KILLED proof of input_d9HQBYn39O.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 341 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 4 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), 88 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 4 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 143 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 90 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 43 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 189 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 26 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 3709 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 753 ms] (50) CpxRNTS (51) CompletionProof [UPPER BOUND(ID), 0 ms] (52) CpxTypedWeightedCompleteTrs (53) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (54) CpxRNTS (55) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (56) CdtProblem (57) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 1811 ms] (134) CdtProblem (135) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: from(X) -> cons(X, n__from(s(X))) length(n__nil) -> 0 length(n__cons(X, Y)) -> s(length1(activate(Y))) length1(X) -> length(activate(X)) from(X) -> n__from(X) nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) activate(n__from(X)) -> from(X) activate(n__nil) -> nil activate(n__cons(X1, X2)) -> cons(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: from(X) -> cons(X, n__from(s(X))) length(n__nil) -> 0' length(n__cons(X, Y)) -> s(length1(activate(Y))) length1(X) -> length(activate(X)) from(X) -> n__from(X) nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) activate(n__from(X)) -> from(X) activate(n__nil) -> nil activate(n__cons(X1, X2)) -> cons(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: from(X) -> cons(X, n__from(s(X))) length(n__nil) -> 0 length(n__cons(X, Y)) -> s(length1(activate(Y))) length1(X) -> length(activate(X)) from(X) -> n__from(X) nil -> n__nil cons(X1, X2) -> n__cons(X1, X2) activate(n__from(X)) -> from(X) activate(n__nil) -> nil activate(n__cons(X1, X2)) -> cons(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: from(X) -> cons(X, n__from(s(X))) [1] length(n__nil) -> 0 [1] length(n__cons(X, Y)) -> s(length1(activate(Y))) [1] length1(X) -> length(activate(X)) [1] from(X) -> n__from(X) [1] nil -> n__nil [1] cons(X1, X2) -> n__cons(X1, X2) [1] activate(n__from(X)) -> from(X) [1] activate(n__nil) -> nil [1] activate(n__cons(X1, X2)) -> cons(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: from(X) -> cons(X, n__from(s(X))) [1] length(n__nil) -> 0 [1] length(n__cons(X, Y)) -> s(length1(activate(Y))) [1] length1(X) -> length(activate(X)) [1] from(X) -> n__from(X) [1] nil -> n__nil [1] cons(X1, X2) -> n__cons(X1, X2) [1] activate(n__from(X)) -> from(X) [1] activate(n__nil) -> nil [1] activate(n__cons(X1, X2)) -> cons(X1, X2) [1] activate(X) -> X [1] The TRS has the following type information: from :: s:0 -> n__from:n__nil:n__cons cons :: s:0 -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons n__from :: s:0 -> n__from:n__nil:n__cons s :: s:0 -> s:0 length :: n__from:n__nil:n__cons -> s:0 n__nil :: n__from:n__nil:n__cons 0 :: s:0 n__cons :: s:0 -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons length1 :: n__from:n__nil:n__cons -> s:0 activate :: n__from:n__nil:n__cons -> n__from:n__nil:n__cons nil :: n__from:n__nil:n__cons 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: length_1 length1_1 (c) The following functions are completely defined: activate_1 nil from_1 cons_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: from(X) -> cons(X, n__from(s(X))) [1] length(n__nil) -> 0 [1] length(n__cons(X, Y)) -> s(length1(activate(Y))) [1] length1(X) -> length(activate(X)) [1] from(X) -> n__from(X) [1] nil -> n__nil [1] cons(X1, X2) -> n__cons(X1, X2) [1] activate(n__from(X)) -> from(X) [1] activate(n__nil) -> nil [1] activate(n__cons(X1, X2)) -> cons(X1, X2) [1] activate(X) -> X [1] The TRS has the following type information: from :: s:0 -> n__from:n__nil:n__cons cons :: s:0 -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons n__from :: s:0 -> n__from:n__nil:n__cons s :: s:0 -> s:0 length :: n__from:n__nil:n__cons -> s:0 n__nil :: n__from:n__nil:n__cons 0 :: s:0 n__cons :: s:0 -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons length1 :: n__from:n__nil:n__cons -> s:0 activate :: n__from:n__nil:n__cons -> n__from:n__nil:n__cons nil :: n__from:n__nil:n__cons Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: from(X) -> cons(X, n__from(s(X))) [1] length(n__nil) -> 0 [1] length(n__cons(X, n__from(X'))) -> s(length1(from(X'))) [2] length(n__cons(X, n__nil)) -> s(length1(nil)) [2] length(n__cons(X, n__cons(X1', X2'))) -> s(length1(cons(X1', X2'))) [2] length(n__cons(X, Y)) -> s(length1(Y)) [2] length1(n__from(X'')) -> length(from(X'')) [2] length1(n__nil) -> length(nil) [2] length1(n__cons(X1'', X2'')) -> length(cons(X1'', X2'')) [2] length1(X) -> length(X) [2] from(X) -> n__from(X) [1] nil -> n__nil [1] cons(X1, X2) -> n__cons(X1, X2) [1] activate(n__from(X)) -> from(X) [1] activate(n__nil) -> nil [1] activate(n__cons(X1, X2)) -> cons(X1, X2) [1] activate(X) -> X [1] The TRS has the following type information: from :: s:0 -> n__from:n__nil:n__cons cons :: s:0 -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons n__from :: s:0 -> n__from:n__nil:n__cons s :: s:0 -> s:0 length :: n__from:n__nil:n__cons -> s:0 n__nil :: n__from:n__nil:n__cons 0 :: s:0 n__cons :: s:0 -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons length1 :: n__from:n__nil:n__cons -> s:0 activate :: n__from:n__nil:n__cons -> n__from:n__nil:n__cons nil :: n__from:n__nil:n__cons 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: n__nil => 0 0 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> nil :|: z = 0 activate(z) -{ 1 }-> from(X) :|: z = 1 + X, X >= 0 activate(z) -{ 1 }-> cons(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 cons(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 from(z) -{ 1 }-> cons(X, 1 + (1 + X)) :|: X >= 0, z = X from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 2 }-> 1 + length1(nil) :|: z = 1 + X + 0, X >= 0 length(z) -{ 2 }-> 1 + length1(from(X')) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0 length(z) -{ 2 }-> 1 + length1(cons(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2') length1(z) -{ 2 }-> length(X) :|: X >= 0, z = X length1(z) -{ 2 }-> length(nil) :|: z = 0 length1(z) -{ 2 }-> length(from(X'')) :|: z = 1 + X'', X'' >= 0 length1(z) -{ 2 }-> length(cons(X1'', X2'')) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0 nil -{ 1 }-> 0 :|: ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: nil -{ 1 }-> 0 :|: cons(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 from(z) -{ 1 }-> cons(X, 1 + (1 + X)) :|: X >= 0, z = X from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z = 1 + X, X >= 0, X' >= 0, X = X' activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X from(z) -{ 2 }-> 1 + X1 + X2 :|: X >= 0, z = X, X1 >= 0, X2 >= 0, X = X1, 1 + (1 + X) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z = 1 + X + 0, X >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 2 }-> length(X) :|: X >= 0, z = X length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z = 1 + X'', X'' >= 0, X >= 0, X'' = X length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z = 1 + X'', X'' >= 0, X >= 0, X'' = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 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: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { nil } { from } { cons } { activate } { length1, length } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: Function symbols to be analyzed: {nil}, {from}, {cons}, {activate}, {length1,length} ---------------------------------------- (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: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: Function symbols to be analyzed: {nil}, {from}, {cons}, {activate}, {length1,length} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: nil after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: Function symbols to be analyzed: {nil}, {from}, {cons}, {activate}, {length1,length} Previous analysis results are: nil: runtime: ?, size: O(1) [0] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: nil after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: Function symbols to be analyzed: {from}, {cons}, {activate}, {length1,length} Previous analysis results are: nil: runtime: O(1) [1], size: O(1) [0] ---------------------------------------- (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: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: Function symbols to be analyzed: {from}, {cons}, {activate}, {length1,length} Previous analysis results are: nil: runtime: O(1) [1], size: O(1) [0] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: from after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 2*z ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: Function symbols to be analyzed: {from}, {cons}, {activate}, {length1,length} Previous analysis results are: nil: runtime: O(1) [1], size: O(1) [0] from: runtime: ?, size: O(n^1) [3 + 2*z] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: from after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: Function symbols to be analyzed: {cons}, {activate}, {length1,length} Previous analysis results are: nil: runtime: O(1) [1], size: O(1) [0] from: runtime: O(1) [2], size: O(n^1) [3 + 2*z] ---------------------------------------- (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: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: Function symbols to be analyzed: {cons}, {activate}, {length1,length} Previous analysis results are: nil: runtime: O(1) [1], size: O(1) [0] from: runtime: O(1) [2], size: O(n^1) [3 + 2*z] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: cons after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: Function symbols to be analyzed: {cons}, {activate}, {length1,length} Previous analysis results are: nil: runtime: O(1) [1], size: O(1) [0] from: runtime: O(1) [2], size: O(n^1) [3 + 2*z] cons: runtime: ?, size: O(n^1) [1 + z + z'] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: cons after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: Function symbols to be analyzed: {activate}, {length1,length} Previous analysis results are: nil: runtime: O(1) [1], size: O(1) [0] from: runtime: O(1) [2], size: O(n^1) [3 + 2*z] cons: runtime: O(1) [1], size: O(n^1) [1 + z + z'] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 }-> s'' :|: s'' >= 0, s'' <= X' + (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 4 }-> 1 + length1(s) :|: s >= 0, s <= X'' + (1 + (1 + X'')) + 1, z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 4 }-> length(s') :|: s' >= 0, s' <= X + (1 + (1 + X)) + 1, z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: Function symbols to be analyzed: {activate}, {length1,length} Previous analysis results are: nil: runtime: O(1) [1], size: O(1) [0] from: runtime: O(1) [2], size: O(n^1) [3 + 2*z] cons: runtime: O(1) [1], size: O(n^1) [1 + z + z'] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: activate after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 2*z ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 }-> s'' :|: s'' >= 0, s'' <= X' + (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 4 }-> 1 + length1(s) :|: s >= 0, s <= X'' + (1 + (1 + X'')) + 1, z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 4 }-> length(s') :|: s' >= 0, s' <= X + (1 + (1 + X)) + 1, z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: Function symbols to be analyzed: {activate}, {length1,length} Previous analysis results are: nil: runtime: O(1) [1], size: O(1) [0] from: runtime: O(1) [2], size: O(n^1) [3 + 2*z] cons: runtime: O(1) [1], size: O(n^1) [1 + z + z'] activate: runtime: ?, size: O(n^1) [1 + 2*z] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: activate after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 10 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 }-> s'' :|: s'' >= 0, s'' <= X' + (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 4 }-> 1 + length1(s) :|: s >= 0, s <= X'' + (1 + (1 + X'')) + 1, z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 4 }-> length(s') :|: s' >= 0, s' <= X + (1 + (1 + X)) + 1, z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: Function symbols to be analyzed: {length1,length} Previous analysis results are: nil: runtime: O(1) [1], size: O(1) [0] from: runtime: O(1) [2], size: O(n^1) [3 + 2*z] cons: runtime: O(1) [1], size: O(n^1) [1 + z + z'] activate: runtime: O(1) [10], size: O(n^1) [1 + 2*z] ---------------------------------------- (45) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 }-> s'' :|: s'' >= 0, s'' <= X' + (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 4 }-> 1 + length1(s) :|: s >= 0, s <= X'' + (1 + (1 + X'')) + 1, z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 4 }-> length(s') :|: s' >= 0, s' <= X + (1 + (1 + X)) + 1, z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: Function symbols to be analyzed: {length1,length} Previous analysis results are: nil: runtime: O(1) [1], size: O(1) [0] from: runtime: O(1) [2], size: O(n^1) [3 + 2*z] cons: runtime: O(1) [1], size: O(n^1) [1 + z + z'] activate: runtime: O(1) [10], size: O(n^1) [1 + 2*z] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: length1 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE 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) -{ 3 }-> s'' :|: s'' >= 0, s'' <= X' + (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 4 }-> 1 + length1(s) :|: s >= 0, s <= X'' + (1 + (1 + X'')) + 1, z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 4 }-> length(s') :|: s' >= 0, s' <= X + (1 + (1 + X)) + 1, z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: Function symbols to be analyzed: {length1,length} Previous analysis results are: nil: runtime: O(1) [1], size: O(1) [0] from: runtime: O(1) [2], size: O(n^1) [3 + 2*z] cons: runtime: O(1) [1], size: O(n^1) [1 + z + z'] activate: runtime: O(1) [10], size: O(n^1) [1 + 2*z] length1: runtime: ?, size: INF length: runtime: ?, size: INF ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: length1 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 }-> s'' :|: s'' >= 0, s'' <= X' + (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 0 :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2' cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 length(z) -{ 4 }-> 1 + length1(s) :|: s >= 0, s <= X'' + (1 + (1 + X'')) + 1, z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0 length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'' length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2 length1(z) -{ 4 }-> length(s') :|: s' >= 0, s' <= X + (1 + (1 + X)) + 1, z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 2 }-> length(z) :|: z >= 0 length1(z) -{ 3 }-> length(0) :|: z = 0 length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2 nil -{ 1 }-> 0 :|: Function symbols to be analyzed: {length1,length} Previous analysis results are: nil: runtime: O(1) [1], size: O(1) [0] from: runtime: O(1) [2], size: O(n^1) [3 + 2*z] cons: runtime: O(1) [1], size: O(n^1) [1 + z + z'] activate: runtime: O(1) [10], size: O(n^1) [1 + 2*z] length1: runtime: INF, size: INF length: runtime: ?, size: INF ---------------------------------------- (51) 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 ---------------------------------------- (52) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: from(X) -> cons(X, n__from(s(X))) [1] length(n__nil) -> 0 [1] length(n__cons(X, Y)) -> s(length1(activate(Y))) [1] length1(X) -> length(activate(X)) [1] from(X) -> n__from(X) [1] nil -> n__nil [1] cons(X1, X2) -> n__cons(X1, X2) [1] activate(n__from(X)) -> from(X) [1] activate(n__nil) -> nil [1] activate(n__cons(X1, X2)) -> cons(X1, X2) [1] activate(X) -> X [1] length(v0) -> null_length [0] The TRS has the following type information: from :: s:0:null_length -> n__from:n__nil:n__cons cons :: s:0:null_length -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons n__from :: s:0:null_length -> n__from:n__nil:n__cons s :: s:0:null_length -> s:0:null_length length :: n__from:n__nil:n__cons -> s:0:null_length n__nil :: n__from:n__nil:n__cons 0 :: s:0:null_length n__cons :: s:0:null_length -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons length1 :: n__from:n__nil:n__cons -> s:0:null_length activate :: n__from:n__nil:n__cons -> n__from:n__nil:n__cons nil :: n__from:n__nil:n__cons null_length :: s:0:null_length Rewrite Strategy: INNERMOST ---------------------------------------- (53) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: n__nil => 0 0 => 0 null_length => 0 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> nil :|: z = 0 activate(z) -{ 1 }-> from(X) :|: z = 1 + X, X >= 0 activate(z) -{ 1 }-> cons(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 cons(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 from(z) -{ 1 }-> cons(X, 1 + (1 + X)) :|: X >= 0, z = X from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 length(z) -{ 1 }-> 1 + length1(activate(Y)) :|: Y >= 0, z = 1 + X + Y, X >= 0 length1(z) -{ 1 }-> length(activate(X)) :|: X >= 0, z = X nil -{ 1 }-> 0 :|: Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (55) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) length(n__nil) -> 0 length(n__cons(z0, z1)) -> s(length1(activate(z1))) length1(z0) -> length(activate(z0)) nil -> n__nil cons(z0, z1) -> n__cons(z0, z1) activate(n__from(z0)) -> from(z0) activate(n__nil) -> nil activate(n__cons(z0, z1)) -> cons(z0, z1) activate(z0) -> z0 Tuples: FROM(z0) -> c(CONS(z0, n__from(s(z0)))) FROM(z0) -> c1 LENGTH(n__nil) -> c2 LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1)) LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0)) NIL -> c5 CONS(z0, z1) -> c6 ACTIVATE(n__from(z0)) -> c7(FROM(z0)) ACTIVATE(n__nil) -> c8(NIL) ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1)) ACTIVATE(z0) -> c10 S tuples: FROM(z0) -> c(CONS(z0, n__from(s(z0)))) FROM(z0) -> c1 LENGTH(n__nil) -> c2 LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1)) LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0)) NIL -> c5 CONS(z0, z1) -> c6 ACTIVATE(n__from(z0)) -> c7(FROM(z0)) ACTIVATE(n__nil) -> c8(NIL) ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1)) ACTIVATE(z0) -> c10 K tuples:none Defined Rule Symbols: from_1, length_1, length1_1, nil, cons_2, activate_1 Defined Pair Symbols: FROM_1, LENGTH_1, LENGTH1_1, NIL, CONS_2, ACTIVATE_1 Compound Symbols: c_1, c1, c2, c3_2, c4_2, c5, c6, c7_1, c8_1, c9_1, c10 ---------------------------------------- (57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ACTIVATE(z0) -> c10 FROM(z0) -> c1 LENGTH(n__nil) -> c2 ACTIVATE(n__from(z0)) -> c7(FROM(z0)) CONS(z0, z1) -> c6 ACTIVATE(n__nil) -> c8(NIL) ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1)) NIL -> c5 FROM(z0) -> c(CONS(z0, n__from(s(z0)))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) length(n__nil) -> 0 length(n__cons(z0, z1)) -> s(length1(activate(z1))) length1(z0) -> length(activate(z0)) nil -> n__nil cons(z0, z1) -> n__cons(z0, z1) activate(n__from(z0)) -> from(z0) activate(n__nil) -> nil activate(n__cons(z0, z1)) -> cons(z0, z1) activate(z0) -> z0 Tuples: LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1)) LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0)) S tuples: LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1)) LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0)) K tuples:none Defined Rule Symbols: from_1, length_1, length1_1, nil, cons_2, activate_1 Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_2, c4_2 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) length(n__nil) -> 0 length(n__cons(z0, z1)) -> s(length1(activate(z1))) length1(z0) -> length(activate(z0)) nil -> n__nil cons(z0, z1) -> n__cons(z0, z1) activate(n__from(z0)) -> from(z0) activate(n__nil) -> nil activate(n__cons(z0, z1)) -> cons(z0, z1) activate(z0) -> z0 Tuples: LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1))) LENGTH1(z0) -> c4(LENGTH(activate(z0))) S tuples: LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1))) LENGTH1(z0) -> c4(LENGTH(activate(z0))) K tuples:none Defined Rule Symbols: from_1, length_1, length1_1, nil, cons_2, activate_1 Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (61) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: length(n__nil) -> 0 length(n__cons(z0, z1)) -> s(length1(activate(z1))) length1(z0) -> length(activate(z0)) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__nil) -> nil activate(n__cons(z0, z1)) -> cons(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) cons(z0, z1) -> n__cons(z0, z1) nil -> n__nil Tuples: LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1))) LENGTH1(z0) -> c4(LENGTH(activate(z0))) S tuples: LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1))) LENGTH1(z0) -> c4(LENGTH(activate(z0))) K tuples:none Defined Rule Symbols: activate_1, from_1, cons_2, nil Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1))) by LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1))) LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__nil) -> nil activate(n__cons(z0, z1)) -> cons(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) cons(z0, z1) -> n__cons(z0, z1) nil -> n__nil Tuples: LENGTH1(z0) -> c4(LENGTH(activate(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1))) LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) S tuples: LENGTH1(z0) -> c4(LENGTH(activate(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1))) LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) K tuples:none Defined Rule Symbols: activate_1, from_1, cons_2, nil Defined Pair Symbols: LENGTH1_1, LENGTH_1 Compound Symbols: c4_1, c3_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LENGTH1(z0) -> c4(LENGTH(activate(z0))) by LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0))) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__nil) -> nil activate(n__cons(z0, z1)) -> cons(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) cons(z0, z1) -> n__cons(z0, z1) nil -> n__nil Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1))) LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0))) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1))) LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0))) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) K tuples:none Defined Rule Symbols: activate_1, from_1, cons_2, nil Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (67) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: activate(n__from(z0)) -> from(z0) activate(n__nil) -> nil activate(n__cons(z0, z1)) -> cons(z0, z1) activate(z0) -> z0 ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) cons(z0, z1) -> n__cons(z0, z1) nil -> n__nil Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1))) LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0))) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1))) LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0))) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) K tuples:none Defined Rule Symbols: from_1, cons_2, nil Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(from(z0))) by LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) cons(z0, z1) -> n__cons(z0, z1) nil -> n__nil Tuples: LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1))) LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0))) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) S tuples: LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1))) LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0))) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) K tuples:none Defined Rule Symbols: from_1, cons_2, nil Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil)) by LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) cons(z0, z1) -> n__cons(z0, z1) nil -> n__nil Tuples: LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1))) LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0))) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) S tuples: LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1))) LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0))) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) K tuples:none Defined Rule Symbols: from_1, cons_2, nil Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1))) by LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) cons(z0, z1) -> n__cons(z0, z1) nil -> n__nil Tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0))) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) S tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0))) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) K tuples:none Defined Rule Symbols: from_1, cons_2, nil Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0))) by LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__from(z0))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) cons(z0, z1) -> n__cons(z0, z1) nil -> n__nil Tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__from(z0))) S tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__from(z0))) K tuples:none Defined Rule Symbols: from_1, cons_2, nil Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: LENGTH1(n__from(z0)) -> c4(LENGTH(n__from(z0))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) cons(z0, z1) -> n__cons(z0, z1) nil -> n__nil Tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) S tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) K tuples:none Defined Rule Symbols: from_1, cons_2, nil Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (79) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: nil -> n__nil cons(z0, z1) -> n__cons(z0, z1) Tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) S tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__nil) -> c4(LENGTH(nil)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) K tuples:none Defined Rule Symbols: nil, cons_2 Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LENGTH1(n__nil) -> c4(LENGTH(nil)) by LENGTH1(n__nil) -> c4(LENGTH(n__nil)) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: nil -> n__nil cons(z0, z1) -> n__cons(z0, z1) Tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) LENGTH1(n__nil) -> c4(LENGTH(n__nil)) S tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) LENGTH1(n__nil) -> c4(LENGTH(n__nil)) K tuples:none Defined Rule Symbols: nil, cons_2 Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: LENGTH1(n__nil) -> c4(LENGTH(n__nil)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: nil -> n__nil cons(z0, z1) -> n__cons(z0, z1) Tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) S tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) K tuples:none Defined Rule Symbols: nil, cons_2 Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (85) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: nil -> n__nil ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: cons(z0, z1) -> n__cons(z0, z1) Tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) S tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) K tuples:none Defined Rule Symbols: cons_2 Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) by LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: cons(z0, z1) -> n__cons(z0, z1) Tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) S tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) K tuples:none Defined Rule Symbols: cons_2 Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) by LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: cons(z0, z1) -> n__cons(z0, z1) Tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) S tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) K tuples:none Defined Rule Symbols: cons_2 Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) by LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: cons(z0, z1) -> n__cons(z0, z1) Tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) S tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) K tuples:none Defined Rule Symbols: cons_2 Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (93) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: cons(z0, z1) -> n__cons(z0, z1) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) S tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH1(z0) -> c4(LENGTH(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH1(z0) -> c4(LENGTH(z0)) by LENGTH1(n__cons(y0, y1)) -> c4(LENGTH(n__cons(y0, y1))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__nil)) -> c4(LENGTH(n__cons(y0, n__nil))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__nil)) -> c4(LENGTH(n__cons(y0, n__nil))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) S tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__nil)) -> c4(LENGTH(n__cons(y0, n__nil))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (97) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil)) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__nil)) -> c4(LENGTH(n__cons(y0, n__nil))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) S tuples: LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__nil)) -> c4(LENGTH(n__cons(y0, n__nil))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (99) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) by LENGTH(n__cons(z0, n__cons(y0, y1))) -> c3(LENGTH1(n__cons(y0, y1))) LENGTH(n__cons(z0, n__from(y0))) -> c3(LENGTH1(n__from(y0))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__nil)) -> c4(LENGTH(n__cons(y0, n__nil))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__nil)) -> c4(LENGTH(n__cons(y0, n__nil))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (101) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: LENGTH1(n__cons(y0, n__nil)) -> c4(LENGTH(n__cons(y0, n__nil))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (103) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) by LENGTH1(n__cons(z0, n__from(y1))) -> c4(LENGTH(n__cons(z0, n__from(y1)))) LENGTH1(n__cons(z0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(z0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil)))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil)))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil)))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH1(n__cons(y0, y1)) -> c4(LENGTH(n__cons(y0, y1))) by LENGTH1(n__cons(z0, n__from(y1))) -> c4(LENGTH(n__cons(z0, n__from(y1)))) LENGTH1(n__cons(z0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(z0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil)))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil)))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil)))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil)))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil)))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (109) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) by LENGTH(n__cons(z0, n__cons(z1, n__from(y1)))) -> c3(LENGTH1(n__cons(z1, n__from(y1)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil)))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil)))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(n__cons(z0, n__cons(y0, y1))) -> c3(LENGTH1(n__cons(y0, y1))) by LENGTH(n__cons(z0, n__cons(z1, n__from(y1)))) -> c3(LENGTH1(n__cons(z1, n__from(y1)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil)))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil)))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (113) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil)))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) by LENGTH1(n__cons(z0, n__cons(z1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH1(n__cons(z0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(z0, n__cons(y1, y2)))) by LENGTH1(n__cons(z0, n__cons(z1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH1(n__cons(z0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(z0, n__cons(y1, y2)))) by LENGTH1(n__cons(z0, n__cons(z1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil)))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (123) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) by LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (125) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, y2)))) by LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (127) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, y2)))) by LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil))))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) by LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, y4)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4))))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5))))))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4))))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5))))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4))))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5))))))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (133) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) We considered the (Usable) Rules:none And the Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4))))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5))))))) The order we found is given by the following interpretation: Matrix interpretation [MATRO]: Non-tuple symbols: <<< M( s_1(x_1) ) = [[2], [0]] + [[1, 2], [0, 0]] * x_1 >>> <<< M( n__nil ) = [[0], [1]] >>> <<< M( n__from_1(x_1) ) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< M( n__cons_2(x_1, x_2) ) = [[1], [1]] + [[0, 0], [0, 0]] * x_1 + [[0, 2], [0, 1]] * x_2 >>> Tuple symbols: <<< M( LENGTH_1(x_1) ) = [[0], [0]] + [[0, 2], [0, 0]] * x_1 >>> <<< M( LENGTH1_1(x_1) ) = [[1], [4]] + [[1, 0], [0, 0]] * x_1 >>> <<< M( c3_1(x_1) ) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< M( c4_1(x_1) ) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> 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. ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4))))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5))))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4))))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5))))))) K tuples: LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1 ---------------------------------------- (135) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(z1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(z1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(z1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, y4)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4))))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5))))))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(z1, n__from(y1)))) -> c3(LENGTH1(n__cons(z1, n__from(y1)))) LENGTH(n__cons(z0, n__cons(z1, n__from(y1)))) -> c3(LENGTH1(n__cons(z1, n__from(y1)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4))))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5))))))) S tuples: LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0))) LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0))))) LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0))))) LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1)))) K tuples: LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1)))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4))))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil)))))) LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5))))))) Defined Rule Symbols:none Defined Pair Symbols: LENGTH_1, LENGTH1_1 Compound Symbols: c3_1, c4_1