MAYBE proof of input_CJRmiko6Jy.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) CompletionProof [UPPER BOUND(ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxTypedWeightedCompleteTrs (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) InliningProof [UPPER BOUND(ID), 121 ms] (20) CpxRNTS (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 62 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 5 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 851 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 253 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 39 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 62 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 1045 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 127 ms] (48) CpxRNTS (49) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (52) CdtProblem (53) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 7 ms] (60) CdtProblem (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 45 ms] (62) CdtProblem (63) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtLeafRemovalProof [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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (96) 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) and(tt, X) -> activate(X) length(nil) -> 0 length(cons(N, L)) -> s(length(activate(L))) take(0, IL) -> nil take(s(M), cons(N, IL)) -> cons(N, n__take(M, activate(IL))) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) 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) and(tt, X) -> activate(X) length(nil) -> 0' length(cons(N, L)) -> s(length(activate(L))) take(0', IL) -> nil take(s(M), cons(N, IL)) -> cons(N, n__take(M, activate(IL))) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) 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) and(tt, X) -> activate(X) length(nil) -> 0 length(cons(N, L)) -> s(length(activate(L))) take(0, IL) -> nil take(s(M), cons(N, IL)) -> cons(N, n__take(M, activate(IL))) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) 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] and(tt, X) -> activate(X) [1] length(nil) -> 0 [1] length(cons(N, L)) -> s(length(activate(L))) [1] take(0, IL) -> nil [1] take(s(M), cons(N, IL)) -> cons(N, n__take(M, activate(IL))) [1] zeros -> n__zeros [1] take(X1, X2) -> n__take(X1, X2) [1] activate(n__zeros) -> zeros [1] activate(n__take(X1, X2)) -> take(X1, X2) [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] and(tt, X) -> activate(X) [1] length(nil) -> 0 [1] length(cons(N, L)) -> s(length(activate(L))) [1] take(0, IL) -> nil [1] take(s(M), cons(N, IL)) -> cons(N, n__take(M, activate(IL))) [1] zeros -> n__zeros [1] take(X1, X2) -> n__take(X1, X2) [1] activate(n__zeros) -> zeros [1] activate(n__take(X1, X2)) -> take(X1, X2) [1] activate(X) -> X [1] The TRS has the following type information: zeros :: n__zeros:cons:nil:n__take cons :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take 0 :: 0:s n__zeros :: n__zeros:cons:nil:n__take and :: tt -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take tt :: tt activate :: n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take length :: n__zeros:cons:nil:n__take -> 0:s nil :: n__zeros:cons:nil:n__take s :: 0:s -> 0:s take :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take n__take :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take Rewrite Strategy: INNERMOST ---------------------------------------- (9) 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 ---------------------------------------- (10) 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] and(tt, X) -> activate(X) [1] length(nil) -> 0 [1] length(cons(N, L)) -> s(length(activate(L))) [1] take(0, IL) -> nil [1] take(s(M), cons(N, IL)) -> cons(N, n__take(M, activate(IL))) [1] zeros -> n__zeros [1] take(X1, X2) -> n__take(X1, X2) [1] activate(n__zeros) -> zeros [1] activate(n__take(X1, X2)) -> take(X1, X2) [1] activate(X) -> X [1] length(v0) -> null_length [0] The TRS has the following type information: zeros :: n__zeros:cons:nil:n__take cons :: 0:s:null_length -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take 0 :: 0:s:null_length n__zeros :: n__zeros:cons:nil:n__take and :: tt -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take tt :: tt activate :: n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take length :: n__zeros:cons:nil:n__take -> 0:s:null_length nil :: n__zeros:cons:nil:n__take s :: 0:s:null_length -> 0:s:null_length take :: 0:s:null_length -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take n__take :: 0:s:null_length -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take null_length :: 0:s:null_length Rewrite Strategy: INNERMOST ---------------------------------------- (11) 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 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> zeros :|: z = 0 activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 and(z, z') -{ 1 }-> activate(X) :|: z' = X, X >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 length(z) -{ 1 }-> 1 + length(activate(L)) :|: z = 1 + N + L, L >= 0, N >= 0 take(z, z') -{ 1 }-> 1 :|: z' = IL, z = 0, IL >= 0 take(z, z') -{ 1 }-> 1 + N + (1 + M + activate(IL)) :|: z = 1 + M, M >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (13) 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: and_2 length_1 (c) The following functions are completely defined: activate_1 take_2 zeros Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (14) 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] and(tt, X) -> activate(X) [1] length(nil) -> 0 [1] length(cons(N, L)) -> s(length(activate(L))) [1] take(0, IL) -> nil [1] take(s(M), cons(N, IL)) -> cons(N, n__take(M, activate(IL))) [1] zeros -> n__zeros [1] take(X1, X2) -> n__take(X1, X2) [1] activate(n__zeros) -> zeros [1] activate(n__take(X1, X2)) -> take(X1, X2) [1] activate(X) -> X [1] The TRS has the following type information: zeros :: n__zeros:cons:nil:n__take cons :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take 0 :: 0:s n__zeros :: n__zeros:cons:nil:n__take and :: tt -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take tt :: tt activate :: n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take length :: n__zeros:cons:nil:n__take -> 0:s nil :: n__zeros:cons:nil:n__take s :: 0:s -> 0:s take :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take n__take :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take Rewrite Strategy: INNERMOST ---------------------------------------- (15) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (16) 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] and(tt, X) -> activate(X) [1] length(nil) -> 0 [1] length(cons(N, n__zeros)) -> s(length(zeros)) [2] length(cons(N, n__take(X1', X2'))) -> s(length(take(X1', X2'))) [2] length(cons(N, L)) -> s(length(L)) [2] take(0, IL) -> nil [1] take(s(M), cons(N, IL)) -> cons(N, n__take(M, activate(IL))) [1] zeros -> n__zeros [1] take(X1, X2) -> n__take(X1, X2) [1] activate(n__zeros) -> zeros [1] activate(n__take(X1, X2)) -> take(X1, X2) [1] activate(X) -> X [1] The TRS has the following type information: zeros :: n__zeros:cons:nil:n__take cons :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take 0 :: 0:s n__zeros :: n__zeros:cons:nil:n__take and :: tt -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take tt :: tt activate :: n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take length :: n__zeros:cons:nil:n__take -> 0:s nil :: n__zeros:cons:nil:n__take s :: 0:s -> 0:s take :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take n__take :: 0:s -> n__zeros:cons:nil:n__take -> n__zeros:cons:nil:n__take Rewrite Strategy: INNERMOST ---------------------------------------- (17) 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 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> zeros :|: z = 0 activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 and(z, z') -{ 1 }-> activate(X) :|: z' = X, X >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 2 }-> 1 + length(zeros) :|: z = 1 + N + 0, N >= 0 length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0 take(z, z') -{ 1 }-> 1 :|: z' = IL, z = 0, IL >= 0 take(z, z') -{ 1 }-> 1 + N + (1 + M + activate(IL)) :|: z = 1 + M, M >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: ---------------------------------------- (19) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: zeros -{ 1 }-> 1 + 0 + 0 :|: zeros -{ 1 }-> 0 :|: ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 and(z, z') -{ 1 }-> activate(X) :|: z' = X, X >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0 length(z) -{ 3 }-> 1 + length(0) :|: z = 1 + N + 0, N >= 0 length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 1 + N + 0, N >= 0 take(z, z') -{ 1 }-> 1 :|: z' = IL, z = 0, IL >= 0 take(z, z') -{ 1 }-> 1 + N + (1 + M + activate(IL)) :|: z = 1 + M, M >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: ---------------------------------------- (21) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 and(z, z') -{ 1 }-> activate(z') :|: z' >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0 length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0 take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0 take(z, z') -{ 1 }-> 1 + N + (1 + (z - 1) + activate(IL)) :|: z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: ---------------------------------------- (23) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { zeros } { take, activate } { and } { length } ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 and(z, z') -{ 1 }-> activate(z') :|: z' >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0 length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0 take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0 take(z, z') -{ 1 }-> 1 + N + (1 + (z - 1) + activate(IL)) :|: z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {zeros}, {take,activate}, {and}, {length} ---------------------------------------- (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: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 and(z, z') -{ 1 }-> activate(z') :|: z' >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0 length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0 take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0 take(z, z') -{ 1 }-> 1 + N + (1 + (z - 1) + activate(IL)) :|: z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {zeros}, {take,activate}, {and}, {length} ---------------------------------------- (27) 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 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 and(z, z') -{ 1 }-> activate(z') :|: z' >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0 length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0 take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0 take(z, z') -{ 1 }-> 1 + N + (1 + (z - 1) + activate(IL)) :|: z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {zeros}, {take,activate}, {and}, {length} Previous analysis results are: zeros: runtime: ?, size: O(1) [1] ---------------------------------------- (29) 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 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 and(z, z') -{ 1 }-> activate(z') :|: z' >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0 length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0 take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0 take(z, z') -{ 1 }-> 1 + N + (1 + (z - 1) + activate(IL)) :|: z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {take,activate}, {and}, {length} Previous analysis results are: zeros: 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: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 and(z, z') -{ 1 }-> activate(z') :|: z' >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0 length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0 take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0 take(z, z') -{ 1 }-> 1 + N + (1 + (z - 1) + activate(IL)) :|: z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {take,activate}, {and}, {length} Previous analysis results are: zeros: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: take after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z' 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 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 and(z, z') -{ 1 }-> activate(z') :|: z' >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0 length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0 take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0 take(z, z') -{ 1 }-> 1 + N + (1 + (z - 1) + activate(IL)) :|: z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {take,activate}, {and}, {length} Previous analysis results are: zeros: runtime: O(1) [1], size: O(1) [1] take: runtime: ?, size: O(n^1) [1 + z + z'] activate: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: take after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 7 + 2*z + 2*z' Computed RUNTIME bound using CoFloCo for: activate after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 6 + 2*z ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 and(z, z') -{ 1 }-> activate(z') :|: z' >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 2 }-> 1 + length(take(X1', X2')) :|: X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0 length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0 take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0 take(z, z') -{ 1 }-> 1 + N + (1 + (z - 1) + activate(IL)) :|: z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {and}, {length} Previous analysis results are: zeros: runtime: O(1) [1], size: O(1) [1] take: runtime: O(n^1) [7 + 2*z + 2*z'], size: O(n^1) [1 + z + z'] activate: runtime: O(n^1) [6 + 2*z], size: O(n^1) [1 + z] ---------------------------------------- (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: activate(z) -{ 8 + 2*X1 + 2*X2 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 and(z, z') -{ 7 + 2*z' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 9 + 2*X1' + 2*X2' }-> 1 + length(s') :|: s' >= 0, s' <= X1' + X2' + 1, X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0 length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0 take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0 take(z, z') -{ 7 + 2*IL }-> 1 + N + (1 + (z - 1) + s'') :|: s'' >= 0, s'' <= IL + 1, z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {and}, {length} Previous analysis results are: zeros: runtime: O(1) [1], size: O(1) [1] take: runtime: O(n^1) [7 + 2*z + 2*z'], size: O(n^1) [1 + z + z'] activate: runtime: O(n^1) [6 + 2*z], size: O(n^1) [1 + z] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 8 + 2*X1 + 2*X2 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 and(z, z') -{ 7 + 2*z' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 9 + 2*X1' + 2*X2' }-> 1 + length(s') :|: s' >= 0, s' <= X1' + X2' + 1, X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0 length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0 take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0 take(z, z') -{ 7 + 2*IL }-> 1 + N + (1 + (z - 1) + s'') :|: s'' >= 0, s'' <= IL + 1, z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {and}, {length} Previous analysis results are: zeros: runtime: O(1) [1], size: O(1) [1] take: runtime: O(n^1) [7 + 2*z + 2*z'], size: O(n^1) [1 + z + z'] activate: runtime: O(n^1) [6 + 2*z], size: O(n^1) [1 + z] and: runtime: ?, size: O(n^1) [1 + z'] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 7 + 2*z' ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 8 + 2*X1 + 2*X2 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 and(z, z') -{ 7 + 2*z' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 9 + 2*X1' + 2*X2' }-> 1 + length(s') :|: s' >= 0, s' <= X1' + X2' + 1, X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0 length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0 take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0 take(z, z') -{ 7 + 2*IL }-> 1 + N + (1 + (z - 1) + s'') :|: s'' >= 0, s'' <= IL + 1, z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {length} Previous analysis results are: zeros: runtime: O(1) [1], size: O(1) [1] take: runtime: O(n^1) [7 + 2*z + 2*z'], size: O(n^1) [1 + z + z'] activate: runtime: O(n^1) [6 + 2*z], size: O(n^1) [1 + z] and: runtime: O(n^1) [7 + 2*z'], size: O(n^1) [1 + z'] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 8 + 2*X1 + 2*X2 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 and(z, z') -{ 7 + 2*z' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 9 + 2*X1' + 2*X2' }-> 1 + length(s') :|: s' >= 0, s' <= X1' + X2' + 1, X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0 length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0 take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0 take(z, z') -{ 7 + 2*IL }-> 1 + N + (1 + (z - 1) + s'') :|: s'' >= 0, s'' <= IL + 1, z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {length} Previous analysis results are: zeros: runtime: O(1) [1], size: O(1) [1] take: runtime: O(n^1) [7 + 2*z + 2*z'], size: O(n^1) [1 + z + z'] activate: runtime: O(n^1) [6 + 2*z], size: O(n^1) [1 + z] and: runtime: O(n^1) [7 + 2*z'], size: O(n^1) [1 + z'] ---------------------------------------- (45) 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: ? ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 8 + 2*X1 + 2*X2 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 and(z, z') -{ 7 + 2*z' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 9 + 2*X1' + 2*X2' }-> 1 + length(s') :|: s' >= 0, s' <= X1' + X2' + 1, X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0 length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0 take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0 take(z, z') -{ 7 + 2*IL }-> 1 + N + (1 + (z - 1) + s'') :|: s'' >= 0, s'' <= IL + 1, z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {length} Previous analysis results are: zeros: runtime: O(1) [1], size: O(1) [1] take: runtime: O(n^1) [7 + 2*z + 2*z'], size: O(n^1) [1 + z + z'] activate: runtime: O(n^1) [6 + 2*z], size: O(n^1) [1 + z] and: runtime: O(n^1) [7 + 2*z'], size: O(n^1) [1 + z'] length: runtime: ?, size: INF ---------------------------------------- (47) 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: ? ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 8 + 2*X1 + 2*X2 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0 and(z, z') -{ 7 + 2*z' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z = 0 length(z) -{ 1 }-> 0 :|: z = 1 length(z) -{ 2 }-> 1 + length(L) :|: z = 1 + N + L, L >= 0, N >= 0 length(z) -{ 9 + 2*X1' + 2*X2' }-> 1 + length(s') :|: s' >= 0, s' <= X1' + X2' + 1, X2' >= 0, z = 1 + N + (1 + X1' + X2'), X1' >= 0, N >= 0 length(z) -{ 3 }-> 1 + length(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length(1 + 0 + 0) :|: z - 1 >= 0 take(z, z') -{ 1 }-> 1 :|: z = 0, z' >= 0 take(z, z') -{ 7 + 2*IL }-> 1 + N + (1 + (z - 1) + s'') :|: s'' >= 0, s'' <= IL + 1, z - 1 >= 0, IL >= 0, z' = 1 + N + IL, N >= 0 take(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zeros -{ 1 }-> 0 :|: zeros -{ 1 }-> 1 + 0 + 0 :|: Function symbols to be analyzed: {length} Previous analysis results are: zeros: runtime: O(1) [1], size: O(1) [1] take: runtime: O(n^1) [7 + 2*z + 2*z'], size: O(n^1) [1 + z + z'] activate: runtime: O(n^1) [6 + 2*z], size: O(n^1) [1 + z] and: runtime: O(n^1) [7 + 2*z'], size: O(n^1) [1 + z'] length: runtime: INF, size: INF ---------------------------------------- (49) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: zeros -> cons(0, n__zeros) zeros -> n__zeros and(tt, z0) -> activate(z0) length(nil) -> 0 length(cons(z0, z1)) -> s(length(activate(z1))) take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 Tuples: ZEROS -> c ZEROS -> c1 AND(tt, z0) -> c2(ACTIVATE(z0)) LENGTH(nil) -> c3 LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1)) TAKE(0, z0) -> c5 TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) TAKE(z0, z1) -> c7 ACTIVATE(n__zeros) -> c8(ZEROS) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) ACTIVATE(z0) -> c10 S tuples: ZEROS -> c ZEROS -> c1 AND(tt, z0) -> c2(ACTIVATE(z0)) LENGTH(nil) -> c3 LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1)) TAKE(0, z0) -> c5 TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) TAKE(z0, z1) -> c7 ACTIVATE(n__zeros) -> c8(ZEROS) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) ACTIVATE(z0) -> c10 K tuples:none Defined Rule Symbols: zeros, and_2, length_1, take_2, activate_1 Defined Pair Symbols: ZEROS, AND_2, LENGTH_1, TAKE_2, ACTIVATE_1 Compound Symbols: c, c1, c2_1, c3, c4_2, c5, c6_1, c7, c8_1, c9_1, c10 ---------------------------------------- (51) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: AND(tt, z0) -> c2(ACTIVATE(z0)) Removed 7 trailing nodes: TAKE(0, z0) -> c5 LENGTH(nil) -> c3 ZEROS -> c1 ACTIVATE(n__zeros) -> c8(ZEROS) ZEROS -> c ACTIVATE(z0) -> c10 TAKE(z0, z1) -> c7 ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: zeros -> cons(0, n__zeros) zeros -> n__zeros and(tt, z0) -> activate(z0) length(nil) -> 0 length(cons(z0, z1)) -> s(length(activate(z1))) take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 Tuples: LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1)) TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) S tuples: LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1)) TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) K tuples:none Defined Rule Symbols: zeros, and_2, length_1, take_2, activate_1 Defined Pair Symbols: LENGTH_1, TAKE_2, ACTIVATE_1 Compound Symbols: c4_2, c6_1, c9_1 ---------------------------------------- (53) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: and(tt, z0) -> activate(z0) length(nil) -> 0 length(cons(z0, z1)) -> s(length(activate(z1))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1)) TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) S tuples: LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1)) TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) K tuples:none Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: LENGTH_1, TAKE_2, ACTIVATE_1 Compound Symbols: c4_2, c6_1, c9_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LENGTH(cons(z0, z1)) -> c4(LENGTH(activate(z1)), ACTIVATE(z1)) by LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros), ACTIVATE(n__zeros)) LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros), ACTIVATE(n__zeros)) LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) S tuples: TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros), ACTIVATE(n__zeros)) LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) K tuples:none Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: TAKE_2, ACTIVATE_1, LENGTH_1 Compound Symbols: c6_1, c9_1, c4_2 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) S tuples: TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) K tuples:none Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: TAKE_2, ACTIVATE_1, LENGTH_1 Compound Symbols: c6_1, c9_1, c4_2, c4_1 ---------------------------------------- (59) 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(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) We considered the (Usable) Rules: take(z0, z1) -> n__take(z0, z1) take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(0, z0) -> nil zeros -> n__zeros zeros -> cons(0, n__zeros) And the Tuples: TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [3] POL(ACTIVATE(x_1)) = 0 POL(LENGTH(x_1)) = x_1 POL(TAKE(x_1, x_2)) = 0 POL(activate(x_1)) = [3] POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(n__take(x_1, x_2)) = x_1 POL(n__zeros) = 0 POL(nil) = [3] POL(s(x_1)) = [1] + x_1 POL(take(x_1, x_2)) = x_1 POL(zeros) = [1] ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) S tuples: TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) K tuples: LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: TAKE_2, ACTIVATE_1, LENGTH_1 Compound Symbols: c6_1, c9_1, c4_2, c4_1 ---------------------------------------- (61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) We considered the (Usable) Rules: take(z0, z1) -> n__take(z0, z1) take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) activate(n__take(z0, z1)) -> take(z0, z1) take(0, z0) -> nil zeros -> n__zeros activate(z0) -> z0 zeros -> cons(0, n__zeros) activate(n__zeros) -> zeros And the Tuples: TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(ACTIVATE(x_1)) = [2]x_1 POL(LENGTH(x_1)) = x_1^2 POL(TAKE(x_1, x_2)) = x_1 + [2]x_2 POL(activate(x_1)) = [2] + x_1 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(n__take(x_1, x_2)) = x_1 + x_2 POL(n__zeros) = [1] POL(nil) = [1] POL(s(x_1)) = [2] + x_1 POL(take(x_1, x_2)) = x_1 + x_2 POL(zeros) = [2] ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) S tuples: ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) K tuples: LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: TAKE_2, ACTIVATE_1, LENGTH_1 Compound Symbols: c6_1, c9_1, c4_2, c4_1 ---------------------------------------- (63) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) K tuples: LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: TAKE_2, ACTIVATE_1, LENGTH_1 Compound Symbols: c6_1, c9_1, c4_2, c4_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) by LENGTH(cons(x0, n__take(0, z0))) -> c4(LENGTH(nil), ACTIVATE(n__take(0, z0))) LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(n__take(z0, z1)), ACTIVATE(n__take(z0, z1))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) LENGTH(cons(x0, n__take(0, z0))) -> c4(LENGTH(nil), ACTIVATE(n__take(0, z0))) LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(n__take(z0, z1)), ACTIVATE(n__take(z0, z1))) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) K tuples: LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: TAKE_2, ACTIVATE_1, LENGTH_1 Compound Symbols: c6_1, c9_1, c4_2, c4_1 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__take(0, z0))) -> c4(ACTIVATE(n__take(0, z0))) LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1))) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) K tuples: LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: TAKE_2, ACTIVATE_1, LENGTH_1 Compound Symbols: c6_1, c9_1, c4_2, c4_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(zeros)) by LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(n__zeros)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__take(0, z0))) -> c4(ACTIVATE(n__take(0, z0))) LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(n__zeros)) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(n__zeros)) K tuples: LENGTH(cons(x0, n__take(z0, z1))) -> c4(LENGTH(take(z0, z1)), ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: TAKE_2, ACTIVATE_1, LENGTH_1 Compound Symbols: c6_1, c9_1, c4_2, c4_1 ---------------------------------------- (71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(n__zeros)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__take(0, z0))) -> c4(ACTIVATE(n__take(0, z0))) LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) K tuples: LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: TAKE_2, ACTIVATE_1, LENGTH_1 Compound Symbols: c6_1, c9_1, c4_2, c4_1 ---------------------------------------- (73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace TAKE(s(z0), cons(z1, z2)) -> c6(ACTIVATE(z2)) by TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__take(0, z0))) -> c4(ACTIVATE(n__take(0, z0))) LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1))) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) K tuples: LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1))) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: ACTIVATE_1, LENGTH_1, TAKE_2 Compound Symbols: c9_1, c4_2, c4_1, c6_1 ---------------------------------------- (75) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__take(z0, z1)) -> c9(TAKE(z0, z1)) by ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__take(0, z0))) -> c4(ACTIVATE(n__take(0, z0))) LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1))) ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3)))) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) K tuples: LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1))) ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3)))) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: LENGTH_1, TAKE_2, ACTIVATE_1 Compound Symbols: c4_2, c4_1, c6_1, c9_1 ---------------------------------------- (77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: LENGTH(cons(x0, n__take(0, z0))) -> c4(ACTIVATE(n__take(0, z0))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1))) ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3)))) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) K tuples: LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1))) ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3)))) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: LENGTH_1, TAKE_2, ACTIVATE_1 Compound Symbols: c4_2, c4_1, c6_1, c9_1 ---------------------------------------- (79) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(cons(x0, z0)) -> c4(LENGTH(z0), ACTIVATE(z0)) by LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)), ACTIVATE(cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))), ACTIVATE(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2))), ACTIVATE(cons(y0, n__take(y1, y2)))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)), ACTIVATE(cons(y0, n__zeros))) LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(LENGTH(n__take(s(y0), cons(y1, n__take(y2, y3)))), ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1))) ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3)))) LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)), ACTIVATE(cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))), ACTIVATE(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2))), ACTIVATE(cons(y0, n__take(y1, y2)))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)), ACTIVATE(cons(y0, n__zeros))) LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(LENGTH(n__take(s(y0), cons(y1, n__take(y2, y3)))), ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) K tuples: TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1))) ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3)))) LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1)), ACTIVATE(cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3)))), ACTIVATE(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2))), ACTIVATE(cons(y0, n__take(y1, y2)))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros)), ACTIVATE(cons(y0, n__zeros))) LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(LENGTH(n__take(s(y0), cons(y1, n__take(y2, y3)))), ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: LENGTH_1, TAKE_2, ACTIVATE_1 Compound Symbols: c4_2, c4_1, c6_1, c9_1 ---------------------------------------- (81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1))) ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3)))) LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2)))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros))) LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) K tuples: TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1))) ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3)))) LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2)))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros))) LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: LENGTH_1, TAKE_2, ACTIVATE_1 Compound Symbols: c4_2, c4_1, c6_1, c9_1 ---------------------------------------- (83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(cons(x0, n__take(z0, z1))) -> c4(ACTIVATE(n__take(z0, z1))) by LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1))) ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3)))) LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2)))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros))) LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) K tuples: TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1))) ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3)))) LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2)))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros))) LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: LENGTH_1, TAKE_2, ACTIVATE_1 Compound Symbols: c4_2, c4_1, c6_1, c9_1 ---------------------------------------- (85) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace TAKE(s(z0), cons(z1, n__take(y0, y1))) -> c6(ACTIVATE(n__take(y0, y1))) by TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3)))) LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2)))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros))) LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) K tuples: ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3)))) LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2)))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros))) LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: LENGTH_1, ACTIVATE_1, TAKE_2 Compound Symbols: c4_2, c4_1, c9_1, c6_1 ---------------------------------------- (87) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3)))) -> c9(TAKE(s(y0), cons(y1, n__take(y2, y3)))) by ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2)))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros))) LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) K tuples: LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1))) LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2)))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros))) LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: LENGTH_1, TAKE_2, ACTIVATE_1 Compound Symbols: c4_2, c4_1, c6_1, c9_1 ---------------------------------------- (89) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(cons(z0, cons(y0, y1))) -> c4(LENGTH(cons(y0, y1))) by LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(z1, n__zeros))) -> c4(LENGTH(cons(z1, n__zeros))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3))))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2)))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros))) LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3))))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) K tuples: LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2)))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros))) LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3))))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: LENGTH_1, TAKE_2, ACTIVATE_1 Compound Symbols: c4_2, c4_1, c6_1, c9_1 ---------------------------------------- (91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(cons(z0, cons(y0, n__take(y1, y2)))) -> c4(LENGTH(cons(y0, n__take(y1, y2)))) by LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros))) LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3))))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) K tuples: LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros))) LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3))))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: LENGTH_1, TAKE_2, ACTIVATE_1 Compound Symbols: c4_2, c4_1, c6_1, c9_1 ---------------------------------------- (93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) by LENGTH(cons(z0, n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) -> c4(ACTIVATE(n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros))) LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3))))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) LENGTH(cons(z0, n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) -> c4(ACTIVATE(n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) K tuples: LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros))) TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3))))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) LENGTH(cons(z0, n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) -> c4(ACTIVATE(n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: LENGTH_1, TAKE_2, ACTIVATE_1 Compound Symbols: c4_2, c4_1, c6_1, c9_1 ---------------------------------------- (95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(cons(z0, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c4(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) by LENGTH(cons(z0, n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) -> c4(ACTIVATE(n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__zeros) -> zeros activate(n__take(z0, z1)) -> take(z0, z1) activate(z0) -> z0 zeros -> cons(0, n__zeros) zeros -> n__zeros take(0, z0) -> nil take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2))) take(z0, z1) -> n__take(z0, z1) Tuples: LENGTH(cons(x0, n__take(s(z0), cons(z1, z2)))) -> c4(LENGTH(cons(z1, n__take(z0, activate(z2)))), ACTIVATE(n__take(s(z0), cons(z1, z2)))) LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros))) TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3))))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) LENGTH(cons(z0, n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) -> c4(ACTIVATE(n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) S tuples: LENGTH(cons(x0, n__zeros)) -> c4(LENGTH(cons(0, n__zeros))) K tuples: LENGTH(cons(z0, cons(y0, n__take(s(y1), cons(y2, y3))))) -> c4(LENGTH(cons(y0, n__take(s(y1), cons(y2, y3))))) LENGTH(cons(z0, cons(y0, n__zeros))) -> c4(LENGTH(cons(y0, n__zeros))) TAKE(s(z0), cons(z1, n__take(s(y0), cons(y1, n__take(y2, y3))))) -> c6(ACTIVATE(n__take(s(y0), cons(y1, n__take(y2, y3))))) ACTIVATE(n__take(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) -> c9(TAKE(s(z0), cons(z1, n__take(s(y2), cons(y3, n__take(y4, y5)))))) LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c4(LENGTH(cons(z1, cons(y1, y2)))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(s(y2), cons(y3, y4)))))) LENGTH(cons(z0, cons(z1, cons(y1, n__take(y2, y3))))) -> c4(LENGTH(cons(z1, cons(y1, n__take(y2, y3))))) LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c4(LENGTH(cons(z1, cons(y1, n__zeros)))) LENGTH(cons(z0, cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) -> c4(LENGTH(cons(z1, n__take(s(y1), cons(y2, n__take(y3, y4)))))) LENGTH(cons(z0, n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) -> c4(ACTIVATE(n__take(s(z1), cons(z2, n__take(s(y2), cons(y3, n__take(y4, y5))))))) Defined Rule Symbols: activate_1, zeros, take_2 Defined Pair Symbols: LENGTH_1, TAKE_2, ACTIVATE_1 Compound Symbols: c4_2, c4_1, c6_1, c9_1