KILLED proof of input_RZXA1D64mm.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) InliningProof [UPPER BOUND(ID), 198 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 185 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 56 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 66 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 3467 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 580 ms] (38) CpxRNTS (39) CompletionProof [UPPER BOUND(ID), 0 ms] (40) CpxTypedWeightedCompleteTrs (41) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (44) CdtProblem (45) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 1348 ms] (82) CdtProblem (83) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 1545 ms] (84) CdtProblem (85) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (86) 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: zeros -> cons(0, n__zeros) U11(tt, L) -> U12(tt, activate(L)) U12(tt, L) -> s(length(activate(L))) length(nil) -> 0 length(cons(N, L)) -> U11(tt, activate(L)) zeros -> n__zeros activate(n__zeros) -> zeros activate(X) -> 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: zeros -> cons(0', n__zeros) U11(tt, L) -> U12(tt, activate(L)) U12(tt, L) -> s(length(activate(L))) length(nil) -> 0' length(cons(N, L)) -> U11(tt, activate(L)) zeros -> n__zeros activate(n__zeros) -> zeros activate(X) -> 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: zeros -> cons(0, n__zeros) U11(tt, L) -> U12(tt, activate(L)) U12(tt, L) -> s(length(activate(L))) length(nil) -> 0 length(cons(N, L)) -> U11(tt, activate(L)) zeros -> n__zeros activate(n__zeros) -> zeros activate(X) -> 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: zeros -> cons(0, n__zeros) [1] U11(tt, L) -> U12(tt, activate(L)) [1] U12(tt, L) -> s(length(activate(L))) [1] length(nil) -> 0 [1] length(cons(N, L)) -> U11(tt, activate(L)) [1] zeros -> n__zeros [1] activate(n__zeros) -> zeros [1] activate(X) -> 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: zeros -> cons(0, n__zeros) [1] U11(tt, L) -> U12(tt, activate(L)) [1] U12(tt, L) -> s(length(activate(L))) [1] length(nil) -> 0 [1] length(cons(N, L)) -> U11(tt, activate(L)) [1] zeros -> n__zeros [1] activate(n__zeros) -> zeros [1] activate(X) -> X [1] The TRS has the following type information: zeros :: n__zeros:cons:nil cons :: 0:s -> n__zeros:cons:nil -> n__zeros:cons:nil 0 :: 0:s n__zeros :: n__zeros:cons:nil U11 :: tt -> n__zeros:cons:nil -> 0:s tt :: tt U12 :: tt -> n__zeros:cons:nil -> 0:s activate :: n__zeros:cons:nil -> n__zeros:cons:nil s :: 0:s -> 0:s length :: n__zeros:cons:nil -> 0:s nil :: n__zeros:cons:nil 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: U11_2 U12_2 length_1 (c) The following functions are completely defined: activate_1 zeros 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: zeros -> cons(0, n__zeros) [1] U11(tt, L) -> U12(tt, activate(L)) [1] U12(tt, L) -> s(length(activate(L))) [1] length(nil) -> 0 [1] length(cons(N, L)) -> U11(tt, activate(L)) [1] zeros -> n__zeros [1] activate(n__zeros) -> zeros [1] activate(X) -> X [1] The TRS has the following type information: zeros :: n__zeros:cons:nil cons :: 0:s -> n__zeros:cons:nil -> n__zeros:cons:nil 0 :: 0:s n__zeros :: n__zeros:cons:nil U11 :: tt -> n__zeros:cons:nil -> 0:s tt :: tt U12 :: tt -> n__zeros:cons:nil -> 0:s activate :: n__zeros:cons:nil -> n__zeros:cons:nil s :: 0:s -> 0:s length :: n__zeros:cons:nil -> 0:s nil :: n__zeros:cons:nil 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: zeros -> cons(0, n__zeros) [1] U11(tt, n__zeros) -> U12(tt, zeros) [2] U11(tt, L) -> U12(tt, L) [2] U12(tt, n__zeros) -> s(length(zeros)) [2] U12(tt, L) -> s(length(L)) [2] length(nil) -> 0 [1] length(cons(N, n__zeros)) -> U11(tt, zeros) [2] length(cons(N, L)) -> U11(tt, L) [2] zeros -> n__zeros [1] activate(n__zeros) -> zeros [1] activate(X) -> X [1] The TRS has the following type information: zeros :: n__zeros:cons:nil cons :: 0:s -> n__zeros:cons:nil -> n__zeros:cons:nil 0 :: 0:s n__zeros :: n__zeros:cons:nil U11 :: tt -> n__zeros:cons:nil -> 0:s tt :: tt U12 :: tt -> n__zeros:cons:nil -> 0:s activate :: n__zeros:cons:nil -> n__zeros:cons:nil s :: 0:s -> 0:s length :: n__zeros:cons:nil -> 0:s nil :: n__zeros:cons:nil 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: 0 => 0 n__zeros => 0 tt => 0 nil => 1 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: U11(z, z') -{ 2 }-> U12(0, L) :|: L >= 0, z = 0, z' = L U11(z, z') -{ 2 }-> U12(0, zeros) :|: z = 0, z' = 0 U12(z, z') -{ 2 }-> 1 + length(L) :|: L >= 0, z = 0, z' = L U12(z, z') -{ 2 }-> 1 + length(zeros) :|: z = 0, z' = 0 activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> zeros :|: z = 0 length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 2 }-> U11(0, zeros) :|: z = 1 + N + 0, N >= 0 length(z) -{ 1 }-> 0 :|: z = 1 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: zeros -{ 1 }-> 1 + 0 + 0 :|: zeros -{ 1 }-> 0 :|: ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: U11(z, z') -{ 2 }-> U12(0, L) :|: L >= 0, z = 0, z' = L U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0 U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0 U12(z, z') -{ 2 }-> 1 + length(L) :|: L >= 0, z = 0, z' = L U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0 U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0 activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 3 }-> U11(0, 0) :|: z = 1 + N + 0, N >= 0 length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z = 1 + N + 0, N >= 0 length(z) -{ 1 }-> 0 :|: z = 1 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0 U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0 U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0 U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0 U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0 U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0 length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0 length(z) -{ 1 }-> 0 :|: z = 1 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { activate } { zeros } { length, U12, U11 } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0 U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0 U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0 U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0 U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0 U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0 length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0 length(z) -{ 1 }-> 0 :|: z = 1 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {activate}, {zeros}, {length,U12,U11} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0 U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0 U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0 U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0 U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0 U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0 length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0 length(z) -{ 1 }-> 0 :|: z = 1 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {activate}, {zeros}, {length,U12,U11} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: activate after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0 U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0 U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0 U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0 U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0 U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0 length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0 length(z) -{ 1 }-> 0 :|: z = 1 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {activate}, {zeros}, {length,U12,U11} Previous analysis results are: activate: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: activate after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0 U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0 U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0 U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0 U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0 U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0 length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0 length(z) -{ 1 }-> 0 :|: z = 1 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {zeros}, {length,U12,U11} Previous analysis results are: activate: runtime: O(1) [2], size: O(n^1) [1 + z] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0 U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0 U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0 U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0 U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0 U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0 length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0 length(z) -{ 1 }-> 0 :|: z = 1 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {zeros}, {length,U12,U11} Previous analysis results are: activate: runtime: O(1) [2], size: O(n^1) [1 + z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: zeros 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: U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0 U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0 U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0 U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0 U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0 U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0 length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0 length(z) -{ 1 }-> 0 :|: z = 1 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {zeros}, {length,U12,U11} Previous analysis results are: activate: runtime: O(1) [2], size: O(n^1) [1 + z] zeros: runtime: ?, size: O(1) [1] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: zeros after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0 U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0 U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0 U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0 U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0 U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0 length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0 length(z) -{ 1 }-> 0 :|: z = 1 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {length,U12,U11} Previous analysis results are: activate: runtime: O(1) [2], size: O(n^1) [1 + z] zeros: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0 U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0 U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0 U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0 U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0 U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0 length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0 length(z) -{ 1 }-> 0 :|: z = 1 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {length,U12,U11} Previous analysis results are: activate: runtime: O(1) [2], size: O(n^1) [1 + z] zeros: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: length after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: U12 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: U11 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0 U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0 U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0 U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0 U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0 U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0 length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0 length(z) -{ 1 }-> 0 :|: z = 1 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {length,U12,U11} Previous analysis results are: activate: runtime: O(1) [2], size: O(n^1) [1 + z] zeros: runtime: O(1) [1], size: O(1) [1] length: runtime: ?, size: INF U12: runtime: ?, size: INF U11: runtime: ?, size: INF ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: length after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0 U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0 U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0 U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0 U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0 U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0 length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0 length(z) -{ 1 }-> 0 :|: z = 1 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {length,U12,U11} Previous analysis results are: activate: runtime: O(1) [2], size: O(n^1) [1 + z] zeros: runtime: O(1) [1], size: O(1) [1] length: runtime: INF, size: INF U12: runtime: ?, size: INF U11: runtime: ?, size: INF ---------------------------------------- (39) 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: length(v0) -> null_length [0] And the following fresh constants: null_length ---------------------------------------- (40) 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: zeros -> cons(0, n__zeros) [1] U11(tt, L) -> U12(tt, activate(L)) [1] U12(tt, L) -> s(length(activate(L))) [1] length(nil) -> 0 [1] length(cons(N, L)) -> U11(tt, activate(L)) [1] zeros -> n__zeros [1] activate(n__zeros) -> zeros [1] activate(X) -> X [1] length(v0) -> null_length [0] The TRS has the following type information: zeros :: n__zeros:cons:nil cons :: 0:s:null_length -> n__zeros:cons:nil -> n__zeros:cons:nil 0 :: 0:s:null_length n__zeros :: n__zeros:cons:nil U11 :: tt -> n__zeros:cons:nil -> 0:s:null_length tt :: tt U12 :: tt -> n__zeros:cons:nil -> 0:s:null_length activate :: n__zeros:cons:nil -> n__zeros:cons:nil s :: 0:s:null_length -> 0:s:null_length length :: n__zeros:cons:nil -> 0:s:null_length nil :: n__zeros:cons:nil null_length :: 0:s:null_length Rewrite Strategy: INNERMOST ---------------------------------------- (41) 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 n__zeros => 0 tt => 0 nil => 1 null_length => 0 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: U11(z, z') -{ 1 }-> U12(0, activate(L)) :|: L >= 0, z = 0, z' = L U12(z, z') -{ 1 }-> 1 + length(activate(L)) :|: L >= 0, z = 0, z' = L activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> zeros :|: z = 0 length(z) -{ 1 }-> U11(0, activate(L)) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (43) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: zeros -> cons(0, n__zeros) zeros -> n__zeros U11(tt, z0) -> U12(tt, activate(z0)) U12(tt, z0) -> s(length(activate(z0))) length(nil) -> 0 length(cons(z0, z1)) -> U11(tt, activate(z1)) activate(n__zeros) -> zeros activate(z0) -> z0 Tuples: ZEROS -> c ZEROS -> c1 U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0)) U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0)) LENGTH(nil) -> c4 LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)), ACTIVATE(z1)) ACTIVATE(n__zeros) -> c6(ZEROS) ACTIVATE(z0) -> c7 S tuples: ZEROS -> c ZEROS -> c1 U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0)) U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0)) LENGTH(nil) -> c4 LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)), ACTIVATE(z1)) ACTIVATE(n__zeros) -> c6(ZEROS) ACTIVATE(z0) -> c7 K tuples:none Defined Rule Symbols: zeros, U11_2, U12_2, length_1, activate_1 Defined Pair Symbols: ZEROS, U11'_2, U12'_2, LENGTH_1, ACTIVATE_1 Compound Symbols: c, c1, c2_2, c3_2, c4, c5_2, c6_1, c7 ---------------------------------------- (45) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing nodes: ZEROS -> c1 ZEROS -> c ACTIVATE(z0) -> c7 ACTIVATE(n__zeros) -> c6(ZEROS) LENGTH(nil) -> c4 ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: zeros -> cons(0, n__zeros) zeros -> n__zeros U11(tt, z0) -> U12(tt, activate(z0)) U12(tt, z0) -> s(length(activate(z0))) length(nil) -> 0 length(cons(z0, z1)) -> U11(tt, activate(z1)) activate(n__zeros) -> zeros activate(z0) -> z0 Tuples: U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0)) U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0)) LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)), ACTIVATE(z1)) S tuples: U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0)) U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0)) LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)), ACTIVATE(z1)) K tuples:none Defined Rule Symbols: zeros, U11_2, U12_2, length_1, activate_1 Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_2, c3_2, c5_2 ---------------------------------------- (47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: zeros -> cons(0, n__zeros) zeros -> n__zeros U11(tt, z0) -> U12(tt, activate(z0)) U12(tt, z0) -> s(length(activate(z0))) length(nil) -> 0 length(cons(z0, z1)) -> U11(tt, activate(z1)) activate(n__zeros) -> zeros activate(z0) -> z0 Tuples: U11'(tt, z0) -> c2(U12'(tt, activate(z0))) U12'(tt, z0) -> c3(LENGTH(activate(z0))) LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1))) S tuples: U11'(tt, z0) -> c2(U12'(tt, activate(z0))) U12'(tt, z0) -> c3(LENGTH(activate(z0))) LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1))) K tuples:none Defined Rule Symbols: zeros, U11_2, U12_2, length_1, activate_1 Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_1, c3_1, c5_1 ---------------------------------------- (49) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: U11(tt, z0) -> U12(tt, activate(z0)) U12(tt, z0) -> s(length(activate(z0))) length(nil) -> 0 length(cons(z0, z1)) -> U11(tt, activate(z1)) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros Tuples: U11'(tt, z0) -> c2(U12'(tt, activate(z0))) U12'(tt, z0) -> c3(LENGTH(activate(z0))) LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1))) S tuples: U11'(tt, z0) -> c2(U12'(tt, activate(z0))) U12'(tt, z0) -> c3(LENGTH(activate(z0))) LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1))) K tuples:none Defined Rule Symbols: activate_1, zeros Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_1, c3_1, c5_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace U11'(tt, z0) -> c2(U12'(tt, activate(z0))) by U11'(tt, n__zeros) -> c2(U12'(tt, zeros)) U11'(tt, z0) -> c2(U12'(tt, z0)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros Tuples: U12'(tt, z0) -> c3(LENGTH(activate(z0))) LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1))) U11'(tt, n__zeros) -> c2(U12'(tt, zeros)) U11'(tt, z0) -> c2(U12'(tt, z0)) S tuples: U12'(tt, z0) -> c3(LENGTH(activate(z0))) LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1))) U11'(tt, n__zeros) -> c2(U12'(tt, zeros)) U11'(tt, z0) -> c2(U12'(tt, z0)) K tuples:none Defined Rule Symbols: activate_1, zeros Defined Pair Symbols: U12'_2, LENGTH_1, U11'_2 Compound Symbols: c3_1, c5_1, c2_1 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace U12'(tt, z0) -> c3(LENGTH(activate(z0))) by U12'(tt, n__zeros) -> c3(LENGTH(zeros)) U12'(tt, z0) -> c3(LENGTH(z0)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros Tuples: LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1))) U11'(tt, n__zeros) -> c2(U12'(tt, zeros)) U11'(tt, z0) -> c2(U12'(tt, z0)) U12'(tt, n__zeros) -> c3(LENGTH(zeros)) U12'(tt, z0) -> c3(LENGTH(z0)) S tuples: LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1))) U11'(tt, n__zeros) -> c2(U12'(tt, zeros)) U11'(tt, z0) -> c2(U12'(tt, z0)) U12'(tt, n__zeros) -> c3(LENGTH(zeros)) U12'(tt, z0) -> c3(LENGTH(z0)) K tuples:none Defined Rule Symbols: activate_1, zeros Defined Pair Symbols: LENGTH_1, U11'_2, U12'_2 Compound Symbols: c5_1, c2_1, c3_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1))) by LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros Tuples: U11'(tt, n__zeros) -> c2(U12'(tt, zeros)) U11'(tt, z0) -> c2(U12'(tt, z0)) U12'(tt, n__zeros) -> c3(LENGTH(zeros)) U12'(tt, z0) -> c3(LENGTH(z0)) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) S tuples: U11'(tt, n__zeros) -> c2(U12'(tt, zeros)) U11'(tt, z0) -> c2(U12'(tt, z0)) U12'(tt, n__zeros) -> c3(LENGTH(zeros)) U12'(tt, z0) -> c3(LENGTH(z0)) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) K tuples:none Defined Rule Symbols: activate_1, zeros Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_1, c3_1, c5_1 ---------------------------------------- (57) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: activate(n__zeros) -> zeros activate(z0) -> z0 ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: zeros -> cons(0, n__zeros) zeros -> n__zeros Tuples: U11'(tt, n__zeros) -> c2(U12'(tt, zeros)) U11'(tt, z0) -> c2(U12'(tt, z0)) U12'(tt, n__zeros) -> c3(LENGTH(zeros)) U12'(tt, z0) -> c3(LENGTH(z0)) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) S tuples: U11'(tt, n__zeros) -> c2(U12'(tt, zeros)) U11'(tt, z0) -> c2(U12'(tt, z0)) U12'(tt, n__zeros) -> c3(LENGTH(zeros)) U12'(tt, z0) -> c3(LENGTH(z0)) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) K tuples:none Defined Rule Symbols: zeros Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_1, c3_1, c5_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace U11'(tt, n__zeros) -> c2(U12'(tt, zeros)) by U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: zeros -> cons(0, n__zeros) zeros -> n__zeros Tuples: U11'(tt, z0) -> c2(U12'(tt, z0)) U12'(tt, n__zeros) -> c3(LENGTH(zeros)) U12'(tt, z0) -> c3(LENGTH(z0)) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) S tuples: U11'(tt, z0) -> c2(U12'(tt, z0)) U12'(tt, n__zeros) -> c3(LENGTH(zeros)) U12'(tt, z0) -> c3(LENGTH(z0)) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) K tuples:none Defined Rule Symbols: zeros Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_1, c3_1, c5_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace U12'(tt, n__zeros) -> c3(LENGTH(zeros)) by U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) U12'(tt, n__zeros) -> c3(LENGTH(n__zeros)) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: zeros -> cons(0, n__zeros) zeros -> n__zeros Tuples: U11'(tt, z0) -> c2(U12'(tt, z0)) U12'(tt, z0) -> c3(LENGTH(z0)) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) U12'(tt, n__zeros) -> c3(LENGTH(n__zeros)) S tuples: U11'(tt, z0) -> c2(U12'(tt, z0)) U12'(tt, z0) -> c3(LENGTH(z0)) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) U12'(tt, n__zeros) -> c3(LENGTH(n__zeros)) K tuples:none Defined Rule Symbols: zeros Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_1, c3_1, c5_1 ---------------------------------------- (63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: U12'(tt, n__zeros) -> c3(LENGTH(n__zeros)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: zeros -> cons(0, n__zeros) zeros -> n__zeros Tuples: U11'(tt, z0) -> c2(U12'(tt, z0)) U12'(tt, z0) -> c3(LENGTH(z0)) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) S tuples: U11'(tt, z0) -> c2(U12'(tt, z0)) U12'(tt, z0) -> c3(LENGTH(z0)) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) K tuples:none Defined Rule Symbols: zeros Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_1, c3_1, c5_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros)) by LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: zeros -> cons(0, n__zeros) zeros -> n__zeros Tuples: U11'(tt, z0) -> c2(U12'(tt, z0)) U12'(tt, z0) -> c3(LENGTH(z0)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) S tuples: U11'(tt, z0) -> c2(U12'(tt, z0)) U12'(tt, z0) -> c3(LENGTH(z0)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) K tuples:none Defined Rule Symbols: zeros Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_1, c3_1, c5_1 ---------------------------------------- (67) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: zeros -> cons(0, n__zeros) zeros -> n__zeros ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: U11'(tt, z0) -> c2(U12'(tt, z0)) U12'(tt, z0) -> c3(LENGTH(z0)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) S tuples: U11'(tt, z0) -> c2(U12'(tt, z0)) U12'(tt, z0) -> c3(LENGTH(z0)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_1, c3_1, c5_1 ---------------------------------------- (69) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace U12'(tt, z0) -> c3(LENGTH(z0)) by U12'(tt, cons(y0, y1)) -> c3(LENGTH(cons(y0, y1))) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: U11'(tt, z0) -> c2(U12'(tt, z0)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, y1)) -> c3(LENGTH(cons(y0, y1))) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) S tuples: U11'(tt, z0) -> c2(U12'(tt, z0)) LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, y1)) -> c3(LENGTH(cons(y0, y1))) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: U11'_2, LENGTH_1, U12'_2 Compound Symbols: c2_1, c5_1, c3_1 ---------------------------------------- (71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace U11'(tt, z0) -> c2(U12'(tt, z0)) by U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U11'(tt, cons(y0, y1)) -> c2(U12'(tt, cons(y0, y1))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, y1)) -> c3(LENGTH(cons(y0, y1))) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, y1)) -> c2(U12'(tt, cons(y0, y1))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) S tuples: LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, y1)) -> c3(LENGTH(cons(y0, y1))) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, y1)) -> c2(U12'(tt, cons(y0, y1))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, U11'_2, U12'_2 Compound Symbols: c5_1, c2_1, c3_1 ---------------------------------------- (73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) by LENGTH(cons(z0, n__zeros)) -> c5(U11'(tt, n__zeros)) LENGTH(cons(z0, cons(y0, y1))) -> c5(U11'(tt, cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, y1)) -> c3(LENGTH(cons(y0, y1))) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, y1)) -> c2(U12'(tt, cons(y0, y1))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(y0, y1))) -> c5(U11'(tt, cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) S tuples: U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, y1)) -> c3(LENGTH(cons(y0, y1))) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, y1)) -> c2(U12'(tt, cons(y0, y1))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(y0, y1))) -> c5(U11'(tt, cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_1, c3_1, c5_1 ---------------------------------------- (75) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace U12'(tt, cons(y0, y1)) -> c3(LENGTH(cons(y0, y1))) by U12'(tt, cons(z0, n__zeros)) -> c3(LENGTH(cons(z0, n__zeros))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, y1)) -> c2(U12'(tt, cons(y0, y1))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(y0, y1))) -> c5(U11'(tt, cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) S tuples: U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, y1)) -> c2(U12'(tt, cons(y0, y1))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(y0, y1))) -> c5(U11'(tt, cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_1, c3_1, c5_1 ---------------------------------------- (77) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace U11'(tt, cons(y0, y1)) -> c2(U12'(tt, cons(y0, y1))) by U11'(tt, cons(z0, n__zeros)) -> c2(U12'(tt, cons(z0, n__zeros))) U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2)))) U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros)))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(y0, y1))) -> c5(U11'(tt, cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2)))) U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros)))) S tuples: U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(y0, y1))) -> c5(U11'(tt, cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2)))) U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros)))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_1, c3_1, c5_1 ---------------------------------------- (79) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(cons(z0, cons(y0, y1))) -> c5(U11'(tt, cons(y0, y1))) by LENGTH(cons(z0, cons(z1, n__zeros))) -> c5(U11'(tt, cons(z1, n__zeros))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros)))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2)))) U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros)))) S tuples: U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2)))) U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros)))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_1, c3_1, c5_1 ---------------------------------------- (81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) We considered the (Usable) Rules:none And the Tuples: U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2)))) U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros)))) The order we found is given by the following interpretation: Matrix interpretation [MATRO]: Non-tuple symbols: <<< M( n__zeros ) = [[0], [0], [0]] >>> <<< M( 0 ) = [[2], [0], [0]] >>> <<< M( cons_2(x_1, x_2) ) = [[0], [0], [3]] + [[0, 0, 0], [0, 0, 0], [0, 0, 0]] * x_1 + [[0, 0, 0], [0, 0, 2], [0, 0, 0]] * x_2 >>> <<< M( tt ) = [[0], [0], [0]] >>> Tuple symbols: <<< M( LENGTH_1(x_1) ) = [[0], [2], [0]] + [[0, 3, 0], [0, 0, 0], [0, 0, 0]] * x_1 >>> <<< M( U11'_2(x_1, x_2) ) = [[0], [0], [0]] + [[0, 0, 0], [0, 0, 0], [0, 0, 0]] * x_1 + [[0, 3, 0], [0, 0, 0], [0, 0, 0]] * x_2 >>> <<< M( c5_1(x_1) ) = [[0], [0], [0]] + [[1, 0, 0], [0, 0, 0], [0, 0, 0]] * x_1 >>> <<< M( c3_1(x_1) ) = [[0], [0], [0]] + [[1, 0, 0], [0, 0, 0], [0, 0, 0]] * x_1 >>> <<< M( c2_1(x_1) ) = [[0], [0], [0]] + [[1, 0, 0], [0, 0, 0], [0, 0, 0]] * x_1 >>> <<< M( U12'_2(x_1, x_2) ) = [[0], [0], [0]] + [[0, 0, 0], [0, 0, 0], [0, 0, 0]] * x_1 + [[0, 3, 0], [0, 0, 0], [0, 0, 0]] * x_2 >>> Matrix type: We used a basic matrix type which is not further parametrizeable. As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order. ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2)))) U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros)))) S tuples: U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2)))) U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros)))) K tuples: LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) Defined Rule Symbols:none Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_1, c3_1, c5_1 ---------------------------------------- (83) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros)))) We considered the (Usable) Rules:none And the Tuples: U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2)))) U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros)))) The order we found is given by the following interpretation: Matrix interpretation [MATRO]: Non-tuple symbols: <<< M( n__zeros ) = [[0], [0], [0]] >>> <<< M( 0 ) = [[2], [0], [0]] >>> <<< M( cons_2(x_1, x_2) ) = [[0], [0], [2]] + [[0, 0, 0], [0, 0, 0], [0, 0, 0]] * x_1 + [[0, 0, 0], [0, 1, 2], [0, 0, 0]] * x_2 >>> <<< M( tt ) = [[0], [0], [0]] >>> Tuple symbols: <<< M( LENGTH_1(x_1) ) = [[0], [0], [0]] + [[0, 2, 0], [0, 0, 0], [0, 0, 0]] * x_1 >>> <<< M( U11'_2(x_1, x_2) ) = [[0], [0], [2]] + [[0, 0, 0], [0, 0, 0], [0, 0, 0]] * x_1 + [[0, 2, 0], [0, 0, 0], [0, 0, 0]] * x_2 >>> <<< M( c5_1(x_1) ) = [[0], [0], [0]] + [[1, 0, 0], [0, 0, 0], [0, 0, 0]] * x_1 >>> <<< M( c3_1(x_1) ) = [[0], [0], [0]] + [[1, 0, 0], [0, 0, 0], [0, 0, 0]] * x_1 >>> <<< M( c2_1(x_1) ) = [[0], [0], [0]] + [[1, 0, 0], [0, 0, 0], [0, 0, 0]] * x_1 >>> <<< M( U12'_2(x_1, x_2) ) = [[0], [0], [2]] + [[0, 0, 0], [0, 0, 0], [0, 0, 0]] * x_1 + [[0, 2, 0], [0, 0, 0], [0, 0, 0]] * x_2 >>> Matrix type: We used a basic matrix type which is not further parametrizeable. As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order. ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2)))) U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros)))) S tuples: U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2)))) U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros)))) K tuples: LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros)))) Defined Rule Symbols:none Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_1, c3_1, c5_1 ---------------------------------------- (85) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2)))) U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros)))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(z1, n__zeros))) -> c5(U11'(tt, cons(z1, n__zeros))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(z1, n__zeros))) -> c5(U11'(tt, cons(z1, n__zeros))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2)))) U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros)))) S tuples: U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros))) U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros)) U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros)) U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros))) U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros))) K tuples: LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros)))) U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2)))) U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros)))) U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2)))) U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros)))) Defined Rule Symbols:none Defined Pair Symbols: U11'_2, U12'_2, LENGTH_1 Compound Symbols: c2_1, c3_1, c5_1