/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- KILLED proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 160 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 697 ms] (12) BOUNDS(1, INF) (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (14) TRS for Loop Detection (15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (16) CpxTRS (17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] (18) CpxRelTRS (19) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxWeightedTrs (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxTypedWeightedTrs (23) CompletionProof [UPPER BOUND(ID), 0 ms] (24) CpxTypedWeightedCompleteTrs (25) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) CompletionProof [UPPER BOUND(ID), 0 ms] (28) CpxTypedWeightedCompleteTrs (29) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CpxTypedWeightedCompleteTrs (31) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (42) CdtProblem (43) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 1 ms] (48) CdtProblem (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 8 ms] (54) CdtProblem (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(f(a, b), x) -> f(b, f(a, f(c, f(b, f(a, x))))) f(x, f(y, z)) -> f(f(x, y), z) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_a -> a encode_b -> b encode_c -> c ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(f(a, b), x) -> f(b, f(a, f(c, f(b, f(a, x))))) f(x, f(y, z)) -> f(f(x, y), z) The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_a -> a encode_b -> b encode_c -> c Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(f(a, b), x) -> f(b, f(a, f(c, f(b, f(a, x))))) f(x, f(y, z)) -> f(f(x, y), z) The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_a -> a encode_b -> b encode_c -> c Rewrite Strategy: INNERMOST ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(f(a, b), x) -> f(b, f(a, f(c, f(b, f(a, x))))) f(x, f(y, z)) -> f(f(x, y), z) The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_a -> a encode_b -> b encode_c -> c Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: f(f(a, b), x) -> f(b, f(a, f(c, f(b, f(a, x))))) f(x, f(y, z)) -> f(f(x, y), z) encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_a -> a encode_b -> b encode_c -> c Types: f :: a:b:c:cons_f -> a:b:c:cons_f -> a:b:c:cons_f a :: a:b:c:cons_f b :: a:b:c:cons_f c :: a:b:c:cons_f encArg :: a:b:c:cons_f -> a:b:c:cons_f cons_f :: a:b:c:cons_f -> a:b:c:cons_f -> a:b:c:cons_f encode_f :: a:b:c:cons_f -> a:b:c:cons_f -> a:b:c:cons_f encode_a :: a:b:c:cons_f encode_b :: a:b:c:cons_f encode_c :: a:b:c:cons_f hole_a:b:c:cons_f1_0 :: a:b:c:cons_f gen_a:b:c:cons_f2_0 :: Nat -> a:b:c:cons_f ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: f, encArg They will be analysed ascendingly in the following order: f < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: f(f(a, b), x) -> f(b, f(a, f(c, f(b, f(a, x))))) f(x, f(y, z)) -> f(f(x, y), z) encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_a -> a encode_b -> b encode_c -> c Types: f :: a:b:c:cons_f -> a:b:c:cons_f -> a:b:c:cons_f a :: a:b:c:cons_f b :: a:b:c:cons_f c :: a:b:c:cons_f encArg :: a:b:c:cons_f -> a:b:c:cons_f cons_f :: a:b:c:cons_f -> a:b:c:cons_f -> a:b:c:cons_f encode_f :: a:b:c:cons_f -> a:b:c:cons_f -> a:b:c:cons_f encode_a :: a:b:c:cons_f encode_b :: a:b:c:cons_f encode_c :: a:b:c:cons_f hole_a:b:c:cons_f1_0 :: a:b:c:cons_f gen_a:b:c:cons_f2_0 :: Nat -> a:b:c:cons_f Generator Equations: gen_a:b:c:cons_f2_0(0) <=> a gen_a:b:c:cons_f2_0(+(x, 1)) <=> cons_f(a, gen_a:b:c:cons_f2_0(x)) The following defined symbols remain to be analysed: f, encArg They will be analysed ascendingly in the following order: f < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_a:b:c:cons_f2_0(n15_0)) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_a:b:c:cons_f2_0(0)) Induction Step: encArg(gen_a:b:c:cons_f2_0(+(n15_0, 1))) ->_R^Omega(0) f(encArg(a), encArg(gen_a:b:c:cons_f2_0(n15_0))) ->_R^Omega(0) f(a, encArg(gen_a:b:c:cons_f2_0(n15_0))) ->_IH f(a, *3_0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (12) BOUNDS(1, INF) ---------------------------------------- (13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (14) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(f(a, b), x) -> f(b, f(a, f(c, f(b, f(a, x))))) f(x, f(y, z)) -> f(f(x, y), z) The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_a -> a encode_b -> b encode_c -> c Rewrite Strategy: INNERMOST ---------------------------------------- (15) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (16) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(f(a, b), x) -> f(b, f(a, f(c, f(b, f(a, x))))) f(x, f(y, z)) -> f(f(x, y), z) encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_a -> a encode_b -> b encode_c -> c S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (17) NonCtorToCtorProof (UPPER BOUND(ID)) transformed non-ctor to ctor-system ---------------------------------------- (18) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, c_f(y, z)) -> f(f(x, y), z) f(c_f(a, b), x) -> f(b, f(a, f(c, f(b, f(a, x))))) The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_a -> a encode_b -> b encode_c -> c f(x0, x1) -> c_f(x0, x1) Rewrite Strategy: INNERMOST ---------------------------------------- (19) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (20) 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: f(x, c_f(y, z)) -> f(f(x, y), z) [1] f(c_f(a, b), x) -> f(b, f(a, f(c, f(b, f(a, x))))) [1] encArg(a) -> a [0] encArg(b) -> b [0] encArg(c) -> c [0] encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0] encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0] encode_a -> a [0] encode_b -> b [0] encode_c -> c [0] f(x0, x1) -> c_f(x0, x1) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (21) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (22) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(x, c_f(y, z)) -> f(f(x, y), z) [1] f(c_f(a, b), x) -> f(b, f(a, f(c, f(b, f(a, x))))) [1] encArg(a) -> a [0] encArg(b) -> b [0] encArg(c) -> c [0] encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0] encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0] encode_a -> a [0] encode_b -> b [0] encode_c -> c [0] f(x0, x1) -> c_f(x0, x1) [0] The TRS has the following type information: f :: c_f:a:b:c:cons_f -> c_f:a:b:c:cons_f -> c_f:a:b:c:cons_f c_f :: c_f:a:b:c:cons_f -> c_f:a:b:c:cons_f -> c_f:a:b:c:cons_f a :: c_f:a:b:c:cons_f b :: c_f:a:b:c:cons_f c :: c_f:a:b:c:cons_f encArg :: c_f:a:b:c:cons_f -> c_f:a:b:c:cons_f cons_f :: c_f:a:b:c:cons_f -> c_f:a:b:c:cons_f -> c_f:a:b:c:cons_f encode_f :: c_f:a:b:c:cons_f -> c_f:a:b:c:cons_f -> c_f:a:b:c:cons_f encode_a :: c_f:a:b:c:cons_f encode_b :: c_f:a:b:c:cons_f encode_c :: c_f:a:b:c:cons_f Rewrite Strategy: INNERMOST ---------------------------------------- (23) 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: encArg(v0) -> null_encArg [0] encode_f(v0, v1) -> null_encode_f [0] encode_a -> null_encode_a [0] encode_b -> null_encode_b [0] encode_c -> null_encode_c [0] f(v0, v1) -> null_f [0] And the following fresh constants: null_encArg, null_encode_f, null_encode_a, null_encode_b, null_encode_c, null_f ---------------------------------------- (24) 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: f(x, c_f(y, z)) -> f(f(x, y), z) [1] f(c_f(a, b), x) -> f(b, f(a, f(c, f(b, f(a, x))))) [1] encArg(a) -> a [0] encArg(b) -> b [0] encArg(c) -> c [0] encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0] encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0] encode_a -> a [0] encode_b -> b [0] encode_c -> c [0] f(x0, x1) -> c_f(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_f(v0, v1) -> null_encode_f [0] encode_a -> null_encode_a [0] encode_b -> null_encode_b [0] encode_c -> null_encode_c [0] f(v0, v1) -> null_f [0] The TRS has the following type information: f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f c_f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f a :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f b :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f c :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f encArg :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f cons_f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f encode_f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f encode_a :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f encode_b :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f encode_c :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_encArg :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_encode_f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_encode_a :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_encode_b :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_encode_c :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f Rewrite Strategy: INNERMOST ---------------------------------------- (25) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: a => 0 b => 1 c => 2 null_encArg => 0 null_encode_f => 0 null_encode_a => 0 null_encode_b => 0 null_encode_c => 0 null_f => 0 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 2 :|: z' = 2 encArg(z') -{ 0 }-> 1 :|: z' = 1 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 encode_a -{ 0 }-> 0 :|: encode_b -{ 0 }-> 1 :|: encode_b -{ 0 }-> 0 :|: encode_c -{ 0 }-> 2 :|: encode_c -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2 encode_f(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 f(z', z'') -{ 1 }-> f(f(x, y), z) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> f(1, f(0, f(2, f(1, f(0, x))))) :|: z' = 1 + 0 + 1, x >= 0, z'' = x f(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 f(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (27) 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: none (c) The following functions are completely defined: encArg_1 encode_f_2 encode_a encode_b encode_c f_2 Due to the following rules being added: encArg(v0) -> null_encArg [0] encode_f(v0, v1) -> null_encode_f [0] encode_a -> null_encode_a [0] encode_b -> null_encode_b [0] encode_c -> null_encode_c [0] f(v0, v1) -> null_f [0] And the following fresh constants: null_encArg, null_encode_f, null_encode_a, null_encode_b, null_encode_c, null_f ---------------------------------------- (28) 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: f(x, c_f(y, z)) -> f(f(x, y), z) [1] f(c_f(a, b), x) -> f(b, f(a, f(c, f(b, f(a, x))))) [1] encArg(a) -> a [0] encArg(b) -> b [0] encArg(c) -> c [0] encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0] encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0] encode_a -> a [0] encode_b -> b [0] encode_c -> c [0] f(x0, x1) -> c_f(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_f(v0, v1) -> null_encode_f [0] encode_a -> null_encode_a [0] encode_b -> null_encode_b [0] encode_c -> null_encode_c [0] f(v0, v1) -> null_f [0] The TRS has the following type information: f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f c_f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f a :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f b :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f c :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f encArg :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f cons_f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f encode_f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f encode_a :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f encode_b :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f encode_c :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_encArg :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_encode_f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_encode_a :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_encode_b :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_encode_c :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f Rewrite Strategy: INNERMOST ---------------------------------------- (29) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (30) 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: f(x, c_f(c_f(y', z'), z)) -> f(f(f(x, y'), z'), z) [2] f(c_f(a, b), c_f(y, z)) -> f(f(b, f(a, f(c, f(b, f(a, y))))), z) [2] f(x, c_f(y, z)) -> f(c_f(x, y), z) [1] f(x, c_f(y, z)) -> f(null_f, z) [1] f(c_f(a, b), c_f(y'', z'')) -> f(b, f(a, f(c, f(b, f(f(a, y''), z''))))) [2] f(c_f(a, b), x) -> f(b, f(a, f(c, f(b, c_f(a, x))))) [1] f(c_f(a, b), x) -> f(b, f(a, f(c, f(b, null_f)))) [1] encArg(a) -> a [0] encArg(b) -> b [0] encArg(c) -> c [0] encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0] encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0] encode_a -> a [0] encode_b -> b [0] encode_c -> c [0] f(x0, x1) -> c_f(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_f(v0, v1) -> null_encode_f [0] encode_a -> null_encode_a [0] encode_b -> null_encode_b [0] encode_c -> null_encode_c [0] f(v0, v1) -> null_f [0] The TRS has the following type information: f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f c_f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f a :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f b :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f c :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f encArg :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f cons_f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f encode_f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f encode_a :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f encode_b :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f encode_c :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_encArg :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_encode_f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_encode_a :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_encode_b :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_encode_c :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f null_f :: c_f:a:b:c:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f Rewrite Strategy: INNERMOST ---------------------------------------- (31) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: a => 0 b => 1 c => 2 null_encArg => 0 null_encode_f => 0 null_encode_a => 0 null_encode_b => 0 null_encode_c => 0 null_f => 0 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: encArg(z1) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z1 = 1 + x_1 + x_2, x_2 >= 0 encArg(z1) -{ 0 }-> 2 :|: z1 = 2 encArg(z1) -{ 0 }-> 1 :|: z1 = 1 encArg(z1) -{ 0 }-> 0 :|: z1 = 0 encArg(z1) -{ 0 }-> 0 :|: z1 = v0, v0 >= 0 encode_a -{ 0 }-> 0 :|: encode_b -{ 0 }-> 1 :|: encode_b -{ 0 }-> 0 :|: encode_c -{ 0 }-> 2 :|: encode_c -{ 0 }-> 0 :|: encode_f(z1, z2) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z1 = x_1, z2 = x_2, x_2 >= 0 encode_f(z1, z2) -{ 0 }-> 0 :|: z1 = v0, v0 >= 0, z2 = v1, v1 >= 0 f(z1, z2) -{ 2 }-> f(f(f(x, y'), z'), z) :|: z >= 0, x >= 0, z' >= 0, y' >= 0, z2 = 1 + (1 + y' + z') + z, z1 = x f(z1, z2) -{ 2 }-> f(f(1, f(0, f(2, f(1, f(0, y))))), z) :|: z2 = 1 + y + z, z >= 0, y >= 0, z1 = 1 + 0 + 1 f(z1, z2) -{ 2 }-> f(1, f(0, f(2, f(1, f(f(0, y''), z''))))) :|: z2 = 1 + y'' + z'', z'' >= 0, z1 = 1 + 0 + 1, y'' >= 0 f(z1, z2) -{ 1 }-> f(1, f(0, f(2, f(1, 0)))) :|: z2 = x, x >= 0, z1 = 1 + 0 + 1 f(z1, z2) -{ 1 }-> f(1, f(0, f(2, f(1, 1 + 0 + x)))) :|: z2 = x, x >= 0, z1 = 1 + 0 + 1 f(z1, z2) -{ 1 }-> f(0, z) :|: z2 = 1 + y + z, z >= 0, x >= 0, y >= 0, z1 = x f(z1, z2) -{ 1 }-> f(1 + x + y, z) :|: z2 = 1 + y + z, z >= 0, x >= 0, y >= 0, z1 = x f(z1, z2) -{ 0 }-> 0 :|: z1 = v0, v0 >= 0, z2 = v1, v1 >= 0 f(z1, z2) -{ 0 }-> 1 + x0 + x1 :|: z1 = x0, x0 >= 0, x1 >= 0, z2 = x1 ---------------------------------------- (33) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: encArg(z1) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z1 = 1 + x_1 + x_2, x_2 >= 0 encArg(z1) -{ 0 }-> 2 :|: z1 = 2 encArg(z1) -{ 0 }-> 1 :|: z1 = 1 encArg(z1) -{ 0 }-> 0 :|: z1 = 0 encArg(z1) -{ 0 }-> 0 :|: z1 >= 0 encode_a -{ 0 }-> 0 :|: encode_b -{ 0 }-> 1 :|: encode_b -{ 0 }-> 0 :|: encode_c -{ 0 }-> 2 :|: encode_c -{ 0 }-> 0 :|: encode_f(z1, z2) -{ 0 }-> f(encArg(z1), encArg(z2)) :|: z1 >= 0, z2 >= 0 encode_f(z1, z2) -{ 0 }-> 0 :|: z1 >= 0, z2 >= 0 f(z1, z2) -{ 2 }-> f(f(f(z1, y'), z'), z) :|: z >= 0, z1 >= 0, z' >= 0, y' >= 0, z2 = 1 + (1 + y' + z') + z f(z1, z2) -{ 2 }-> f(f(1, f(0, f(2, f(1, f(0, y))))), z) :|: z2 = 1 + y + z, z >= 0, y >= 0, z1 = 1 + 0 + 1 f(z1, z2) -{ 2 }-> f(1, f(0, f(2, f(1, f(f(0, y''), z''))))) :|: z2 = 1 + y'' + z'', z'' >= 0, z1 = 1 + 0 + 1, y'' >= 0 f(z1, z2) -{ 1 }-> f(1, f(0, f(2, f(1, 0)))) :|: z2 >= 0, z1 = 1 + 0 + 1 f(z1, z2) -{ 1 }-> f(1, f(0, f(2, f(1, 1 + 0 + z2)))) :|: z2 >= 0, z1 = 1 + 0 + 1 f(z1, z2) -{ 1 }-> f(0, z) :|: z2 = 1 + y + z, z >= 0, z1 >= 0, y >= 0 f(z1, z2) -{ 1 }-> f(1 + z1 + y, z) :|: z2 = 1 + y + z, z >= 0, z1 >= 0, y >= 0 f(z1, z2) -{ 0 }-> 0 :|: z1 >= 0, z2 >= 0 f(z1, z2) -{ 0 }-> 1 + z1 + z2 :|: z1 >= 0, z2 >= 0 ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a encode_b -> b encode_c -> c f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: ENCARG(a) -> c1 ENCARG(b) -> c2 ENCARG(c) -> c3 ENCARG(cons_f(z0, z1)) -> c4(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_F(z0, z1) -> c5(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_A -> c6 ENCODE_B -> c7 ENCODE_C -> c8 F(f(a, b), z0) -> c9(F(b, f(a, f(c, f(b, f(a, z0))))), F(a, f(c, f(b, f(a, z0)))), F(c, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0)) F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) S tuples: F(f(a, b), z0) -> c9(F(b, f(a, f(c, f(b, f(a, z0))))), F(a, f(c, f(b, f(a, z0)))), F(c, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0)) F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a, encode_b, encode_c Defined Pair Symbols: ENCARG_1, ENCODE_F_2, ENCODE_A, ENCODE_B, ENCODE_C, F_2 Compound Symbols: c1, c2, c3, c4_3, c5_3, c6, c7, c8, c9_5, c10_2 ---------------------------------------- (37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing nodes: ENCARG(c) -> c3 ENCODE_B -> c7 ENCARG(a) -> c1 ENCODE_C -> c8 ENCODE_A -> c6 ENCARG(b) -> c2 ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a encode_b -> b encode_c -> c f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: ENCARG(cons_f(z0, z1)) -> c4(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_F(z0, z1) -> c5(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(f(a, b), z0) -> c9(F(b, f(a, f(c, f(b, f(a, z0))))), F(a, f(c, f(b, f(a, z0)))), F(c, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0)) F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) S tuples: F(f(a, b), z0) -> c9(F(b, f(a, f(c, f(b, f(a, z0))))), F(a, f(c, f(b, f(a, z0)))), F(c, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0)) F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a, encode_b, encode_c Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c4_3, c5_3, c9_5, c10_2 ---------------------------------------- (39) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a encode_b -> b encode_c -> c f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: ENCARG(cons_f(z0, z1)) -> c4(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(f(a, b), z0) -> c9(F(b, f(a, f(c, f(b, f(a, z0))))), F(a, f(c, f(b, f(a, z0)))), F(c, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0)) F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1))) ENCODE_F(z0, z1) -> c1(ENCARG(z0)) ENCODE_F(z0, z1) -> c1(ENCARG(z1)) S tuples: F(f(a, b), z0) -> c9(F(b, f(a, f(c, f(b, f(a, z0))))), F(a, f(c, f(b, f(a, z0)))), F(c, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0)) F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a, encode_b, encode_c Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c4_3, c9_5, c10_2, c1_1 ---------------------------------------- (41) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ENCODE_F(z0, z1) -> c1(ENCARG(z0)) ENCODE_F(z0, z1) -> c1(ENCARG(z1)) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a encode_b -> b encode_c -> c f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: ENCARG(cons_f(z0, z1)) -> c4(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(f(a, b), z0) -> c9(F(b, f(a, f(c, f(b, f(a, z0))))), F(a, f(c, f(b, f(a, z0)))), F(c, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0)) F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1))) S tuples: F(f(a, b), z0) -> c9(F(b, f(a, f(c, f(b, f(a, z0))))), F(a, f(c, f(b, f(a, z0)))), F(c, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0)) F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a, encode_b, encode_c Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c4_3, c9_5, c10_2, c1_1 ---------------------------------------- (43) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a encode_b -> b encode_c -> c ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: ENCARG(cons_f(z0, z1)) -> c4(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(f(a, b), z0) -> c9(F(b, f(a, f(c, f(b, f(a, z0))))), F(a, f(c, f(b, f(a, z0)))), F(c, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0)) F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1))) S tuples: F(f(a, b), z0) -> c9(F(b, f(a, f(c, f(b, f(a, z0))))), F(a, f(c, f(b, f(a, z0)))), F(c, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0)) F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c4_3, c9_5, c10_2, c1_1 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(z0, z1)) -> c4(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by ENCARG(cons_f(x0, a)) -> c4(F(encArg(x0), a), ENCARG(x0), ENCARG(a)) ENCARG(cons_f(x0, b)) -> c4(F(encArg(x0), b), ENCARG(x0), ENCARG(b)) ENCARG(cons_f(x0, c)) -> c4(F(encArg(x0), c), ENCARG(x0), ENCARG(c)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(a, x1)) -> c4(F(a, encArg(x1)), ENCARG(a), ENCARG(x1)) ENCARG(cons_f(b, x1)) -> c4(F(b, encArg(x1)), ENCARG(b), ENCARG(x1)) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(c), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c4(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(f(a, b), z0) -> c9(F(b, f(a, f(c, f(b, f(a, z0))))), F(a, f(c, f(b, f(a, z0)))), F(c, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0)) F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1))) ENCARG(cons_f(x0, a)) -> c4(F(encArg(x0), a), ENCARG(x0), ENCARG(a)) ENCARG(cons_f(x0, b)) -> c4(F(encArg(x0), b), ENCARG(x0), ENCARG(b)) ENCARG(cons_f(x0, c)) -> c4(F(encArg(x0), c), ENCARG(x0), ENCARG(c)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(a, x1)) -> c4(F(a, encArg(x1)), ENCARG(a), ENCARG(x1)) ENCARG(cons_f(b, x1)) -> c4(F(b, encArg(x1)), ENCARG(b), ENCARG(x1)) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(c), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c4(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) S tuples: F(f(a, b), z0) -> c9(F(b, f(a, f(c, f(b, f(a, z0))))), F(a, f(c, f(b, f(a, z0)))), F(c, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0)) F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCODE_F_2, ENCARG_1 Compound Symbols: c9_5, c10_2, c1_1, c4_3 ---------------------------------------- (47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(f(a, b), z0) -> c9(F(b, f(a, f(c, f(b, f(a, z0))))), F(a, f(c, f(b, f(a, z0)))), F(c, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0)) F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c4(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c4(F(encArg(x0), a), ENCARG(x0)) ENCARG(cons_f(x0, b)) -> c4(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(x0, c)) -> c4(F(encArg(x0), c), ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c4(F(a, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(b, x1)) -> c4(F(b, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(x1)) S tuples: F(f(a, b), z0) -> c9(F(b, f(a, f(c, f(b, f(a, z0))))), F(a, f(c, f(b, f(a, z0)))), F(c, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0)) F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCODE_F_2, ENCARG_1 Compound Symbols: c9_5, c10_2, c1_1, c4_3, c4_2 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(f(a, b), z0) -> c9(F(b, f(a, f(c, f(b, f(a, z0))))), F(a, f(c, f(b, f(a, z0)))), F(c, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0)) by F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c4(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c4(F(encArg(x0), a), ENCARG(x0)) ENCARG(cons_f(x0, b)) -> c4(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(x0, c)) -> c4(F(encArg(x0), c), ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c4(F(a, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(b, x1)) -> c4(F(b, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(x1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCODE_F_2, ENCARG_1 Compound Symbols: c10_2, c1_1, c4_3, c4_2, c9_5 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1))) by ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c4(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c4(F(encArg(x0), a), ENCARG(x0)) ENCARG(cons_f(x0, b)) -> c4(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(x0, c)) -> c4(F(encArg(x0), c), ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c4(F(a, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(b, x1)) -> c4(F(b, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(x1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c4_2, c9_5, c1_1 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(cons_f(z0, z1), x1)) -> c4(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) by ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1)), ENCARG(a)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1)), ENCARG(b)) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1)), ENCARG(c)) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, a)) -> c4(F(encArg(x0), a), ENCARG(x0)) ENCARG(cons_f(x0, b)) -> c4(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(x0, c)) -> c4(F(encArg(x0), c), ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c4(F(a, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(b, x1)) -> c4(F(b, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(x1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1)), ENCARG(a)) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1)), ENCARG(b)) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1)), ENCARG(c)) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c4_2, c9_5, c1_1 ---------------------------------------- (55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, a)) -> c4(F(encArg(x0), a), ENCARG(x0)) ENCARG(cons_f(x0, b)) -> c4(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(x0, c)) -> c4(F(encArg(x0), c), ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c4(F(a, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(b, x1)) -> c4(F(b, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(x1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c4_2, c9_5, c1_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(x0, a)) -> c4(F(encArg(x0), a), ENCARG(x0)) by ENCARG(cons_f(a, a)) -> c4(F(a, a), ENCARG(a)) ENCARG(cons_f(b, a)) -> c4(F(b, a), ENCARG(b)) ENCARG(cons_f(c, a)) -> c4(F(c, a), ENCARG(c)) ENCARG(cons_f(cons_f(z0, z1), a)) -> c4(F(f(encArg(z0), encArg(z1)), a), ENCARG(cons_f(z0, z1))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, b)) -> c4(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(x0, c)) -> c4(F(encArg(x0), c), ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c4(F(a, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(b, x1)) -> c4(F(b, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(x1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, a)) -> c4(F(a, a), ENCARG(a)) ENCARG(cons_f(b, a)) -> c4(F(b, a), ENCARG(b)) ENCARG(cons_f(c, a)) -> c4(F(c, a), ENCARG(c)) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c4_2, c9_5, c1_1 ---------------------------------------- (59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: ENCARG(cons_f(a, a)) -> c4(F(a, a), ENCARG(a)) ENCARG(cons_f(b, a)) -> c4(F(b, a), ENCARG(b)) ENCARG(cons_f(c, a)) -> c4(F(c, a), ENCARG(c)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, b)) -> c4(F(encArg(x0), b), ENCARG(x0)) ENCARG(cons_f(x0, c)) -> c4(F(encArg(x0), c), ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c4(F(a, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(b, x1)) -> c4(F(b, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(x1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c4_2, c9_5, c1_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(x0, b)) -> c4(F(encArg(x0), b), ENCARG(x0)) by ENCARG(cons_f(a, b)) -> c4(F(a, b), ENCARG(a)) ENCARG(cons_f(b, b)) -> c4(F(b, b), ENCARG(b)) ENCARG(cons_f(c, b)) -> c4(F(c, b), ENCARG(c)) ENCARG(cons_f(cons_f(z0, z1), b)) -> c4(F(f(encArg(z0), encArg(z1)), b), ENCARG(cons_f(z0, z1))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, c)) -> c4(F(encArg(x0), c), ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c4(F(a, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(b, x1)) -> c4(F(b, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(x1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, b)) -> c4(F(a, b), ENCARG(a)) ENCARG(cons_f(b, b)) -> c4(F(b, b), ENCARG(b)) ENCARG(cons_f(c, b)) -> c4(F(c, b), ENCARG(c)) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c4_2, c9_5, c1_1 ---------------------------------------- (63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: ENCARG(cons_f(c, b)) -> c4(F(c, b), ENCARG(c)) ENCARG(cons_f(b, b)) -> c4(F(b, b), ENCARG(b)) ENCARG(cons_f(a, b)) -> c4(F(a, b), ENCARG(a)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(x0, c)) -> c4(F(encArg(x0), c), ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c4(F(a, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(b, x1)) -> c4(F(b, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(x1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c4_2, c9_5, c1_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(x0, c)) -> c4(F(encArg(x0), c), ENCARG(x0)) by ENCARG(cons_f(a, c)) -> c4(F(a, c), ENCARG(a)) ENCARG(cons_f(b, c)) -> c4(F(b, c), ENCARG(b)) ENCARG(cons_f(c, c)) -> c4(F(c, c), ENCARG(c)) ENCARG(cons_f(cons_f(z0, z1), c)) -> c4(F(f(encArg(z0), encArg(z1)), c), ENCARG(cons_f(z0, z1))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(a, x1)) -> c4(F(a, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(b, x1)) -> c4(F(b, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(x1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, c)) -> c4(F(a, c), ENCARG(a)) ENCARG(cons_f(b, c)) -> c4(F(b, c), ENCARG(b)) ENCARG(cons_f(c, c)) -> c4(F(c, c), ENCARG(c)) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c4_2, c9_5, c1_1 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: ENCARG(cons_f(a, c)) -> c4(F(a, c), ENCARG(a)) ENCARG(cons_f(c, c)) -> c4(F(c, c), ENCARG(c)) ENCARG(cons_f(b, c)) -> c4(F(b, c), ENCARG(b)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(a, x1)) -> c4(F(a, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(b, x1)) -> c4(F(b, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(x1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c4_2, c9_5, c1_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(a, x1)) -> c4(F(a, encArg(x1)), ENCARG(x1)) by ENCARG(cons_f(a, a)) -> c4(F(a, a), ENCARG(a)) ENCARG(cons_f(a, b)) -> c4(F(a, b), ENCARG(b)) ENCARG(cons_f(a, c)) -> c4(F(a, c), ENCARG(c)) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, x1)) -> c4(F(b, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(x1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, a)) -> c4(F(a, a), ENCARG(a)) ENCARG(cons_f(a, b)) -> c4(F(a, b), ENCARG(b)) ENCARG(cons_f(a, c)) -> c4(F(a, c), ENCARG(c)) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c4_2, c9_5, c1_1 ---------------------------------------- (71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: ENCARG(cons_f(a, b)) -> c4(F(a, b), ENCARG(b)) ENCARG(cons_f(a, a)) -> c4(F(a, a), ENCARG(a)) ENCARG(cons_f(a, c)) -> c4(F(a, c), ENCARG(c)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, x1)) -> c4(F(b, encArg(x1)), ENCARG(x1)) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(x1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c4_2, c9_5, c1_1 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(b, x1)) -> c4(F(b, encArg(x1)), ENCARG(x1)) by ENCARG(cons_f(b, a)) -> c4(F(b, a), ENCARG(a)) ENCARG(cons_f(b, b)) -> c4(F(b, b), ENCARG(b)) ENCARG(cons_f(b, c)) -> c4(F(b, c), ENCARG(c)) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(x1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, a)) -> c4(F(b, a), ENCARG(a)) ENCARG(cons_f(b, b)) -> c4(F(b, b), ENCARG(b)) ENCARG(cons_f(b, c)) -> c4(F(b, c), ENCARG(c)) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c4_2, c9_5, c1_1 ---------------------------------------- (75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: ENCARG(cons_f(b, c)) -> c4(F(b, c), ENCARG(c)) ENCARG(cons_f(b, b)) -> c4(F(b, b), ENCARG(b)) ENCARG(cons_f(b, a)) -> c4(F(b, a), ENCARG(a)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(x1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c4_2, c9_5, c1_1 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(c, x1)) -> c4(F(c, encArg(x1)), ENCARG(x1)) by ENCARG(cons_f(c, a)) -> c4(F(c, a), ENCARG(a)) ENCARG(cons_f(c, b)) -> c4(F(c, b), ENCARG(b)) ENCARG(cons_f(c, c)) -> c4(F(c, c), ENCARG(c)) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, a)) -> c4(F(c, a), ENCARG(a)) ENCARG(cons_f(c, b)) -> c4(F(c, b), ENCARG(b)) ENCARG(cons_f(c, c)) -> c4(F(c, c), ENCARG(c)) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c9_5, c1_1, c4_2 ---------------------------------------- (79) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: ENCARG(cons_f(c, a)) -> c4(F(c, a), ENCARG(a)) ENCARG(cons_f(c, b)) -> c4(F(c, b), ENCARG(b)) ENCARG(cons_f(c, c)) -> c4(F(c, c), ENCARG(c)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c9_5, c1_1, c4_2 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(b, f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) by F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c9_5, c1_1, c4_2, c9_2 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(f(a, b), x0) -> c9(F(b, f(a, f(f(c, b), f(a, x0)))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) by F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c9_5, c1_1, c4_2, c9_2, c9_3 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(f(a, b), z2) -> c9(F(b, f(a, f(c, f(f(b, a), z2)))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) by F(f(a, b), x0) -> c9(F(b, f(f(a, c), f(f(b, a), x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c9_5, c1_1, c4_2, c9_2, c9_3, c9_4 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(b, f(f(a, z1), z2))))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) by F(f(a, b), f(x0, x1)) -> c9(F(b, f(f(a, c), f(b, f(f(a, x0), x1)))), F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) F(f(a, b), f(x0, x1)) -> c9(F(b, f(a, f(f(c, b), f(f(a, x0), x1)))), F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c1_1, c4_2, c9_5, c9_2, c9_3, c9_4 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(x0, a) -> c1(F(encArg(x0), a)) by ENCODE_F(a, a) -> c1(F(a, a)) ENCODE_F(b, a) -> c1(F(b, a)) ENCODE_F(c, a) -> c1(F(c, a)) ENCODE_F(cons_f(z0, z1), a) -> c1(F(f(encArg(z0), encArg(z1)), a)) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) ENCODE_F(a, a) -> c1(F(a, a)) ENCODE_F(b, a) -> c1(F(b, a)) ENCODE_F(c, a) -> c1(F(c, a)) ENCODE_F(cons_f(z0, z1), a) -> c1(F(f(encArg(z0), encArg(z1)), a)) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c1_1, c4_2, c9_5, c9_2, c9_3, c9_4 ---------------------------------------- (91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: ENCODE_F(a, a) -> c1(F(a, a)) ENCODE_F(b, a) -> c1(F(b, a)) ENCODE_F(c, a) -> c1(F(c, a)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c1(F(f(encArg(z0), encArg(z1)), a)) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c1_1, c4_2, c9_5, c9_2, c9_3, c9_4 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(x0, b) -> c1(F(encArg(x0), b)) by ENCODE_F(a, b) -> c1(F(a, b)) ENCODE_F(b, b) -> c1(F(b, b)) ENCODE_F(c, b) -> c1(F(c, b)) ENCODE_F(cons_f(z0, z1), b) -> c1(F(f(encArg(z0), encArg(z1)), b)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c1(F(f(encArg(z0), encArg(z1)), a)) ENCODE_F(a, b) -> c1(F(a, b)) ENCODE_F(b, b) -> c1(F(b, b)) ENCODE_F(c, b) -> c1(F(c, b)) ENCODE_F(cons_f(z0, z1), b) -> c1(F(f(encArg(z0), encArg(z1)), b)) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c1_1, c4_2, c9_5, c9_2, c9_3, c9_4 ---------------------------------------- (95) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: ENCODE_F(a, b) -> c1(F(a, b)) ENCODE_F(c, b) -> c1(F(c, b)) ENCODE_F(b, b) -> c1(F(b, b)) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c1(F(f(encArg(z0), encArg(z1)), a)) ENCODE_F(cons_f(z0, z1), b) -> c1(F(f(encArg(z0), encArg(z1)), b)) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c1_1, c4_2, c9_5, c9_2, c9_3, c9_4 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(x0, c) -> c1(F(encArg(x0), c)) by ENCODE_F(a, c) -> c1(F(a, c)) ENCODE_F(b, c) -> c1(F(b, c)) ENCODE_F(c, c) -> c1(F(c, c)) ENCODE_F(cons_f(z0, z1), c) -> c1(F(f(encArg(z0), encArg(z1)), c)) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c1(F(f(encArg(z0), encArg(z1)), a)) ENCODE_F(cons_f(z0, z1), b) -> c1(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(a, c) -> c1(F(a, c)) ENCODE_F(b, c) -> c1(F(b, c)) ENCODE_F(c, c) -> c1(F(c, c)) ENCODE_F(cons_f(z0, z1), c) -> c1(F(f(encArg(z0), encArg(z1)), c)) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c1_1, c4_2, c9_5, c9_2, c9_3, c9_4 ---------------------------------------- (99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: ENCODE_F(c, c) -> c1(F(c, c)) ENCODE_F(b, c) -> c1(F(b, c)) ENCODE_F(a, c) -> c1(F(a, c)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c1(F(f(encArg(z0), encArg(z1)), a)) ENCODE_F(cons_f(z0, z1), b) -> c1(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(cons_f(z0, z1), c) -> c1(F(f(encArg(z0), encArg(z1)), c)) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c1_1, c4_2, c9_5, c9_2, c9_3, c9_4 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(a, x1) -> c1(F(a, encArg(x1))) by ENCODE_F(a, a) -> c1(F(a, a)) ENCODE_F(a, b) -> c1(F(a, b)) ENCODE_F(a, c) -> c1(F(a, c)) ENCODE_F(a, cons_f(z0, z1)) -> c1(F(a, f(encArg(z0), encArg(z1)))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c1(F(f(encArg(z0), encArg(z1)), a)) ENCODE_F(cons_f(z0, z1), b) -> c1(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(cons_f(z0, z1), c) -> c1(F(f(encArg(z0), encArg(z1)), c)) ENCODE_F(a, a) -> c1(F(a, a)) ENCODE_F(a, b) -> c1(F(a, b)) ENCODE_F(a, c) -> c1(F(a, c)) ENCODE_F(a, cons_f(z0, z1)) -> c1(F(a, f(encArg(z0), encArg(z1)))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c1_1, c4_2, c9_5, c9_2, c9_3, c9_4 ---------------------------------------- (103) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: ENCODE_F(a, c) -> c1(F(a, c)) ENCODE_F(a, b) -> c1(F(a, b)) ENCODE_F(a, a) -> c1(F(a, a)) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c1(F(f(encArg(z0), encArg(z1)), a)) ENCODE_F(cons_f(z0, z1), b) -> c1(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(cons_f(z0, z1), c) -> c1(F(f(encArg(z0), encArg(z1)), c)) ENCODE_F(a, cons_f(z0, z1)) -> c1(F(a, f(encArg(z0), encArg(z1)))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c1_1, c4_2, c9_5, c9_2, c9_3, c9_4 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(b, x1) -> c1(F(b, encArg(x1))) by ENCODE_F(b, a) -> c1(F(b, a)) ENCODE_F(b, b) -> c1(F(b, b)) ENCODE_F(b, c) -> c1(F(b, c)) ENCODE_F(b, cons_f(z0, z1)) -> c1(F(b, f(encArg(z0), encArg(z1)))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c1(F(f(encArg(z0), encArg(z1)), a)) ENCODE_F(cons_f(z0, z1), b) -> c1(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(cons_f(z0, z1), c) -> c1(F(f(encArg(z0), encArg(z1)), c)) ENCODE_F(a, cons_f(z0, z1)) -> c1(F(a, f(encArg(z0), encArg(z1)))) ENCODE_F(b, a) -> c1(F(b, a)) ENCODE_F(b, b) -> c1(F(b, b)) ENCODE_F(b, c) -> c1(F(b, c)) ENCODE_F(b, cons_f(z0, z1)) -> c1(F(b, f(encArg(z0), encArg(z1)))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c1_1, c4_2, c9_5, c9_2, c9_3, c9_4 ---------------------------------------- (107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: ENCODE_F(b, a) -> c1(F(b, a)) ENCODE_F(b, c) -> c1(F(b, c)) ENCODE_F(b, b) -> c1(F(b, b)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c1(F(f(encArg(z0), encArg(z1)), a)) ENCODE_F(cons_f(z0, z1), b) -> c1(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(cons_f(z0, z1), c) -> c1(F(f(encArg(z0), encArg(z1)), c)) ENCODE_F(a, cons_f(z0, z1)) -> c1(F(a, f(encArg(z0), encArg(z1)))) ENCODE_F(b, cons_f(z0, z1)) -> c1(F(b, f(encArg(z0), encArg(z1)))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c1_1, c4_2, c9_5, c9_2, c9_3, c9_4 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(c, x1) -> c1(F(c, encArg(x1))) by ENCODE_F(c, a) -> c1(F(c, a)) ENCODE_F(c, b) -> c1(F(c, b)) ENCODE_F(c, c) -> c1(F(c, c)) ENCODE_F(c, cons_f(z0, z1)) -> c1(F(c, f(encArg(z0), encArg(z1)))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c1(F(f(encArg(z0), encArg(z1)), a)) ENCODE_F(cons_f(z0, z1), b) -> c1(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(cons_f(z0, z1), c) -> c1(F(f(encArg(z0), encArg(z1)), c)) ENCODE_F(a, cons_f(z0, z1)) -> c1(F(a, f(encArg(z0), encArg(z1)))) ENCODE_F(b, cons_f(z0, z1)) -> c1(F(b, f(encArg(z0), encArg(z1)))) ENCODE_F(c, a) -> c1(F(c, a)) ENCODE_F(c, b) -> c1(F(c, b)) ENCODE_F(c, c) -> c1(F(c, c)) ENCODE_F(c, cons_f(z0, z1)) -> c1(F(c, f(encArg(z0), encArg(z1)))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c1_1, c4_2, c9_5, c9_2, c9_3, c9_4 ---------------------------------------- (111) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: ENCODE_F(c, c) -> c1(F(c, c)) ENCODE_F(c, b) -> c1(F(c, b)) ENCODE_F(c, a) -> c1(F(c, a)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c1(F(f(encArg(z0), encArg(z1)), a)) ENCODE_F(cons_f(z0, z1), b) -> c1(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(cons_f(z0, z1), c) -> c1(F(f(encArg(z0), encArg(z1)), c)) ENCODE_F(a, cons_f(z0, z1)) -> c1(F(a, f(encArg(z0), encArg(z1)))) ENCODE_F(b, cons_f(z0, z1)) -> c1(F(b, f(encArg(z0), encArg(z1)))) ENCODE_F(c, cons_f(z0, z1)) -> c1(F(c, f(encArg(z0), encArg(z1)))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c1_1, c4_2, c9_5, c9_2, c9_3, c9_4 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(cons_f(z0, z1), x1) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1))) by ENCODE_F(cons_f(x0, x1), a) -> c1(F(f(encArg(x0), encArg(x1)), a)) ENCODE_F(cons_f(x0, x1), b) -> c1(F(f(encArg(x0), encArg(x1)), b)) ENCODE_F(cons_f(x0, x1), c) -> c1(F(f(encArg(x0), encArg(x1)), c)) ENCODE_F(cons_f(x0, x1), cons_f(z0, z1)) -> c1(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(x0, a), x2) -> c1(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, b), x2) -> c1(F(f(encArg(x0), b), encArg(x2))) ENCODE_F(cons_f(x0, c), x2) -> c1(F(f(encArg(x0), c), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c1(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c1(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(b, x1), x2) -> c1(F(f(b, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(c, x1), x2) -> c1(F(f(c, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(b) -> b encArg(c) -> c encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(f(a, b), z0) -> f(b, f(a, f(c, f(b, f(a, z0))))) f(z0, f(z1, z2)) -> f(f(z0, z1), z2) Tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c4(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c4(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, b), x2)) -> c4(F(f(encArg(x0), b), encArg(x2)), ENCARG(cons_f(x0, b)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, c), x2)) -> c4(F(f(encArg(x0), c), encArg(x2)), ENCARG(cons_f(x0, c)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c4(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_f(x0, cons_f(z0, z1))), ENCARG(x2)) ENCARG(cons_f(cons_f(a, x1), x2)) -> c4(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(b, x1), x2)) -> c4(F(f(b, encArg(x1)), encArg(x2)), ENCARG(cons_f(b, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(c, x1), x2)) -> c4(F(f(c, encArg(x1)), encArg(x2)), ENCARG(cons_f(c, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c4(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, x1), a)) -> c4(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), b)) -> c4(F(f(encArg(x0), encArg(x1)), b), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), c)) -> c4(F(f(encArg(x0), encArg(x1)), c), ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c4(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(b, cons_f(z0, z1))) -> c4(F(b, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(c, cons_f(z0, z1))) -> c4(F(c, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) ENCODE_F(cons_f(z0, z1), a) -> c1(F(f(encArg(z0), encArg(z1)), a)) ENCODE_F(cons_f(z0, z1), b) -> c1(F(f(encArg(z0), encArg(z1)), b)) ENCODE_F(cons_f(z0, z1), c) -> c1(F(f(encArg(z0), encArg(z1)), c)) ENCODE_F(a, cons_f(z0, z1)) -> c1(F(a, f(encArg(z0), encArg(z1)))) ENCODE_F(b, cons_f(z0, z1)) -> c1(F(b, f(encArg(z0), encArg(z1)))) ENCODE_F(c, cons_f(z0, z1)) -> c1(F(c, f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(x0, x1), cons_f(z0, z1)) -> c1(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(x0, a), x2) -> c1(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, b), x2) -> c1(F(f(encArg(x0), b), encArg(x2))) ENCODE_F(cons_f(x0, c), x2) -> c1(F(f(encArg(x0), c), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c1(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c1(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(b, x1), x2) -> c1(F(f(b, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(c, x1), x2) -> c1(F(f(c, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) S tuples: F(z0, f(z1, z2)) -> c10(F(f(z0, z1), z2), F(z0, z1)) F(f(a, b), x0) -> c9(F(b, f(f(f(a, c), b), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(f(a, c), f(f(b, a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(f(a, c), f(b, f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0)))) F(f(a, b), x0) -> c9(F(b, f(f(a, f(c, b)), f(a, x0))), F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), z2) -> c9(F(b, f(a, f(f(f(c, b), a), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(f(c, b), f(f(a, z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0))) F(f(a, b), z2) -> c9(F(b, f(a, f(f(c, f(b, a)), z2))), F(a, f(c, f(b, f(a, z2)))), F(c, f(b, f(a, z2))), F(b, f(a, z2)), F(a, z2)) F(f(a, b), f(z1, z2)) -> c9(F(b, f(a, f(c, f(f(f(b, a), z1), z2)))), F(a, f(c, f(b, f(a, f(z1, z2))))), F(c, f(b, f(a, f(z1, z2)))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) F(f(a, b), x0) -> c9(F(a, f(c, f(b, f(a, x0)))), F(c, f(b, f(a, x0))), F(b, f(a, x0)), F(a, x0)) F(f(a, b), f(x0, z2)) -> c9(F(b, f(a, f(c, f(f(b, f(a, x0)), z2)))), F(a, f(c, f(b, f(a, f(x0, z2))))), F(c, f(b, f(a, f(x0, z2)))), F(b, f(a, f(x0, z2))), F(a, f(x0, z2))) F(f(a, b), f(b, z0)) -> c9(F(b, f(a, f(c, f(b, f(b, f(a, f(c, f(b, f(a, z0))))))))), F(a, f(c, f(b, f(a, f(b, z0))))), F(c, f(b, f(a, f(b, z0)))), F(b, f(a, f(b, z0))), F(a, f(b, z0))) F(f(a, b), f(f(z1, z2), x1)) -> c9(F(b, f(a, f(c, f(b, f(f(f(a, z1), z2), x1))))), F(a, f(c, f(b, f(a, f(f(z1, z2), x1))))), F(c, f(b, f(a, f(f(z1, z2), x1)))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1))) F(f(a, b), f(x0, x1)) -> c9(F(a, f(c, f(b, f(a, f(x0, x1))))), F(c, f(b, f(a, f(x0, x1)))), F(b, f(a, f(x0, x1))), F(a, f(x0, x1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c10_2, c4_3, c1_1, c4_2, c9_5, c9_2, c9_3, c9_4