symbols: EQ, INF_1, TAKE_2, LENGTH_1 Compound Symbols: c, c1_1, c2, c3, c4, c5, c6, c7 ---------------------------------------- (49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: LENGTH(nil) -> c6 TAKE(s, cons) -> c5 LENGTH(cons) -> c7 TAKE(0, z0) -> c4 EQ -> c2 EQ -> c INF(z0) -> c3 ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: eq -> true eq -> eq eq -> false inf(z0) -> cons take(0, z0) -> nil take(s, cons) -> cons length(nil) -> 0 length(cons) -> s Tuples: EQ -> c1(EQ) S tuples: EQ -> c1(EQ) K tuples:none Defined Rule Symbols: eq, inf_1, take_2, length_1 Defined Pair Symbols: EQ Compound Symbols: c1_1 ---------------------------------------- (51) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: eq -> true eq -> eq eq -> false inf(z0) -> cons take(0, z0) -> nil take(s, cons) -> cons length(nil) -> 0 length(cons) -> s ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: EQ -> c1(EQ) S tuples: EQ -> c1(EQ) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: EQ Compound Symbols: c1_1